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            source: "// Numbas version: finer_feedback_settings\n{\"name\": \"Skills Audit for Maths and Stats - Physics programmes (PHY129)\", \"metadata\": {\"description\": \"\u003cp\u003eDraft Skills Audit for Physics Students.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"duration\": 0, \"percentPass\": 0, \"showQuestionGroupNames\": false, \"shuffleQuestionGroups\": false, \"showstudentname\": true, \"question_groups\": [{\"name\": \"Group\", \"pickingStrategy\": \"all-ordered\", \"pickQuestions\": 1, \"questionNames\": [\"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\", \"\"], \"variable_overrides\": [[], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []], \"questions\": [{\"name\": \"NU02 Convert Units (m/s and km/h)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eUnit conversion between two compound units.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eA Cheetah runs at a speed of {speedkm} kilometres per hour (km/h). What is the Cheetah\u0027s speed in metres per second (m/s)?\u003c/p\u003e\\n\u003cp\u003eGive your ansswer to 2 decimal places where appropriate.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eThere are a number of ways to work out the conversion. Here are a couple of suggestions.\u003c/p\u003e\\n\u003cp\u003e\u003cstrong\u003eMETHOD 1\u003c/strong\u003e\u003c/p\u003e\\n\u003cp\u003eSince there are $1000$m in $1$km we first multiply by $1000$ to get the speed in metres per hour:\u003c/p\u003e\\n\u003cp\u003e\\\\begin{equation} 1000*\\\\var{speedkm} = \\\\var{step1}\\\\end{equation}\u003c/p\u003e\\n\u003cp\u003eThen we divide by $3600$ since that is the number of seconds in an hour to get the speed in metres per second:\u003c/p\u003e\\n\u003cp\u003e\\\\begin{equation} \\\\frac{\\\\var{step1}}{3600} = \\\\var{speedms} \\\\end{equation}\u003c/p\u003e\\n\u003cp\u003eFinally we round off to 2 decimal places as required, $\\\\var{roundanswer}$m/s.\u003c/p\u003e\\n\u003cp\u003e\u003cstrong\u003eMETHOD 2\u003c/strong\u003e\u003c/p\u003e\\n\u003cp\u003eWe can actually do all of the above in one step of working by using a single conversion factor. Since there are $1000$m in a km and $3600$ seconds in an hour, we can calaculate the conversion factor:\u003c/p\u003e\\n\u003cp\u003e\\\\begin{equation} \\\\frac{3600}{1000} = 3.6 \\\\end{equation}\u003c/p\u003e\\n\u003cp\u003eand then simply divide by that conversion factor:\u003c/p\u003e\\n\u003cp\u003e\\\\begin{equation} \\\\frac{\\\\var{speedkm}}{3.6} = \\\\var{speedms} \\\\end{equation}\u003c/p\u003e\\n\u003cp\u003efinally rounding off as before, $\\\\var{roundanswer}$m/s.\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Conversions\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"speedkm\": {\"name\": \"speedkm\", \"group\": \"Ungrouped 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\"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"speedms\", \"precisionType\": \"dp\", \"precision\": \"2\", \"precisionPartialCredit\": 0, \"precisionMessage\": \"You have not given your answer to the correct precision.\", \"strictPrecision\": false, \"showPrecisionHint\": false, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"NF03 Rounding SF (decimal)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Christian Lawson-Perfect\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/7/\"}, {\"name\": \"Stanislav Duris\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1590/\"}, {\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}, {\"name\": \"Oliver Spenceley\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23557/\"}], \"tags\": [\"rounding\"], \"metadata\": {\"description\": \"\u003cp\u003eRound numbers to\u0026nbsp;a given number of\u0026nbsp;significant figures.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\", \"advice\": \"\u003cp\u003eThe\u0026nbsp;first\u0026nbsp;thing to do\u0026nbsp;when we are rounding numbers is to identify the last digit we are keeping.\u003c/p\u003e\\n\u003cp\u003eWhen you\u0027re asked to \u003cstrong\u003eround\u003c/strong\u003e your answer to a number of significant figures, you need to decide whether to keep the last\u0026nbsp;digit same (rounding down) or increase it by 1 (rounding up). If the following digit is less than 5 we round down and we round up\u0026nbsp;when the next digit is 5 or more.\u003c/p\u003e\\n\u003cp\u003eTo write it down in steps:\u003c/p\u003e\\n\u003col\u003e\\n\u003col\u003e\\n\u003col\u003e\\n\u003col\u003e\\n\u003cli style=\\\"text-align: left;\\\"\u003eIdentify the last digit we need to keep.\u003c/li\u003e\\n\u003cli style=\\\"text-align: left;\\\"\u003eLook at the following digit.\u003c/li\u003e\\n\u003cli style=\\\"text-align: left;\\\"\u003eIf it\u0027s 5 or more, increase the previous digit by one.\u003c/li\u003e\\n\u003cli class=\\\"c-list__ordered-item\\\" style=\\\"text-align: left;\\\"\u003eIf it\u0027s 4 or less, keep the previous digit the same.\u003c/li\u003e\\n\u003cli class=\\\"c-list__ordered-item\\\" style=\\\"text-align: left;\\\"\u003eFill any spaces to the right of the digit with zeros if needed.\u003c/li\u003e\\n\u003c/ol\u003e\\n\u003c/ol\u003e\\n\u003c/ol\u003e\\n\u003c/ol\u003e\\n\u003cp class=\\\"c-list__ordered-item\\\"\u003eIt is important to keep in mind that if the digit we are increasing is 9, it becomes zero and we increase the previous digit instead. If this digit is 9 as well, we\u0026nbsp;move along\u0026nbsp;to the left side until we find a digit less than 9.\u003c/p\u003e\\n\u003cp class=\\\"c-list__ordered-item\\\"\u003eThe last digit we need to keep will depend on how many zeros there are. \u0026nbsp;We don\u0027t consider leading zeros to be significant, \u003cbr/\u003ei.e.\u0026nbsp;0.03 and 0.3 both have 1 significant figure (but 0.30 has two significant figures, since the second zero isn\u0027t a \u0027leading\u0027 zero).\u003c/p\u003e\\n\u003cp\u003e\u003cem\u003ei)\u003c/em\u003e\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 30px;\\\"\u003eWe round $\\\\var{e1}$ to 1 significant figure. The first non-zero digit is $\\\\var{edig[4]}$, followed by $\\\\var{edig[3]}$. This is \u003cspan data-jme-visible=\\\"edig[3] \u0026lt; 5\\\"\u003elower than 5 so we round down\u003c/span\u003e\u003cspan data-jme-visible=\\\"edig[3] \u0026gt; 4\\\"\u003emore than 5 so we round up\u003c/span\u003e to get $\\\\var{sigformat(e1,1)}$.\u003c/p\u003e\\n\u003cp\u003e\u003cem\u003eii)\u003c/em\u003e\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 30px;\\\"\u003eWe round $\\\\var{e1}$ to\u0026nbsp;{sf} significant figures. The first non-zero digit is $\\\\var{edig[4]}$. The second following digit is $\\\\var{edig[3]}$\u003cspan data-jme-visible=\\\"sf \u0026gt; 2\\\"\u003e, the third following digit is $\\\\var{edig[2]}$\u003c/span\u003e\u003cspan data-jme-visible=\\\"sf = 4\\\"\u003e and the fourth following digit is $\\\\var{edig[1]}$\u003c/span\u003e. The\u0026nbsp;digit\u0026nbsp;following the last digit we are keeping is \u003cspan data-jme-visible=\\\"sf = 2\\\"\u003e$\\\\var{edig[2]}$\u003c/span\u003e\u003cspan data-jme-visible=\\\"sf = 3\\\"\u003e$\\\\var{edig[1]}$\u003c/span\u003e\u003cspan data-jme-visible=\\\"sf = 4\\\"\u003e$\\\\var{edig[0]}$\u003c/span\u003e, so we round to\u0026nbsp;get $\\\\var{sigformat(e1, sf)}$. These are our {sf} significant figures.\u0026nbsp;\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 30px;\\\"\u003e\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 30px;\\\"\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#number\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 30px;\\\"\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"edig\": {\"name\": \"edig\", \"group\": \"Ungrouped variables\", \"definition\": \"repeat(random(1..9), 5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"d1\": {\"name\": \"d1\", \"group\": \"Ungrouped variables\", \"definition\": \"n_from_digits(ddig)\", \"description\": \"\u003cp\u003eRandom integer.\u003c/p\u003e\", \"templateType\": \"anything\", \"can_override\": false}, \"e1\": {\"name\": \"e1\", \"group\": \"Ungrouped variables\", \"definition\": \"n_from_digits(edig)*10^(random(-6,-7,-8))\", \"description\": \"\u003cp\u003eRandom number with\u0026nbsp;7 decimal places.\u003c/p\u003e\", \"templateType\": \"anything\", \"can_override\": false}, \"ddig\": {\"name\": \"ddig\", \"group\": \"Ungrouped variables\", \"definition\": \"repeat(random(1..9), 6)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sf\": {\"name\": \"sf\", \"group\": \"Ungrouped variables\", \"definition\": \"3\", \"description\": \"\u003cp\u003eNumber of significant figures to round.\u003c/p\u003e\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"sf\", \"ddig\", \"edig\", \"d1\", \"e1\"], \"variable_groups\": [], \"functions\": {\"n_from_digits\": {\"parameters\": [[\"digits\", \"list\"]], \"type\": \"number\", \"language\": \"jme\", \"definition\": \"if(\\n  len(digits)=0,\\n    0,\\n    digits[0]+10*n_from_digits(digits[1..len(digits)])\\n)\"}}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003eRound $\\\\var{e1}$\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 60px;\\\"\u003e\u003cem\u003eiii)\u003c/em\u003e $\\\\var{e1}$ rounded\u0026nbsp;to 1 significant figure is:\u0026nbsp; [[0]]\u003c/p\u003e\\n\u003cp style=\\\"padding-left: 60px;\\\"\u003e\u003cem\u003eiv)\u003c/em\u003e $\\\\var{e1}$ rounded\u0026nbsp;to {sf} significant figures is:\u0026nbsp; [[1]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"minValue\": \"siground(e1, 1)\", \"maxValue\": \"siground(e1, 1)\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}, {\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"minValue\": \"siground(e1, sf)\", \"maxValue\": \"siground(e1, sf)\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"NF07 Percentage change (decrease then increase)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCompound percentage change.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eThe value of a car is initially {StartingPrice}. If the value decreases by {dec}%, and then increases by {inc}%, what is the final value?\u003c/p\u003e\\n\u003cp\u003eGive your answer correct to two decimal places.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eThere is a {dec}% decrease in price. This means that price after the decrease will be {100-dec}% of the old price.\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{\\\\var{100-dec}}{100} \\\\times \\\\var{StartingPrice} = \\\\var{(100-dec)/100*StartingPrice}\\\\]\u003c/p\u003e\\n\u003cp\u003eThen there is a {inc}% increase in price. This means the final price will be {100+inc}% of the price after the decrease.\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\\\\[\\\\frac{\\\\var{100+inc}}{100} \\\\times \\\\var{(100-dec)/100*StartingPrice} = \u0026pound;\\\\var{dpformat((100+inc)/100*(100-dec)/100*StartingPrice,2)}\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Percentages\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"dec\": {\"name\": \"dec\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..50)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"inc\": {\"name\": \"inc\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..50)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"FinalPrice\": {\"name\": \"FinalPrice\", \"group\": \"Ungrouped variables\", \"definition\": \"StartingPrice*(1-dec/100)*(1+inc/100)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"StartingPrice\": {\"name\": \"StartingPrice\", \"group\": \"Ungrouped variables\", \"definition\": \"random(600..8000 # 10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"dec\", \"inc\", \"FinalPrice\", \"StartingPrice\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u0026pound;[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"minValue\": \"FinalPrice\", \"maxValue\": \"FinalPrice\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"precisionType\": \"dp\", \"precision\": \"2\", \"precisionPartialCredit\": 0, \"precisionMessage\": \"You have not given your answer to the correct precision.\", \"strictPrecision\": false, \"showPrecisionHint\": false, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"NF24 Dividing amounts in ratios\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eDividing amounts in ratios\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eThe ratio of ethanol to water is {a}:{b} for an experiment. If I have {volWater}ml of water, how much ethanol do I need?\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf there is a ratio of {a}:{b} for ethanol:water then that means for every {b}ml of water we need {a}ml of ethanol.\u003c/p\u003e\\n\u003cp\u003eIn our experiment there is {volwater}ml of water so to find the amount of ethanol we divide by {b} and then multiply by {a}.\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\var{volwater}\\\\text{ml}\\\\times\\\\frac{\\\\var{a}}{\\\\var{b}}=\\\\var{volwater*a/b}\\\\text{ml}\\\\]\u003cbr/\u003e\u003cbr/\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Ratio\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..12)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..12 except a)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"volwater\": {\"name\": \"volwater\", \"group\": \"Ungrouped variables\", \"definition\": \"UsedforVol-mod(UsedforVol,b)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"UsedforVol\": {\"name\": \"UsedforVol\", \"group\": \"Ungrouped variables\", \"definition\": \"random(70..1000)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"volwater\", \"UsedforVol\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e[[0]]ml\u003c/p\u003e\", \"gaps\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"minValue\": \"volwater/b*a\", \"maxValue\": \"volwater/b*a\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"NS03 Standard Form (Calculations)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculations involving Standard form.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\", \"advice\": \"\u003cp\u003eTo divide two numbers in standard form we can calculate the division of each part of the standard form number separately. In general we have,\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{x\\\\times10^j}{y\\\\times10^k}=\\\\frac xy\\\\times\\\\frac{10^j}{10^k}=\\\\frac xy\\\\times 10^{j-k}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eIn this question we therefore have,\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{\\\\var{a}\\\\times10^{\\\\var{n}}}{\\\\var{b}\\\\times10^{\\\\var{m}}}=\\\\frac{\\\\var{a}}{\\\\var{b}}\\\\times\\\\frac{10^{\\\\var{n}}}{10^{\\\\var{m}}}=\\\\var{aDivBRound}\\\\times10^\\\\var{n-m}.\\\\]\u003cbr/\u003e\u003cbr/\u003e\u003c/p\u003e\\n\u003cdiv data-jme-visible=\\\"IsADivBLessOne\\\"\u003e\\n\u003cp\u003eSince {aDivBRound} is less than 1 then our answer isn\u0027t in standard form. In this case we need to reduce the exponent by 1 so the final answer is\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\var{MantAnsRound}\\\\times10^{\\\\var{ExponentAns}}.\\\\]\u003c/p\u003e\\n\u003c/div\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Standard\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..9.9 # 0.1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..9.9 # 0.1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"m\": {\"name\": \"m\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"IsADivBLessOne\": {\"name\": \"IsADivBLessOne\", \"group\": \"Ungrouped variables\", \"definition\": \"a/b\u003c1\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"ExponentAns\": {\"name\": \"ExponentAns\", \"group\": \"Ungrouped variables\", \"definition\": \"if(IsADivBLessOne,n-m-1,n-m)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"MantAns\": {\"name\": \"MantAns\", \"group\": \"Ungrouped variables\", \"definition\": \"if(IsADivBLessOne, a/b*10, a/b)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"aDivBRound\": {\"name\": \"aDivBRound\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(a/b,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"MantAnsRound\": {\"name\": \"MantAnsRound\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(MantAns,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"n\", \"m\", \"IsADivBLessOne\", \"ExponentAns\", \"MantAns\", \"aDivBRound\", \"MantAnsRound\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003eFor the equation\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{\\\\var{a}\\\\times10^{\\\\var{n}}}{\\\\var{b}\\\\times10^{\\\\var{m}}}=a\\\\times10^n\\\\]\u003c/p\u003e\\n\u003cp\u003efind the values of $a$ and $n$ which keep the answer in standard form.\u003c/p\u003e\\n\u003cp\u003eGive $a$ to two decimal places.\u003c/p\u003e\\n\u003cp\u003e$a=$[[0]]\u003cbr/\u003e\u003cbr/\u003e$n=$[[1]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"minValue\": \"MantAns\", \"maxValue\": \"MantAns\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"precisionType\": \"dp\", \"precision\": \"2\", \"precisionPartialCredit\": 0, \"precisionMessage\": \"You have not given your answer to the correct precision.\", \"strictPrecision\": false, \"showPrecisionHint\": false, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}, {\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"minValue\": \"ExponentAns\", \"maxValue\": \"ExponentAns\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AC08 Collecting terms (higher powers)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Michael Proudman\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/269/\"}, {\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eSimple exercise in collecting terms in different powers of \\\\(x\\\\)\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003eSimplify the following expression by combining \\\"like\\\" terms.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eFirst we expand the minus sign in the bracket.\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4-({f}x+{e}x^3)}=\\\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4+{-f}x+{-e}x^3}\\\\]\u003c/p\u003e\\n\u003cp\u003eThe idea is to collect together and combine any terms that are the same kind of term so:\u003c/p\u003e\\n\u003cp\u003e$\\\\var{b}x$ and $\\\\var{-f}x$ both have an $x$ term. We can combine them to get $\\\\var{b-f}x$\u003c/p\u003e\\n\u003cp\u003eWe can combine $\\\\var{a}x^4$ and $\\\\var{d}x^4$ to get $\\\\var{a+d}x^4$.\u003c/p\u003e\\n\u003cp\u003eWe combine $\\\\var{c}x^3$ and $\\\\var{-e}x^3$ to get $\\\\var{c-e}x^3$. So our answer is:\u003c/p\u003e\\n\u003cp\u003e$\\\\simplify{{a+d}x^4+{c+e}x^3+{b+f}}$\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Collect\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"d\": {\"name\": \"d\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"f\": {\"name\": \"f\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..10 except 0 except b)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"e\": {\"name\": \"e\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..10 except 0 except c)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"c\", \"b\", \"d\", \"f\", \"e\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4-({f}x+{e}x^3)}$\u003c/p\u003e\", \"answer\": \"({a}+{d})x^4+({c}-{e})x^3+({b}-{f})x\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"mustmatchpattern\": {\"pattern\": \"`+-$n`?*x^4+`+-$n`?*x^3+`+-$n`?*x\", \"partialCredit\": 0, \"message\": \"\", \"nameToCompare\": \"\", \"warningTime\": \"submission\"}, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AC09 Multiply algebraic terms\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Lauren Desoysa\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21504/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFilling in the blanks from the answer to a simplified expression involving indices.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eFind the missing factor in the following statement:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\var{3*a}x^{\\\\var{b}}y^{\\\\var{c}} = \\\\var{a}x^{\\\\var{d}}(?)\\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eWe can divide the left handside of the expression by the factor given on the right hand side of the expression to work out the missing factor:\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\\\\[\\\\begin{split}\u003cbr/\u003e\\\\var{3*a}x^{\\\\var{b}}y^{\\\\var{c}}\u0026amp;=\\\\var{a}x^{\\\\var{d}}(?)\\\\\\\\\u003cbr/\u003e\\\\Rightarrow \\\\frac{\\\\var{3*a}x^{\\\\var{b}}y^{\\\\var{c}}}{\\\\var{a}x^{\\\\var{d}}}\u0026amp;=(?)\\\\\\\\\u003cbr/\u003e\\\\Rightarrow 3x^{\\\\var{b}-\\\\var{d}}y^{\\\\var{c}} \u0026amp;=(?),\u003cbr/\u003e\\\\end{split}\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003ewhich after simplifying gives the answer:\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\\\\[(?) = 3x^{\\\\var{b-d}}y^{\\\\var{c}}\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Collect\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"Random(-3..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"d\": {\"name\": \"d\", \"group\": \"Ungrouped variables\", \"definition\": \"Random(-5..8 except [b,0])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"Random(-3..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"Random(2..9)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"b\", \"d\", \"c\", \"a\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"3x^{b-d}y^{c}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}, {\"name\": \"y\", \"value\": \"\"}]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AC13 Expand Double Brackets (Hard)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Poppy Jeffries\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21275/\"}, {\"name\": \"Lauren Desoysa\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21504/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eExpand two brackets involving powers of $x$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eExpand the brackets and simplify\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo expand the brackets $\\\\simplify{({a[1]}x^{b[1]}+{a[2]}x^{b[2]})({a[3]}x^{b[3]}+{c[1]}x^{b[4]})}$ We first multiply all the terms in the left bracket by all the terms in the right bracket. This gives us\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\var{a[1]}x^\\\\var{b[1]}\\\\times\\\\var{a[3]}x^\\\\var{b[3]}+\\\\var{a[1]}x^\\\\var{b[1]}\\\\times\\\\var{c[1]}x^\\\\var{b[4]}+\\\\var{a[2]}x^\\\\var{b[2]}\\\\times\\\\var{a[3]}x^\\\\var{b[3]}+\\\\var{a[2]}x^\\\\var{b[2]}\\\\times\\\\var{c[1]}x^\\\\var{b[4]}\\\\]\u003c/p\u003e\\n\u003cp\u003eWe can then simplify to give us the final answer of\u003c/p\u003e\\n\u003cp\u003e$\\\\simplify{{a[1]*a[3]}*x^{b[1]+b[3]}+{a[1]*c[1]}*x^{b[1]+b[4]}+{a[2]*a[3]}*x^{b[2]+b[3]}+{a[2]*c[1]}*x^{b[2]+b[4]}}.$\u003c/p\u003e\\n\u003cp\u003e\u003cbr/\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factorise\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"repeat(random(-5..10 except [0]),5\\n)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"repeat(random([-5,-9/2,-4,7/2,-3,-5/2,-2,-3/2,-1,-1/2,1/2,1,3/2,2,5/2,3,7/2,4,9/2,5]),5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"shuffle(1..6)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"c\", \"b\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\simplify{({a[1]}x^{b[1]}+{a[2]}x^{b[2]})({a[3]}x^{b[3]}+{c[1]}x^{b[4]})}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{a[1]*a[3]}*x^{b[1]+b[3]}+{a[1]*c[1]}*x^{b[1]+b[4]}+{a[2]*a[3]}*x^{b[2]+b[3]}+{a[2]*c[1]}*x^{b[2]+b[4]}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AS03 Simultaneous Equations (2 linear)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [\"Category: Simultaneous equations\"], \"metadata\": {\"description\": \"\u003cp\u003eSolving a pair of linear simultaneous equations, giving answers as integers or fractions.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eSolve the simultaneous equations for x and y, giving your answers as integers or fractions, but not decimals.\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split}\u0026nbsp;\\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} \u0026amp;\\\\,=\\\\var{c} \\\\\\\\ \\\\simplify[!noLeadingminus,unitFactor]{{a1}x +{b1}y}\u0026nbsp; \u0026amp;\\\\,=\\\\var{c1} \\\\end{split}\\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003e\\\\[\\\\begin{split}\\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} \u0026amp;\\\\,=\\\\var{c} \\\\qquad\\\\qquad\u0026amp;(1)\\\\\\\\ \\\\simplify[!noLeadingminus,unitFactor]{{a1}x +{b1}y}\u0026nbsp; \u0026amp;\\\\,=\\\\var{c1} \\\\qquad\\\\qquad\u0026amp;(2)\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e{advice1}\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Rearranging\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sim\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-2..8 except [0,1])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-8..8 except [0,1,a])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a1\": {\"name\": \"a1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..8 except [0,1])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b1\": {\"name\": \"b1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..10 except [round(a1*b/a),b,0,1])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c1\": {\"name\": \"c1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-7..7 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"aorsb\": {\"name\": \"aorsb\", \"group\": \"Ungrouped variables\", \"definition\": \"if(b*abs(b1)=b1*abs(b),\u0027subtract\u0027,\u0027add\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"torfb\": {\"name\": \"torfb\", \"group\": \"Ungrouped variables\", \"definition\": \"if(b*abs(b1)=b1*abs(b),\u0027from\u0027,\u0027to\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sgn\": {\"name\": \"sgn\", \"group\": \"Ungrouped variables\", \"definition\": \"if(b*abs(b1)=b1*abs(b),-1,1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"xn\": {\"name\": \"xn\", \"group\": \"Ungrouped variables\", \"definition\": \"c*abs(b1)+sgn*c1*abs(b)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"xd\": {\"name\": \"xd\", \"group\": \"Ungrouped variables\", \"definition\": \"a*abs(b1)+sgn*a1*abs(b)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"xsimp\": {\"name\": \"xsimp\", \"group\": \"Ungrouped variables\", \"definition\": \"xn/xd\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"samex\": {\"name\": \"samex\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eFor these equations, it is easiest to get a solution for $y$ first, due to the $x$-terms having {eqoroppa} coefficients.\u003c/p\u003e\\\\n\u003cp\u003eIf we {aorsa} equation (2) {torfa} equation (1) this eliminates the $x$-terms leaving us with one equation in terms of $y$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{({b}+{sgna*(b1)})y} \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\simplify[!collectNumbers, !noLeadingminus]{{c}+{sgna*(c1)}}\\\\\\\\\\\\\\\\ \\\\\\\\simplify{{b+sgna*(b1)}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{c+sgna*(c1)}} \\\\\\\\\\\\\\\\ y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, fractionNumbers]{{c+sgna*(c1)}/{b+sgna*(b1)}} \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\var{a}x + \\\\\\\\var{b} \\\\\\\\times \\\\\\\\simplify[all, fractionNumbers]{{c+sgna*(c1)}/{b+sgna*(b1)}} \u0026amp;\\\\\\\\,= \\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\var{a}x \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !collectNumbers, !noLeadingminus]{{c} - {c*b+b*sgna*(c1)}/{b+sgna*(b1)}} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\\\\\,= \\\\\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"eqoroppb\": {\"name\": \"eqoroppb\", \"group\": \"Ungrouped variables\", \"definition\": \"if(abs(b)*b1=abs(b1)*b,\u0027equal\u0027,\u0027equal and opposite\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"eqoroppa\": {\"name\": \"eqoroppa\", \"group\": \"Ungrouped variables\", \"definition\": \"if(abs(a)*a1=abs(a1)*a,\u0027equal\u0027,\u0027equal and opposite\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"samey\": {\"name\": \"samey\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eFor these equations, it is easiest to get a solution for $x$ first, due to the $y$-terms having {eqoroppb} coefficients.\u003c/p\u003e\\\\n\u003cp\u003eIf we {aorsb} equation (2) {torfb} equation (1) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{({a}+{sgn*(a1)})x} \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\simplify[!collectNumbers, !noLeadingminus]{{c}+{sgn*(c1)}}\\\\\\\\\\\\\\\\ \\\\\\\\simplify{{a+sgn*(a1)}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{c+sgn*(c1)}} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, fractionNumbers]{{c+sgn*(c1)}/{a+sgn*(a1)}} \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\var{a} \\\\\\\\times\\\\\\\\simplify[fractionNumbers]{{c+sgn*(c1)}/{a+sgn*(a1)}} + \\\\\\\\var{b}y \u0026amp;\\\\\\\\,= \\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\var{b}y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{{c} - {c*a+a*sgn*(c1)}/{a+sgn*(a1)}} \\\\\\\\\\\\\\\\ y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"lcmb\": {\"name\": \"lcmb\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo get a solution for $x$, if we multiply equation (2) by $\\\\\\\\simplify{{abs(b/b1)}}$ we will have two equations with {eqoroppb} $y$-coefficients:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split}\u0026nbsp;\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} \u0026amp;\\\\\\\\,=\\\\\\\\var{c} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(3)\\\\\\\\\\\\\\\\ \\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(b/b1)}x +{b1*abs(b/b1)}y}\u0026nbsp; \u0026amp;\\\\\\\\,=\\\\\\\\var{c1*abs(b/b1)} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(4)\\\\\\\\end{split}\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eIf we {aorsb} equation (4) {torfb} equation (3) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{({a}+{sgn*(a1*abs(b/b1))})x} \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\simplify[all, !collectNumbers, !noLeadingminus]{{c}+{sgn*(c1*abs(b/b1))}}\\\\\\\\\\\\\\\\ \\\\\\\\simplify{{a+sgn*(a1*abs(b/b1))}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{c+sgn*(c1*abs(b/b1))}} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all,fractionNumbers]{{c+sgn*(c1*abs(b/b1))}/{a+sgn*(a1*abs(b/b1))}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\var{a}\\\\\\\\times\\\\\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c+sgn*c1*abs(b/b1)}/{(a)+sgn*a1*abs(b/b1)}) + {b}y} \u0026amp;\\\\\\\\,= \\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{b}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(a*c)+a*sgn*c1*abs(b/b1)}/{(a)+sgn*a1*abs(b/b1)})} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{b}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(a*c+a*sgn*c1*abs(b/b1))/(a+sgn*a1*abs(b/b1))}} \\\\\\\\\\\\\\\\ y \u0026amp;\\\\\\\\,=\\\\\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"lcmb1\": {\"name\": \"lcmb1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo get a solution for $x$, if we multiply equation (1) by $\\\\\\\\simplify{{abs(b1/b)}}$ we will have two equations with {eqoroppb} $y$-coefficients:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!noLeadingminus,unitFactor]{{a*abs(b1/b)}x +{b*abs(b1/b)}y}\u0026nbsp; \u0026amp;\\\\\\\\,=\\\\\\\\var{c*abs(b1/b)} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(3) \\\\\\\\\\\\\\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1}x+{b1}y} \u0026amp;\\\\\\\\,=\\\\\\\\var{c1} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(4)\\\\\\\\\\\\\\\\ \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eIf we {aorsb} equation (4) {torfb} equation (3) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{({(a*abs(b1/b))}+{sgn*a1})x} \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\simplify[!collectNumbers, !noLeadingminus]{{(c*abs(b1/b))}+{sgn*c1}}\\\\\\\\\\\\\\\\ \\\\\\\\simplify{{(a*abs(b1/b))+sgn*a1}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{(c*abs(b1/b))+sgn*c1}} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, fractionNumbers]{{(c*abs(b1/b))+sgn*c1}/{(a*abs(b1/b))+sgn*a1}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\var{a}\\\\\\\\times\\\\\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({(c*abs(b1/b))+sgn*c1}/{(a*abs(b1/b))+sgn*a1}) + {b}y} \u0026amp;\\\\\\\\,= \\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{b}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(a*c*abs(b1/b))+a*sgn*c1}/{(a*abs(b1/b))+sgn*a1})} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{b}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(a*c*abs(b1/b)+a*sgn*c1)/(a*abs(b1/b)+sgn*a1)}} \\\\\\\\\\\\\\\\ y \u0026amp;\\\\\\\\,=\\\\\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"full\": {\"name\": \"full\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo get a solution for $x$, if we multiply equation (1) by $\\\\\\\\var{abs(b1)}$ and equation (2) by $\\\\\\\\var{abs(b)}$, we will have two equations with {eqoroppb} $y$-coefficients:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split}\u0026nbsp;\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a*abs(b1)}x+{b*abs(b1)}y} \u0026amp;\\\\\\\\,=\\\\\\\\var{c*abs(b1)} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(3)\\\\\\\\\\\\\\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(b)}x +{b1*abs(b)}y}\u0026nbsp; \u0026amp;\\\\\\\\,=\\\\\\\\var{c1*abs(b)} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(4) \\\\\\\\end{split}\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eNow, {aorsb} equation (4) {torfb} equation (3) to eliminate the $y$ terms:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} (\\\\\\\\simplify[!collectNumbers]{{a*abs(b1)} +{sgn*a1*abs(b)}}) x \u0026amp;\\\\\\\\,= \\\\\\\\simplify[!collectNumbers]{{c*abs(b1)}+{sgn*c1*abs(b)}}\u0026nbsp;\\\\\\\\\\\\\\\\\u0026nbsp;\u0026nbsp;\\\\\\\\simplify{{a*abs(b1)+sgn*a1*abs(b)}} x \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{c*abs(b1)+sgn*c1*abs(b)}} .\\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eSo the solution for $x$ is \\\\\\\\[ x=\\\\\\\\simplify{{c*abs(b1)+sgn*c1*abs(b)}/{a*abs(b1)+sgn*a1*abs(b)}}.\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $y$ we can substitute this value of $x$ into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[noLeadingminus,fractionNumbers,unitFactor]{{a}\u0026nbsp; {xsimp} + {b}y} \u0026amp;\\\\\\\\,=\\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\var{b}y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[!collectNumbers,fractionNumbers]{{c}-{a*xsimp}} \\\\\\\\\\\\\\\\\\\\\\\\var{b}y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[fractionNumbers]{{c-a*xsimp}} \\\\\\\\\\\\\\\\y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}} \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"aorsa\": {\"name\": \"aorsa\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a*abs(a1)=abs(a)*a1,\u0027subtract\u0027,\u0027add\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"torfa\": {\"name\": \"torfa\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a*abs(a1)=abs(a)*a1,\u0027from\u0027,\u0027to\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sgna\": {\"name\": \"sgna\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a*abs(a1)=abs(a)*a1,-1,1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"lcma\": {\"name\": \"lcma\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo get a solution for $y$, if we multiply equation (2) by $\\\\\\\\simplify{{abs(a/a1)}}$ we will have two equations with {eqoroppa} $x$-coefficients:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split}\u0026nbsp;\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} \u0026amp;\\\\\\\\,=\\\\\\\\var{c} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(3)\\\\\\\\\\\\\\\\ \\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(a/a1)}x +{b1*abs(a/a1)}y}\u0026nbsp; \u0026amp;\\\\\\\\,=\\\\\\\\var{c1*abs(a/a1)} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(4)\\\\\\\\end{split}\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eIf we {aorsa} equation (4) {torfa} equation (3) this eliminates the $x$-terms, leaving us with one equation in terms of $y$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{({b}+{sgna*(b1*abs(a/a1))})y} \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\simplify[all, !collectNumbers, !noLeadingminus]{{c}+{sgna*(c1*abs(a/a1))}}\\\\\\\\\\\\\\\\ \\\\\\\\simplify{{b+sgna*(b1*abs(a/a1))}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{c+sgna*(c1*abs(a/a1))}} \\\\\\\\\\\\\\\\ y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all,fractionNumbers]{{c+sgna*(c1*abs(a/a1))}/{b+sgna*(b1*abs(a/a1))}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{{a}x + {b}}\\\\\\\\times \\\\\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c+sgna*c1*abs(a/a1)}/{(b)+sgna*b1*abs(a/a1)})} \u0026amp;\\\\\\\\,= \\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{a}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(b*c)+b*sgna*c1*abs(a/a1)}/{(b)+sgna*b1*abs(a/a1)})} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{a}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(b*c+b*sgna*c1*abs(a/a1))/(b+sgna*b1*abs(a/a1))}} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\\\\\,=\\\\\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"lcma1\": {\"name\": \"lcma1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo get a solution for $y$, if we multiply equation (1) by $\\\\\\\\simplify{{abs(a1/a)}}$ we will have two equations with {eqoroppa} $x$-coefficients:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!noLeadingminus,unitFactor]{{a*abs(a1/a)}x +{b*abs(a1/a)}y}\u0026nbsp; \u0026amp;\\\\\\\\,=\\\\\\\\var{c*abs(a1/a)} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(3) \\\\\\\\\\\\\\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1}x+{b1}y} \u0026amp;\\\\\\\\,=\\\\\\\\var{c1} \\\\\\\\qquad\\\\\\\\qquad\u0026amp;(4) \\\\\\\\end{split}\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eIf we {aorsa} equation (4) {torfa} equation (3) this eliminates the $x$-terms, leaving us with one equation in terms of $y$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[!collectNumbers, !noLeadingminus]{({(b*abs(a1/a))}+{sgna*b1})y} \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\simplify[!collectNumbers, !noLeadingminus]{{(c*abs(a1/a))}+{sgna*c1}}\\\\\\\\\\\\\\\\ \\\\\\\\simplify{{(b*abs(a1/a))+sgna*b1}y} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{{(c*abs(a1/a))+sgna*c1}} \\\\\\\\\\\\\\\\ y \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, fractionNumbers]{{(c*abs(a1/a))+sgna*c1}/{(b*abs(a1/a))+sgna*b1}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\\n\u003cp\u003eTo obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{{a}x + {b}}\\\\\\\\times \\\\\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c*abs(a1/a)+sgna*c1}/{(b*abs(a1/a))+sgna*b1})} \u0026amp;\\\\\\\\,= \\\\\\\\var{c} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{a}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(b*c*abs(a1/a))+b*sgna*c1}/{(b*abs(a1/a))+sgna*b1})} \\\\\\\\\\\\\\\\ \\\\\\\\simplify{{a}x} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(b*c*abs(a1/a)+b*sgna*c1)/(b*abs(a1/a)+sgna*b1)}} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\\\\\,=\\\\\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"advice1\": {\"name\": \"advice1\", \"group\": \"Ungrouped variables\", \"definition\": \"if(abs(b)=abs(b1), {samey},if(abs(a)=abs(a1),{samex},if(lcm(abs(b),abs(b1))=abs(b),{lcmb},if(lcm(abs(b),abs(b1))=abs(b1),{lcmb1},if(lcm(abs(a),abs(a1))=abs(a),{lcma},if(lcm(abs(a),abs(a1))=abs(a1),{lcma1},{full}))))))\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"abs(b-b1)\u003e1 and\\nabs(a-a1)\u003e1 and\\ngcd(a,c)=1 and\\ngcd(a1,c1)=1\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"a1\", \"b1\", \"c\", \"c1\", \"aorsa\", \"torfa\", \"aorsb\", \"torfb\", \"sgna\", \"sgn\", \"xn\", \"xd\", \"xsimp\", \"eqoroppa\", \"eqoroppb\", \"advice1\", \"samey\", \"samex\", \"lcmb\", \"lcmb1\", \"lcma\", \"lcma1\", \"full\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$x=$ [[0]]\u003c/p\u003e\\n\u003cp\u003e$y=$ [[1]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}\", \"answerSimplification\": \"fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{(c-a*xsimp)/b}\", \"answerSimplification\": \"fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AS04 Simultaneous Equations (one non-linear)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eSolving a pair of simultaneous equations of the form $a_1x+y=c_1$ and $a_2x^2+b_2xy=c_2$ by forming a quadratic equation.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eSolve the following simultaneous equations:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\simplify{{a1}x+y} \u0026amp;\\\\,= \\\\var{c1} \\\\\\\\ \\\\simplify{{a2}x^2+{b2}x*y} \u0026amp;\\\\,= \\\\var{c2} \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cem\u003eGive your answers to 2 decimal places where necessary.\u003c/em\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo solve a pair of simultaneous equations of this type we want to rearrange the linear equation such that $y$ is the subject, which we can then substitute into the equation with the quadratic $x$-term. This will result in a quadratic equation in terms of $x$ only.\u003c/p\u003e\\n\u003cp\u003eFor the equations\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\simplify{{a1}x+y} \u0026amp;\\\\,= \\\\var{c1} \\\\qquad \\\\qquad \u0026amp;(1) \\\\\\\\\\\\simplify{{a2}x^2+{b2}x*y} \u0026amp;\\\\,= \\\\var{c2} \\\\qquad \\\\qquad \u0026amp;(2) \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003ewe can rearrange equation (1)\u0026nbsp;to make $y$ the subject:\u003c/p\u003e\\n\u003cp\u003e\\\\[ y = \\\\simplify{{c1}-{a1}x}. \\\\qquad\\\\qquad (3)\\\\]\u003c/p\u003e\\n\u003cp\u003eSubstituting this into equation (2):\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split}\\\\simplify{{a2}x^2+{b2}x({c1}-{a1}x)} \u0026amp;\\\\,=\\\\var{c2} \\\\\\\\ \\\\simplify[!cancelTerms,unitFactor]{{a2}x^2+{b2*c1}x-{b2*a1}x^2} \u0026amp;\\\\,=\\\\var{c2}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eCollecting similar terms:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{({a2}-{b2*a1})x^2+{b2*c1}x-{c2}} =0. \\\\qquad\\\\qquad (4) \\\\]\u003c/p\u003e\\n\u003cp\u003eUsing the quadratic formula, we find two solutions for $x$:\u003c/p\u003e\\n\u003cp\u003e{check}\u003c/p\u003e\\n\u003cp\u003eTherefore, the 2 pairs of solutions for these simultaneous equations are\u003c/p\u003e\\n\u003cp\u003e\\\\[ (x_1,y_1) = (\\\\var{x1dp},\\\\var{y1dp}) \\\\] and\u0026nbsp;\\\\[ (x_2,y_2) = (\\\\var{x2dp},\\\\var{y2dp}). \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Rearranging\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\u003c/span\u003e\u003c/a\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sim\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a1\": {\"name\": \"a1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..-1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c1\": {\"name\": \"c1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..6)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a2\": {\"name\": \"a2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..3)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b2\": {\"name\": \"b2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c2\": {\"name\": \"c2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(0..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solx1\": {\"name\": \"solx1\", \"group\": \"Ungrouped variables\", \"definition\": \"(-b2*c1-sqrt((b2^2*c1^2)+4(a2-a1*b2)*c2))/(2(a2-b2*a1))\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solx2\": {\"name\": \"solx2\", \"group\": \"Ungrouped variables\", \"definition\": \"(-b2*c1+sqrt((b2^2*c1^2)+4(a2-a1*b2)*c2))/(2(a2-b2*a1))\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"soly1\": {\"name\": \"soly1\", \"group\": \"Ungrouped variables\", \"definition\": \"c1-a1*solx1\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"soly2\": {\"name\": \"soly2\", \"group\": \"Ungrouped variables\", \"definition\": \"c1-a1*solx2\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"x2dp\": {\"name\": \"x2dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(solx2,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"y1dp\": {\"name\": \"y1dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(soly1,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"y2dp\": {\"name\": \"y2dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(soly2,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"x1dp\": {\"name\": \"x1dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(solx1,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solutions1\": {\"name\": \"solutions1\", \"group\": \"Ungrouped variables\", \"definition\": \"matrix([x1dp,y1dp])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solutions2\": {\"name\": \"solutions2\", \"group\": \"Ungrouped variables\", \"definition\": \"matrix([x2dp,y2dp])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"check\": {\"name\": \"check\", \"group\": \"Ungrouped variables\", \"definition\": \"if(x1dp=round(x1dp) and x2dp=round(x2dp),\u0027{text}\u0027, if(x1dp=round(x1dp),\u0027{text1}\u0027,if(x2dp=round(x2dp),\u0027{text2}\u0027,\u0027{text3}\u0027)))\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"text1\": {\"name\": \"text1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ x_1 = \\\\\\\\var{x1dp} \\\\\\\\,\u0026nbsp; \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad x_2=\\\\\\\\var{x2dp} \\\\\\\\, \\\\\\\\text{ (2 d.p.)} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ y_1 = \\\\\\\\var{y1dp} \\\\\\\\,\u0026nbsp; \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad y_2=\\\\\\\\var{y2dp} \\\\\\\\, \\\\\\\\text{(2 d.p.)} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"text2\": {\"name\": \"text2\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ x_1 = \\\\\\\\var{x1dp} \\\\\\\\, \\\\\\\\text{ (2 d.p.)}\\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad x_2=\\\\\\\\var{x2dp} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ y_1 = \\\\\\\\var{y1dp} \\\\\\\\, \\\\\\\\text{(2 d.p.)} \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad y_2=\\\\\\\\var{y2dp} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"text3\": {\"name\": \"text3\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ x_1 = \\\\\\\\var{x1dp} \\\\\\\\,\u0026nbsp; \\\\\\\\text{ (2 d.p.)}\\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad x_2=\\\\\\\\var{x2dp} \\\\\\\\, \\\\\\\\text{(2 d.p.)} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ y_1 = \\\\\\\\var{y1dp} \\\\\\\\, \\\\\\\\text{(2 d.p.)} \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad y_2=\\\\\\\\var{y2dp} \\\\\\\\,\u0026nbsp;\\\\\\\\text{(2 d.p.)} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"text\": {\"name\": \"text\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ x_1 = \\\\\\\\var{x1dp} \\\\\\\\,\u0026nbsp; \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad x_2=\\\\\\\\var{x2dp}\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ y_1 = \\\\\\\\var{y1dp} \\\\\\\\,\u0026nbsp; \\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad y_2=\\\\\\\\var{y2dp} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"(a2-a1*b2)\u003e0 and (b2^2*c1^2+4(a2-a1*b2)*c2)\u003e0 and gcd(a2,b2)=1\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a1\", \"c1\", \"a2\", \"b2\", \"c2\", \"solx1\", \"solx2\", \"soly1\", \"soly2\", \"x1dp\", \"y1dp\", \"x2dp\", \"y2dp\", \"solutions1\", \"solutions2\", \"check\", \"text\", \"text1\", \"text2\", \"text3\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$(x_1,y_1)=$[[0]]\u003c/p\u003e\\n\u003cp\u003e$(x_2,y_2)=$[[1]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"alternatives\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"correctAnswer\": \"solutions2\", \"correctAnswerFractions\": false, \"numRows\": 1, \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": false, \"minColumns\": 1, \"maxColumns\": 0, \"minRows\": 1, \"maxRows\": 0, \"prefilledCells\": \"\"}], \"correctAnswer\": \"solutions1\", \"correctAnswerFractions\": false, \"numRows\": 1, \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": false, \"minColumns\": 1, \"maxColumns\": 0, \"minRows\": 1, \"maxRows\": 0, \"prefilledCells\": \"\"}, {\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"alternatives\": [{\"type\": \"matrix\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"correctAnswer\": \"solutions1\", \"correctAnswerFractions\": false, \"numRows\": 1, \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": false, \"minColumns\": 1, \"maxColumns\": 0, \"minRows\": 1, \"maxRows\": 0, \"prefilledCells\": \"\"}], \"correctAnswer\": \"solutions2\", \"correctAnswerFractions\": false, \"numRows\": 1, \"numColumns\": \"2\", \"allowResize\": false, \"tolerance\": 0, \"markPerCell\": true, \"allowFractions\": false, \"minColumns\": 1, \"maxColumns\": 0, \"minRows\": 1, \"maxRows\": 0, \"prefilledCells\": \"\"}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AS07 Quadratics - solve\", \"extensions\": [], \"custom_part_types\": [{\"source\": {\"pk\": 2, \"author\": {\"name\": \"Christian Lawson-Perfect\", \"pk\": 7}, \"edit_page\": \"/part_type/2/edit\"}, \"name\": \"List of numbers\", \"short_name\": \"list-of-numbers\", \"description\": \"\u003cp\u003eThe answer is a comma-separated list of numbers.\u003c/p\u003e\\n\u003cp\u003eThe list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.\u003c/p\u003e\\n\u003cp\u003eYou can optionally treat the answer as a set, so the number of occurrences doesn\u0027t matter, only whether each number is included or not.\u003c/p\u003e\", \"help_url\": \"\", \"input_widget\": \"string\", \"input_options\": {\"correctAnswer\": \"join(\\n  if(settings[\\\"correctAnswerFractions\\\"],\\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\\\"correctAnswer\\\"])\\n  ,\\n    settings[\\\"correctAnswer\\\"]\\n  ),\\n  settings[\\\"separator\\\"] + \\\" \\\"\\n)\", \"hint\": {\"static\": false, \"value\": \"if(settings[\\\"show_input_hint\\\"],\\n  \\\"Enter a list of numbers separated by \u003ccode\u003e{settings[\u0027separator\u0027]}\u003c/code\u003e.\\\",\\n  \\\"\\\"\\n)\"}, \"allowEmpty\": {\"static\": true, \"value\": true}}, \"can_be_gap\": true, \"can_be_step\": true, \"marking_script\": \"bits:\\nlet(b,filter(x\u003c\u003e\\\"\\\",x,split(studentAnswer,settings[\\\"separator\\\"])),\\n    if(isSet,list(set(b)),b)\\n)\\n\\nexpected_numbers:\\nlet(l,settings[\\\"correctAnswer\\\"] as \\\"list\\\",\\n    if(isSet,list(set(l)),l)\\n)\\n\\nvalid_numbers:\\nif(all(map(not isnan(x),x,interpreted_answer)),\\n    true,\\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\\n      warn(wrong+\\\" is not a valid number\\\");\\n      fail(wrong+\\\" is not a valid number.\\\")\\n    )\\n  )\\n\\nis_sorted:\\nassert(sort(interpreted_answer)=interpreted_answer,\\n    multiply_credit(0.5,\\\"Not in order\\\")\\n  )\\n\\nincluded:\\nmap(\\n      let(\\n        num_student,len(filter(x=y,y,interpreted_answer)),\\n        num_expected,len(filter(x=y,y,expected_numbers)),\\n        switch(\\n          num_student=num_expected,\\n            true,\\n          num_student\u003cnum_expected,\\n            incorrect(\\n              if(isSet,\\\"Your answer does not contain \\\"+x+\\\" but should.\\\",\\n                \\\"Your answer does not contain \\\"+x+\\\" enough times.\\\"\\n              )\\n            );false\\n          ,\\n            incorrect(\\n              if(isSet,\\\"Your answer contains \\\"+x+\\\" but should not.\\\",\\n                \\\"Your answer contains \\\"+x+\\\" too many times.\\\"\\n              )\\n            );false\\n        )\\n      )\\n      ,\\n      x,\\n      sort(list(set(expected_numbers)))\\n  )\\n\\nall_included:\\nall(included)\\n\\nno_extras:\\nif(all(map(x in expected_numbers, x, interpreted_answer)),\\n    true\\n  ,\\n    incorrect(\\\"Your answer contains \\\"+extra_numbers[0]+\\\" but should not.\\\");\\n    false\\n  )\\n\\ninterpreted_answer:\\nif(lower(studentAnswer) in [\\\"empty\\\",\\\"\\u2205\\\"],[],\\n  map(\\n    if(settings[\\\"allowFractions\\\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\\n    ,x\\n    ,bits\\n  )\\n)\\n\\nmark:\\nif(studentanswer=\\\"\\\",fail(\\\"You have not entered an answer\\\"),false);\\napply(valid_numbers);\\napply(included);\\napply(no_extras);\\ncorrectif(all_included and no_extras)\\n\\nnotationStyles:\\n[\\\"en\\\"]\\n\\nisSet:\\nsettings[\\\"isSet\\\"]\\n\\nextra_numbers:\\nfilter(not (x in expected_numbers),x,interpreted_answer)\", \"marking_notes\": [{\"name\": \"bits\", \"description\": \"\u003cp\u003eThe separate items in the student\u0027s answer\u003c/p\u003e\", \"definition\": \"let(b,filter(x\u003c\u003e\\\"\\\",x,split(studentAnswer,settings[\\\"separator\\\"])),\\n    if(isSet,list(set(b)),b)\\n)\"}, {\"name\": \"expected_numbers\", \"description\": \"\", \"definition\": \"let(l,settings[\\\"correctAnswer\\\"] as \\\"list\\\",\\n    if(isSet,list(set(l)),l)\\n)\"}, {\"name\": \"valid_numbers\", \"description\": \"\u003cp\u003eIs every number in the student\u0027s list valid?\u003c/p\u003e\", \"definition\": \"if(all(map(not isnan(x),x,interpreted_answer)),\\n    true,\\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\\n      warn(wrong+\\\" is not a valid number\\\");\\n      fail(wrong+\\\" is not a valid number.\\\")\\n    )\\n  )\"}, {\"name\": \"is_sorted\", \"description\": \"\u003cp\u003eAre the student\u0027s answers in ascending order?\u003c/p\u003e\", \"definition\": \"assert(sort(interpreted_answer)=interpreted_answer,\\n    multiply_credit(0.5,\\\"Not in order\\\")\\n  )\"}, {\"name\": \"included\", \"description\": \"\u003cp\u003eIs each number in the expected answer present in the student\u0027s list the correct number of times?\u003c/p\u003e\", \"definition\": \"map(\\n      let(\\n        num_student,len(filter(x=y,y,interpreted_answer)),\\n        num_expected,len(filter(x=y,y,expected_numbers)),\\n        switch(\\n          num_student=num_expected,\\n            true,\\n          num_student\u003cnum_expected,\\n            incorrect(\\n              if(isSet,\\\"Your answer does not contain \\\"+x+\\\" but should.\\\",\\n                \\\"Your answer does not contain \\\"+x+\\\" enough times.\\\"\\n              )\\n            );false\\n          ,\\n            incorrect(\\n              if(isSet,\\\"Your answer contains \\\"+x+\\\" but should not.\\\",\\n                \\\"Your answer contains \\\"+x+\\\" too many times.\\\"\\n              )\\n            );false\\n        )\\n      )\\n      ,\\n      x,\\n      sort(list(set(expected_numbers)))\\n  )\"}, {\"name\": \"all_included\", \"description\": \"\u003cp\u003eHas every number been included the right number of times?\u003c/p\u003e\", \"definition\": \"all(included)\"}, {\"name\": \"no_extras\", \"description\": \"\u003cp\u003eTrue if the student\u0027s list doesn\u0027t contain any numbers that aren\u0027t in the expected answer.\u003c/p\u003e\", \"definition\": \"if(all(map(x in expected_numbers, x, interpreted_answer)),\\n    true\\n  ,\\n    incorrect(\\\"Your answer contains \\\"+extra_numbers[0]+\\\" but should not.\\\");\\n    false\\n  )\"}, {\"name\": \"interpreted_answer\", \"description\": \"A value representing the student\u0027s answer to this part.\", \"definition\": \"if(lower(studentAnswer) in [\\\"empty\\\",\\\"\\u2205\\\"],[],\\n  map(\\n    if(settings[\\\"allowFractions\\\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\\n    ,x\\n    ,bits\\n  )\\n)\"}, {\"name\": \"mark\", \"description\": \"This is the main marking note. It should award credit and provide feedback based on the student\u0027s answer.\", \"definition\": \"if(studentanswer=\\\"\\\",fail(\\\"You have not entered an answer\\\"),false);\\napply(valid_numbers);\\napply(included);\\napply(no_extras);\\ncorrectif(all_included and no_extras)\"}, {\"name\": \"notationStyles\", \"description\": \"\", \"definition\": \"[\\\"en\\\"]\"}, {\"name\": \"isSet\", \"description\": \"\u003cp\u003eShould the answer be considered as a set, so the number of times an element occurs doesn\u0027t matter?\u003c/p\u003e\", \"definition\": \"settings[\\\"isSet\\\"]\"}, {\"name\": \"extra_numbers\", \"description\": \"\u003cp\u003eNumbers included in the student\u0027s answer that are not in the expected list.\u003c/p\u003e\", \"definition\": \"filter(not (x in expected_numbers),x,interpreted_answer)\"}], \"settings\": [{\"name\": \"correctAnswer\", \"label\": \"Correct answer\", \"help_url\": \"\", \"hint\": \"The list of numbers that the student should enter. The order does not matter.\", \"input_type\": \"code\", \"default_value\": \"\", \"evaluate\": true}, {\"name\": \"allowFractions\", \"label\": \"Allow the student to enter fractions?\", \"help_url\": \"\", \"hint\": \"\", \"input_type\": \"checkbox\", \"default_value\": false}, {\"name\": \"correctAnswerFractions\", \"label\": \"Display the correct answers as fractions?\", \"help_url\": \"\", \"hint\": \"\", \"input_type\": \"checkbox\", \"default_value\": false}, {\"name\": \"isSet\", \"label\": \"Is the answer a set?\", \"help_url\": \"\", \"hint\": \"If ticked, the number of times an element occurs doesn\u0027t matter, only whether it\u0027s included at all.\", \"input_type\": \"checkbox\", \"default_value\": false}, {\"name\": \"show_input_hint\", \"label\": \"Show the input hint?\", \"help_url\": \"\", \"hint\": \"\", \"input_type\": \"checkbox\", \"default_value\": true}, {\"name\": \"separator\", \"label\": \"Separator\", \"help_url\": \"\", \"hint\": \"The substring that should separate items in the student\u0027s list\", \"input_type\": \"string\", \"default_value\": \",\", \"subvars\": false}], \"public_availability\": \"always\", \"published\": true, \"extensions\": []}], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Megan Oliver\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23526/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eSolving a quadratic equation via factorisation (or otherwise) with the $x^2$-term having a coefficient of 1.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eSolve the following quadratic equation by factorisation or otherwise:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify[unitFactor]{x^2+{b}x+{c}=0} \\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo solve a quadratic equation of the form \\\\[ x^2+bx+c=0\\\\] by factorisation, we want to factorise the equation into the form \\\\[(x+p)(x+q)=0,\\\\] where $p+q=b$ and $p \\\\times q = c$.\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003eHence, for the equation \\\\[\\\\simplify{x^2+{b}x+{c}=0}, \\\\]\u003c/p\u003e\\n\u003cp\u003ethis can be factorised to \\\\[\\\\simplify{(x+{p})(x+{q})=0}.\\\\] This equation is satisfied when either \\\\[\\\\simplify{x+{p}=0} \\\\quad \\\\text{or} \\\\quad \\\\simplify{x+{q}=0}, \\\\] which implies the solutions to this quadratic equation are \\\\[ \\\\simplify{x={-p}} \\\\quad \\\\text{and} \\\\quad \\\\simplify{x={-q}} .\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find resources to help you revise how to solve quadratic equations by factorising the expression.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"{p+q}\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"{p*q}\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"p\": {\"name\": \"p\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"q\": {\"name\": \"q\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except [0,p])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol\": {\"name\": \"sol\", \"group\": \"Ungrouped variables\", \"definition\": \"[-p,-q]\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"abs(p+q)\u003e0\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"b\", \"c\", \"p\", \"q\", \"sol\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$x= $[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"list-of-numbers\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"settings\": {\"correctAnswer\": \"{sol}\", \"allowFractions\": false, \"correctAnswerFractions\": false, \"isSet\": false, \"show_input_hint\": true, \"separator\": \",\"}}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AS08 Completing the square\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Christian Lawson-Perfect\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/7/\"}, {\"name\": \"Hannah Aldous\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/1594/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [\"complete the square\", \"completing the square\", \"taxonomy\"], \"metadata\": {\"description\": \"\u003cp\u003eRearrange expressions in the form $ax^2+bx+c$ to $a(x+b)^2+c$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003eWe can rewrite quadratic equations given in the form $ax^2+bx+c$ as a\u0026nbsp;square plus another term - this is called \\\"completing the square\\\".\u003c/p\u003e\\n\u003cp\u003eThis can be useful when it isn\u0027t obvious how to fully factorise a quadratic equation.\u003c/p\u003e\\n\u003cp\u003eRewrite the following expressions in the form \\\\[(x+b)^2-c\\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eCompleting the square works by noticing that\u003c/p\u003e\\n\u003cp\u003e\\\\[ (x+a)^2 = x^2 + 2ax + a^2 \\\\]\u003c/p\u003e\\n\u003cp\u003eSo when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eReplace $x^2+\\\\var{evens2}x$ with $(x+\\\\var{evens2/2})^2 - \\\\var{evens2/2}^2$. Remember\u0026nbsp;to keep the $\\\\var{evens2-evens1}$ term on the end!\u003c/p\u003e\\n\u003cp\u003e\\\\begin{align}\u003cbr/\u003e\\\\simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} \u0026nbsp;\u0026amp;= \\\\simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\\\\\\\\u003cbr/\u003e\u0026amp;= \\\\simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} }\u003cbr/\u003e\\\\end{align}\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\\n\u003ch4\u003e\u003cem\u003e\u003c/em\u003e\u003c/h4\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"multiall2\": {\"name\": \"multiall2\", \"group\": \"Ungrouped variables\", \"definition\": \"all2*random(2..10 #2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"odds3\": {\"name\": \"odds3\", \"group\": \"Odds and Evens\", \"definition\": \"random(11..30 #2 except odds odds2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"evens3\": {\"name\": \"evens3\", \"group\": \"Odds and Evens\", \"definition\": \"random(2..30 #2 except evens1 evens2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"evens2\": {\"name\": \"evens2\", \"group\": \"Odds and Evens\", \"definition\": \"random(10..30 #2 except evens1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"multiall\": {\"name\": \"multiall\", \"group\": \"Ungrouped variables\", \"definition\": \"all*random(2..10#2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"evens1\": {\"name\": \"evens1\", \"group\": \"Odds and Evens\", \"definition\": 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0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\simplify {x^2+ {evens2}x +{evens2-evens1}} =$\u0026nbsp;[[0]]\u003cem\u003e\u003cbr/\u003e\u003c/em\u003e\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"(x+{evens2/2})^2+{-(evens2/2)^2+evens2-evens1}\", \"answerSimplification\": \"basic, fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"musthave\": {\"strings\": [\"(x\", \")^2\"], \"showStrings\": false, \"partialCredit\": 0, \"message\": \"\u003cp\u003eIt doesn\u0027t look like you\u0027ve completed the square.\u003c/p\u003e\"}, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AC18 Algebraic Fractions - addition (harder)\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eSimplify the sum of two algebraic fractions where spotting factorising of both numerators and denominators can reduce the work massively.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eWrite the following as a single fraction $\\\\frac{\\\\var{num1}}{\\\\var{den1}}+\\\\frac{\\\\var{num2}}{\\\\var{den2}}$ simplifying as much as possible. Your answer should be in the form $\\\\frac{\\\\alpha\\\\var{v}+\\\\beta}{\\\\delta\\\\var{v}^2-\\\\gamma}.$\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo write the following as a single fraction $\\\\frac{\\\\var{num1}}{\\\\var{den1}}+\\\\frac{\\\\var{num2}}{\\\\var{den2}}$ first factorise as much as possible and look for any cancellations:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\begin{split}\u003cbr/\u003e\u0026amp;\\\\frac{\\\\var{a}\\\\times\\\\var{b}}{\\\\var{den1fact}} + \\\\frac{\\\\var{num2}}{\\\\var{den2fact}}\\\\\\\\\u003cbr/\u003e\u0026amp; = \\\\frac{\\\\var{b}}{\\\\var{den1simp}} + \\\\frac{1}{\\\\var{f1c}}.\u003cbr/\u003e\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003eThen get a common denominator for the two fractions and combine into a single fraction:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\begin{split}\u003cbr/\u003e\u0026amp;\\\\frac{\\\\var{b}}{\\\\var{den1simp}} + \\\\frac{\\\\var{f1}}{\\\\var{den1simp}}\\\\\\\\\u003cbr/\u003e\u0026amp; = \\\\frac{\\\\var{b}+\\\\var{f1}}{\\\\var{den1simp}}\\\\\\\\\u003cbr/\u003e\u0026amp; = \\\\var{ans}.\u003cbr/\u003e\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#AlgFrac\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Set up\", \"definition\": \"random(2 .. 6#1)\", \"description\": \"\", \"templateType\": \"randrange\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Set up\", \"definition\": \"random(2 .. 5#1)\", \"description\": \"\", \"templateType\": \"randrange\", \"can_override\": false}, \"v\": {\"name\": \"v\", \"group\": \"Set up\", \"definition\": \"random(\\\"a\\\",\\\"b\\\",\\\"c\\\",\\\"d\\\",\\\"f\\\",\\\"g\\\",\\\"h\\\",\\\"k\\\",\\\"m\\\",\\\"n\\\",\\\"p\\\",\\\"q\\\",\\\"r\\\",\\\"s\\\",\\\"t\\\",\\\"u\\\",\\\"v\\\",\\\"w\\\",\\\"x\\\",\\\"y\\\",\\\"z\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"cf1\": {\"name\": \"cf1\", \"group\": \"Set up\", \"definition\": \"repeat(random(2..4),2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"f1\": {\"name\": \"f1\", \"group\": \"Set up\", \"definition\": \"simplify(cf1[0]+\\\"*\\\"+v+\\\"+\\\"+cf1[1],\\\"all\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"f1c\": {\"name\": \"f1c\", \"group\": \"Set up\", \"definition\": \"simplify(cf1[0]+\\\"*\\\"+v+\\\"-\\\"+cf1[1],\\\"all\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"cf2\": {\"name\": \"cf2\", \"group\": \"Set up\", \"definition\": \"repeat(random(2..5),2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"f2\": {\"name\": \"f2\", \"group\": \"Set up\", \"definition\": \"simplify(cf2[0]+\\\"*\\\"+v+\\\"+\\\"+cf2[1],\\\"all\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"den1fact\": {\"name\": \"den1fact\", \"group\": \"Advice\", \"definition\": \"simplify(a+\\\"*\\\"+\\\"(\\\"+string(f1)+\\\")*(\\\"+string(f1c)+\\\")\\\",\\\"all\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"num1\": {\"name\": \"num1\", \"group\": \"Question\", \"definition\": \"a*b\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"den2\": {\"name\": \"den2\", \"group\": \"Question\", \"definition\": \"simplify(den2fact,[\\\"expandBrackets\\\",\\\"all\\\"])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"num2\": {\"name\": \"num2\", \"group\": \"Question\", \"definition\": 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\"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"ans\": {\"name\": \"ans\", \"group\": \"Question\", \"definition\": \"simplify(expression(\\\"(\\\"+string(ansn)+\\\")\\\"+\\\"/\\\"+\\\"(\\\"+string(ansd)+\\\")\\\"),\\\"all\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"den1simp\": {\"name\": \"den1simp\", \"group\": \"Advice\", \"definition\": \"simplify(\\\"(\\\"+string(f1)+\\\")*(\\\"+string(f1c)+\\\")\\\",\\\"all\\\")\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"f1\u003c\u003ef2 AND f1c\u003c\u003ef2 AND cf1[0]\u003c\u003ecf1[1] AND cf2[0]\u003c\u003ecf2[1]\", \"maxRuns\": 100}, \"ungrouped_variables\": [], \"variable_groups\": [{\"name\": \"Set up\", \"variables\": [\"a\", \"b\", \"v\", \"cf1\", \"f1\", \"f1c\", \"cf2\", \"f2\"]}, {\"name\": \"Question\", \"variables\": [\"num1\", \"den1\", \"num2\", \"den2\", \"ansn\", \"ansd\", \"ans\"]}, {\"name\": \"Advice\", \"variables\": [\"den1fact\", \"den2fact\", \"den1simp\"]}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{ans}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"mustmatchpattern\": {\"pattern\": \"?`+/?`+\", \"partialCredit\": 0, \"message\": \"\", \"nameToCompare\": \"\", \"warningTime\": \"submission\"}, \"valuegenerators\": []}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AC19 Cancelling algebraic fractions\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Luke Park\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/826/\"}, {\"name\": \"Anna Strzelecka\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/2945/\"}, {\"name\": \"heike hoffmann\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/2960/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eA question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003eSimplify the following algebraic expression.\u003cem\u003e\u003c/em\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003e\\\\[\\\\frac{{\\\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}}{{\\\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}}\\\\]\u003c/p\u003e\\n\u003cp\u003eIn this question there is a quadratic expression which needs to be factorised into the products of binomials in both the numerator and denominator.\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{({\\\\simplify{n+{e1}}})({\\\\simplify{n+{e2}}})}{({\\\\simplify{n+{e1}}})({\\\\simplify{n+{e3}}})}\\\\]\u003c/p\u003e\\n\u003cp\u003eThe repeated binomials in the numerator and denominator cancel, leaving:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{({\\\\simplify{n+{e2}}})}{({\\\\simplify{n+{e3}}})}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#AlgFrac\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"e2\": {\"name\": \"e2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"e1\": {\"name\": \"e1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"e3\": {\"name\": \"e3\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except 0 except e2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"e1\", \"e2\", \"e3\"], \"variable_groups\": [], \"functions\": {\"\": {\"parameters\": [], \"type\": \"number\", \"language\": \"jme\", \"definition\": \"\"}}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e\\\\[\\\\frac{\\\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}{\\\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}\\\\]\u003c/p\u003e\", \"answer\": \"(n+{e2})/(n+{e3})\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"notallowed\": {\"strings\": [\"^2\", \"^\"], \"showStrings\": false, \"partialCredit\": 0, \"message\": \"\"}, \"valuegenerators\": [{\"name\": \"n\", \"value\": \"\"}]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AC22 Partial Fractions\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}, {\"name\": \"Oliver Spenceley\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23557/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eRewrite the expression $\\\\frac{mx^2+nx+k}{(x+a)(x^2+bx+c)}$ as partial fractions in the form $\\\\frac{A}{x+a}+\\\\frac{Bx+C}{x^2+bx+c}$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eRewrite the following expression as partial fractions:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))}. \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo express \\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} \\\\] as partial fractions, we want to set this equal to the sum of two fractions with denominators $\\\\simplify{x+{a}}$ and $\\\\simplify{x^2+{b}x+{c}}$. Since we have a linear factor and a quadratic factor, this tells us that the form of the partial fractions will be\u003c/p\u003e\\n\u003cp\u003e\\\\[\u0026nbsp;\\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\simplify{A/(x+{a}) + (B*x+C)/(x^2+{b}x+{c})},\\\\]\u003c/p\u003e\\n\u003cp\u003ewhere $A$, $B$, and $C$ are constants.\u003c/p\u003e\\n\u003cp\u003eTo find the values of $A$, $B$, and $C$, we want to first multiply this equation by the denominator of the left-hand side. This gives\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{{m}x^2+{n}x+{k}=A(x^2+{b}x+{c})+B*x(x+{a}) + C(x+{a})}.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003cstrong\u003e(Note: To find $A$, $B$, and $C$, we will use a combination of choosing suitable values of $x$ to eliminate terms, and equating coefficients. It can be solved by \u003cem\u003eonly\u003c/em\u003e equating coefficients, but this is a more efficient process.)\u003c/strong\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eTo find $A$, we can eliminate $B$ and $C$ by setting $x=\\\\var{-a}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{{m*a^2-n*a+k}=A{(a^2-b*a+c)}} \\\\implies A=\\\\simplify[fractionNumbers]{{Asol}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eTo find $C$, we can eliminate $B$ by setting $x=0$ and substituting in the result of $A$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{{k}={c}A+{a}C} \\\\implies C=\\\\simplify[all,fractionNumbers]{({k}-{c}A)/{a}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eHence,\u003c/p\u003e\\n\u003cp\u003e\\\\[ C = \\\\simplify[fractionNumbers]{{Csol}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eFinally, by equating coefficients of the $x^2$-terms we can find $B$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ (x^2): \\\\quad \\\\var{m} = \\\\simplify{A+B} \\\\implies B=\\\\var{m}-A. \\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, \\\\[ B=\\\\simplify[fractionNumbers]{{Bsol}}, \\\\]\u003c/p\u003e\\n\u003cp\u003eand\u003c/p\u003e\\n\u003cp\u003e{check}\u003c/p\u003e\\n\u003cp\u003e\u003cstrong\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/directory#P\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/strong\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"pairs[index][1]\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"if(k=1,random(-1,1)*random([1,3,4,5]),if (k=2,random(-1,1)*random([1,2,4,5]),if(k=3,random(-1,1)*random([1,2,3,5]),if(k=5,random(-1,1)*random([1,2,3,4,5,7]),random(-1,1)*random([1,2,3,4,5,7])))))\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Asol\": {\"name\": \"Asol\", \"group\": \"Ungrouped variables\", \"definition\": \"(m*a^2-n*a+k)/(a^2-b*a+c)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Bsol\": {\"name\": \"Bsol\", \"group\": \"Ungrouped variables\", \"definition\": \"(m*c-m*b*a+n*a-k)/(a^2-b*a+c)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Csol\": {\"name\": \"Csol\", \"group\": \"Ungrouped variables\", \"definition\": \"(k*(a-b)-m*a*c+n*c)/(a^2-a*b+c)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"check\": {\"name\": \"check\", \"group\": \"Ungrouped variables\", \"definition\": \"if(Asol=round(Asol) and Bsol=round(Bsol),\u0027{sol1}\u0027,if(simp2=1,\u0027{sol2}\u0027,\u0027{sol3}\u0027))\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol1\": {\"name\": \"sol1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\\\\\simplify{{Asol}/(x+{a})+({Bsol}x+{Csol})/(x^2+{b}x+{c})}.\\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"sol2\": {\"name\": \"sol2\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\\\\\simplify[all,fractionNumbers]{{m*a^2-n*a+k}/({a^2-a*b+c}(x+{a}))+({m*c-m*b*a+n*a-k}x+{k*(a-b)-m*a*c+n*c})/({a^2-a*b+c}(x^2+{b}x+{c}))}.\\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"pairs[index][0]\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"simp1\": {\"name\": \"simp1\", \"group\": \"Ungrouped variables\", \"definition\": \"gcd(k*(a-b)-m*a*c+n*c,m*c-m*b*a+n*a-k)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"simp2\": {\"name\": \"simp2\", \"group\": \"Ungrouped variables\", \"definition\": \"gcd(simp1,a^2-a*b+c)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol3\": {\"name\": \"sol3\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\\\\\simplify[all,fractionNumbers]{{m*a^2-n*a+k}/({a^2-a*b+c}(x+{a}))+({(m*c-m*b*a+n*a-k)/simp2}x+{(k*(a-b)-m*a*c+n*c)/simp2})/({(a^2-a*b+c)/simp2}(x^2+{b}x+{c}))}.\\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"k\": {\"name\": \"k\", \"group\": \"Ungrouped variables\", \"definition\": \"random([1,2,3,5,7])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"m\": {\"name\": \"m\", \"group\": \"Ungrouped variables\", \"definition\": \"1\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"pairs\": {\"name\": \"pairs\", \"group\": \"Ungrouped variables\", \"definition\": \"[[1,random(-1,1)*random([1,3,4,5])],[2,random(-1,1)*random([1,2,4,5])],[3,random(-1,1)*random([1,2,3,5])],[5,random(-1,1)*random([1,2,3,4,5,7])],[7,random(-1,1)*random([1,2,3,4,5,7])]]\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"index\": {\"name\": \"index\", \"group\": \"Ungrouped variables\", \"definition\": \"random(0..4)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"a^2-a*b+c\u003e0 or a^2-a*b+c\u003c0\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"pairs\", \"index\", \"b\", \"c\", \"m\", \"k\", \"n\", \"Asol\", \"Bsol\", \"Csol\", \"check\", \"sol1\", \"sol2\", \"sol3\", \"simp1\", \"simp2\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\\n\u003cp\u003e[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{(m*a^2-n*a+k)}/({a^2-a*b+c}(x+{a}))+({(m*c-m*b*a+n*a-k)/simp2}x+{(k*(a-b)-m*a*c+n*c)/simp2})/({(a^2-a*b+c)/simp2}(x^2+{b}x+{c}))\", \"answerSimplification\": \"all\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"mustmatchpattern\": {\"pattern\": \"`! (((`+-$n`?*x^2+`+-$n`?*x+`+-$n)/((x+`+-$n)(x^2+`+-$n*x+`+-$n))))\", \"partialCredit\": 0, \"message\": \"\", \"nameToCompare\": \"\", \"warningTime\": \"submission\"}, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AF01 Sigma Notation\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}, {\"name\": \"Megan Oliver\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23526/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eBasic calculation from a sum given in Sigma notation.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eCalculate:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\displaystyle{\\\\Sigma_{n=1}^3} \\\\var{b}n.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eThe sigma notation $\\\\displaystyle\u003cspan class=\\\"enlighter-k0\\\"\u003e\\\\sum_\u003c/span\u003e\u003cspan class=\\\"enlighter-g1\\\"\u003e{\u003c/span\u003e\u003cspan class=\\\"enlighter-text\\\"\u003en=\u003c/span\u003e\u003cspan class=\\\"enlighter-n1\\\"\u003e1\u003c/span\u003e\u003cspan class=\\\"enlighter-g1\\\"\u003e}\u003c/span\u003e\u003cspan class=\\\"enlighter-text\\\"\u003e^\u003c/span\u003e\u003cspan class=\\\"enlighter-g1\\\"\u003e{\u003c/span\u003e\u003cspan class=\\\"enlighter-k0\\\"\u003e3\u003c/span\u003e\u003cspan class=\\\"enlighter-g1\\\"\u003e}\\\\var{b}n$ is asking us to find the \u003c/span\u003esum of the first three terms of the sequence $\\\\var{b}n$\u003cspan class=\\\"enlighter-g1\\\"\u003e.\u0026nbsp;\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\begin{split}\\\\Sigma_{n=1}^3 \\\\var{b}n \u0026amp;\\\\, = (\\\\var{b}\\\\times 1) + (\\\\var{b}\\\\times 2) + (\\\\var{b}\\\\times 3) \\\\\\\\ \u0026amp;\\\\, = \\\\var{b1} + \\\\var{b2} + \\\\var{b3} \\\\\\\\ \u0026amp;\\\\, = \\\\var{sum}.\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sigma\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find resources to help you revise sigma notation.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2 .. 9#1)\", \"description\": \"\", \"templateType\": \"randrange\", \"can_override\": false}, \"b1\": {\"name\": \"b1\", \"group\": \"Ungrouped variables\", \"definition\": \"b*1\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b2\": {\"name\": \"b2\", \"group\": \"Ungrouped variables\", \"definition\": \"b*2\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b3\": {\"name\": \"b3\", \"group\": \"Ungrouped variables\", \"definition\": \"b*3\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sum\": {\"name\": \"sum\", \"group\": \"Ungrouped variables\", \"definition\": \"b1+b2+b3\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"b\", \"b1\", \"b2\", \"b3\", \"sum\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{sum}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"GA09 sec/cosec/cot\", \"extensions\": [\"geogebra\"], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eMatch the graphs to the functions. No randomisation. Multiple choice.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\", \"advice\": \"\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eThis is about knowledge of graphs. Generally with trigonometric graphs it is best to start with making sure you know and understand the graphs of the functionts $\\\\sin(x)$, $\\\\cos(x)$ and $\\\\tan(x)$. From there you can use knowledge of where they are zero to work out the position of the asymptotes in the graphs of $\\\\sec(x)$, $\\\\text{cosec}(x)$ and $\\\\cot(x)$. However, you still need really to be able to recall the shape of each graph for some purposes and be confident about where the zeros and turning points are.\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#TrigFun\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources to help you familiarise yourself with these graphs.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"m_n_x\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, 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\"{geogebra_applet(\u0027https://www.geogebra.org/m/xm44vcwe\u0027)}\"]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"GA08 Trigonometric Identities 2\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Oliver Spenceley\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23557/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eRewriting a trigonometric expression of the form $A\\\\cos(\\\\theta)\\\\pm B\\\\sin(\\\\theta)$ to either $R\\\\sin(\\\\theta+\\\\alpha)$ or $R\\\\cos(\\\\theta+\\\\alpha)$ by calculating $R$ and $\\\\alpha$.\u0026nbsp;\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eIf\u003c/p\u003e\\n\u003cp\u003e{question}\u003c/p\u003e\\n\u003cp\u003efind the values for $R$ and $\\\\alpha$, given $R\u0026gt;0$ and $0\u0026lt;\\\\alpha\u0026lt;\\\\frac{\\\\pi}{2}$.\u003c/p\u003e\", \"advice\": \"\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e{answer}\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Trig\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"A\": {\"name\": \"A\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"B\": {\"name\": \"B\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..5 except A)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"R\": {\"name\": \"R\", \"group\": \"Ungrouped variables\", \"definition\": \"sqrt(A^2+B^2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Rround\": {\"name\": \"Rround\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(R,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"alpha\": {\"name\": \"alpha\", \"group\": \"Ungrouped variables\", \"definition\": \"arctan(B/A)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Rsol\": {\"name\": \"Rsol\", \"group\": \"Ungrouped variables\", \"definition\": \"if(R=round(R),\u0027{Rsol1}\u0027,\u0027{Rsol2}\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Rsol1\": {\"name\": \"Rsol1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} R^2\\\\\\\\cos^2(\\\\\\\\alpha) + R^2 \\\\\\\\sin^2(\\\\\\\\alpha) \u0026amp;\\\\\\\\,= \\\\\\\\var{A}^2+\\\\\\\\var{B}^2 \\\\\\\\\\\\\\\\ R^2 (\\\\\\\\cos^2(\\\\\\\\alpha) +\\\\\\\\sin^2(\\\\\\\\alpha)) \u0026amp;\\\\\\\\,= \\\\\\\\var{A^2+B^2} \\\\\\\\\\\\\\\\ R \u0026amp;\\\\\\\\,= \\\\\\\\var{R}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"Rsol2\": {\"name\": \"Rsol2\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split} R^2\\\\\\\\cos^2(\\\\\\\\alpha) + R^2 \\\\\\\\sin^2(\\\\\\\\alpha) \u0026amp;\\\\\\\\,= \\\\\\\\var{A}^2+\\\\\\\\var{B}^2 \\\\\\\\\\\\\\\\ R^2 (\\\\\\\\cos^2(\\\\\\\\alpha) +\\\\\\\\sin^2(\\\\\\\\alpha)) \u0026amp;\\\\\\\\,= \\\\\\\\var{A^2+B^2} \\\\\\\\\\\\\\\\ R \u0026amp;\\\\\\\\,=\u0026nbsp;\\\\\\\\sqrt{\\\\\\\\var{A^2+B^2}}\\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,=\\\\\\\\var{Rround} \\\\\\\\text{ (2 d.p.)}.\u0026nbsp;\\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"alpharound\": {\"name\": \"alpharound\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(alpha,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"question\": {\"name\": \"question\", \"group\": \"Ungrouped variables\", \"definition\": \"if(Q=1,\u0027{q1}\u0027,\u0027{q2}\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Q\": {\"name\": \"Q\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sign\": {\"name\": \"sign\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1,-1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"q1\": {\"name\": \"q1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\simplify[unitFactor]{{A}sin(theta)+{sign*B}cos(theta)} = \\\\\\\\simplify[unitFactor]{R sin (theta+{sign}*alpha)},\\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"q2\": {\"name\": \"q2\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003e\\\\\\\\[ \\\\\\\\simplify[unitFactor]{{A}cos(theta)-{sign*B}sin(theta)} = \\\\\\\\simplify[unitFactor]{R cos (theta+{sign}*alpha)},\\\\\\\\]\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"a1\": {\"name\": \"a1\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo find $R$ and $\\\\\\\\alpha$ we want to first rewrite our equation using the double-angle formula, $\\\\\\\\simplify[unitFactor]{sin(a+{sign}*b)=sin(a)cos(b)+{sign}*sin(b)cos(a)}$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split}\\\\\\\\simplify[unitFactor]{{A}sin(theta)+{sign*B}cos(theta)} \u0026amp;\\\\\\\\,= \\\\\\\\simplify{R sin(theta+{sign}*alpha)} \\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,= \\\\\\\\simplify{R(sin(theta)cos(alpha) + {sign}*sin(alpha)cos(theta))} \\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,= \\\\\\\\simplify{Rsin(theta)cos(alpha) + {sign}*R sin(alpha)cos(theta)}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eBy comparing the coefficients of $\\\\\\\\sin(\\\\\\\\theta)$ and $\\\\\\\\cos(\\\\\\\\theta)$, we find that\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ R\\\\\\\\cos(\\\\\\\\alpha) = \\\\\\\\var{A},\\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad R\\\\\\\\sin(\\\\\\\\alpha) = \\\\\\\\var{B}. \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo calculate $R$, we want to square these results and add them together, allowing us to make use of $\\\\\\\\sin^2(\\\\\\\\alpha)+\\\\\\\\cos^2(\\\\\\\\alpha) = 1$:\u003c/p\u003e\\\\n\u003cp\u003e{Rsol}\u003c/p\u003e\\\\n\u003cp\u003eSimilarly, to find $\\\\\\\\alpha$ we can divide $R\\\\\\\\sin(\\\\\\\\alpha) = \\\\\\\\var{B}$ by $R\\\\\\\\cos(\\\\\\\\alpha) = \\\\\\\\var{A}$, and use the identity $\\\\\\\\tan(\\\\\\\\alpha) = \\\\\\\\frac{\\\\\\\\sin(\\\\\\\\alpha)}{\\\\\\\\cos(\\\\\\\\alpha)}$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\frac{R\\\\\\\\sin(\\\\\\\\alpha)}{R\\\\\\\\cos(\\\\\\\\alpha)} = \\\\\\\\frac{\\\\\\\\var{B}}{\\\\\\\\var{A}} \\\\\\\\implies \\\\\\\\tan(\\\\\\\\alpha) = \\\\\\\\simplify[fractionNumbers]{{B/A}}.\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTherefore, \\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\alpha \u0026amp;\\\\\\\\,= \\\\\\\\tan^{-1}\\\\\\\\left(\\\\\\\\simplify[fractionNumbers]{{B/A}}\\\\\\\\right) \\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,= \\\\\\\\var{alpharound} \\\\\\\\text{ (2 d.p.)}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"a2\": {\"name\": \"a2\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eTo find $R$ and $\\\\\\\\alpha$ we want to first rewrite our equation using the double-angle formula, $\\\\\\\\simplify{cos(a+{sign}*b)=cos(a)cos(b)-{sign}*sin(a)sin(b)}$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\begin{split}\\\\\\\\simplify[unitFactor]{{A}cos(theta)-{sign*B}sin(theta)} \u0026amp;\\\\\\\\,= \\\\\\\\simplify[unitFactor]{R cos (theta + {sign}*alpha)} \\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,= \\\\\\\\simplify{R(cos(theta)cos(alpha) - {sign}*sin(theta)sin(alpha))} \\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,= \\\\\\\\simplify{Rcos(theta)cos(alpha) - {sign}*R sin(theta)sin(alpha)}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eBy comparing the coefficients of $\\\\\\\\cos(\\\\\\\\theta)$ and $\\\\\\\\sin(\\\\\\\\theta)$, we find that\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ R\\\\\\\\cos(\\\\\\\\alpha) = \\\\\\\\var{A},\\\\\\\\quad \\\\\\\\text{and} \\\\\\\\quad R\\\\\\\\sin(\\\\\\\\alpha) = \\\\\\\\var{B}. \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTo calculate $R$, we want to square these results and add them together, allowing us to make use of $\\\\\\\\sin^2(\\\\\\\\alpha)+\\\\\\\\cos^2(\\\\\\\\alpha) = 1$:\u003c/p\u003e\\\\n\u003cp\u003e{Rsol}\u003c/p\u003e\\\\n\u003cp\u003eSimilarly, to find $\\\\\\\\alpha$ we can divide $R\\\\\\\\sin(\\\\\\\\alpha) = \\\\\\\\var{B}$ by $R\\\\\\\\cos(\\\\\\\\alpha) = \\\\\\\\var{A}$, and use the identity $\\\\\\\\tan(\\\\\\\\alpha) = \\\\\\\\frac{\\\\\\\\sin(\\\\\\\\alpha)}{\\\\\\\\cos(\\\\\\\\alpha)}$:\u003c/p\u003e\\\\n\u003cp\u003e\\\\\\\\[ \\\\\\\\frac{R\\\\\\\\sin(\\\\\\\\alpha)}{R\\\\\\\\cos(\\\\\\\\alpha)} = \\\\\\\\frac{\\\\\\\\var{B}}{\\\\\\\\var{A}} \\\\\\\\implies \\\\\\\\tan(\\\\\\\\alpha) = \\\\\\\\simplify[fractionNumbers]{{B/A}}.\\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003eTherefore, \\\\\\\\[ \\\\\\\\begin{split} \\\\\\\\alpha \u0026amp;\\\\\\\\,= \\\\\\\\tan^{-1}\\\\\\\\left(\\\\\\\\simplify[fractionNumbers]{{B/A}}\\\\\\\\right) \\\\\\\\\\\\\\\\ \u0026amp;\\\\\\\\,= \\\\\\\\var{alpharound} \\\\\\\\text{ (2 d.p.)}. \\\\\\\\end{split} \\\\\\\\]\u003c/p\u003e\\\\n\u003cp\u003e\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"answer\": {\"name\": \"answer\", \"group\": \"Ungrouped variables\", \"definition\": \"if(Q=1,\u0027{a1}\u0027,\u0027{a2}\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"Q\", \"A\", \"B\", \"sign\", \"R\", \"Rround\", \"alpha\", \"alpharound\", \"Rsol\", \"Rsol1\", \"Rsol2\", \"question\", \"q1\", \"q2\", \"answer\", \"a1\", \"a2\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$R=$[[0]]\u003c/p\u003e\\n\u003cp\u003e$\\\\alpha=$[[1]]\u003c/p\u003e\\n\u003cp\u003e\u003cem\u003e(Give your answers to 2 decimal places where necessary.)\u003c/em\u003e\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{Rround}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{alpharound}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AF04 Graphs of trig functions (sin, cos, tan)\", \"extensions\": [\"geogebra\"], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Lauren Desoysa\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21504/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eMatch the relevant graph (sin, cos, tan) with its equation.\u0026nbsp;\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\", \"advice\": \"\u003cp\u003eThis is about core knowledge of graphs. You should know the shapes of the fundamental trig graphs, if you don\u0027t familiarize yourself with them from the resources linked below. In this setting the $x$-axis is given with a scale in radians but you might also find some where it is given in degrees. You should also be aware of the difference between those two different units of angles.\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#TrigFun\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources to help you familiarise yourself with these graphs.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"m_n_x\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003eMatch the graph to its function.\u003c/p\u003e\", \"minMarks\": 0, \"maxMarks\": 0, \"minAnswers\": 0, \"maxAnswers\": 0, \"shuffleChoices\": true, \"shuffleAnswers\": true, \"displayType\": \"radiogroup\", \"warningType\": \"none\", \"showCellAnswerState\": true, \"markingMethod\": \"sum ticked cells\", \"choices\": [\"$\\\\sin(x)$\", \"$\\\\cos(x)$\", \"$\\\\tan(x)$\"], \"matrix\": [[\"1\", 0, 0], [0, \"1\", 0], [0, 0, \"1\"]], \"layout\": {\"type\": \"all\", \"expression\": \"\"}, \"answers\": [\"{geogebra_applet(\u0027https://www.geogebra.org/m/ntqvuwqr\u0027)}\", \"{geogebra_applet(\u0027https://www.geogebra.org/m/fsqmnhsc\u0027)}\", \"{geogebra_applet(\u0027https://www.geogebra.org/m/yg6f9eqz\u0027)}\"]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AF05 Function notation\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Megan Oliver\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23526/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eEvaluating a linear function for a given value of $x$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eGiven $f(x)=\\\\simplify{{m}x+{c}}$, find $f(\\\\var{n})$.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf $f(x)=\\\\simplify{{m}x+{c}}$, to find $f(\\\\var{n})$ we need to evaluate $f(x)$ when $x=\\\\var{n}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} f(\\\\var{n}) \u0026amp;\\\\,= \\\\simplify[alwaysTimes]{{m}({n})+{c}} \\\\\\\\ \u0026amp;\\\\,= \\\\simplify[!collectNumbers]{{m*n}+{c}} \\\\\\\\ \u0026amp;\\\\,= \\\\simplify{{m*n+c}}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Functions\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find resources to help you revise function notation.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"m\": {\"name\": \"m\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..9)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-9..9 except [0,m])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-9..9 except [0,1])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"m\", \"c\", \"n\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$f(\\\\var{n})=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{m*n+c}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AF07 Inverse functions\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Megan Oliver\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23526/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFinding the inverse of a function of the form $f(x)=\\\\frac{mx+c}{x+a},\\\\,x\\\\neq-a$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eIf $f(x)=\\\\simplify{({m}x+{c})/(x+{a})},\\\\,x\\\\neq \\\\simplify{{-a}}$, find the inverse function, $f^{-1}(x)$.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo find $f^{-1}x$, it can help to first set $f(x)$ to a different variable, which we will call $y$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ y = f(x) = \\\\simplify{({m}x+{c})/(x+{a})}\\\\]\u003c/p\u003e\\n\u003cp\u003eSince the function $f(x)$ takes us from $x$ to $y$, the inverse function $f^{-1}$ will take us from $y$ to $x$. So to obtain $f^{-1}$, we want to rearrange $y=\\\\simplify{({m}x+{c})/(x+{a})}$ so that it is $x$ as a function of $y$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} y \u0026amp;\\\\,= \\\\simplify{({m}x+{c})/(x+{a})} \\\\\\\\\\\\\\\\ \\\\simplify{(x+{a})y} \u0026amp;\\\\,= \\\\simplify{{m}x+{c}} \\\\\\\\\\\\\\\\ \\\\simplify{x*y+{a}y} \u0026amp;\\\\,= \\\\simplify{{m}x+{c}} \\\\\\\\\\\\\\\\ \\\\simplify{x*y - {m}x} \u0026amp;\\\\,= \\\\simplify{{c}- {a}y}\u0026nbsp; \u0026nbsp;\\\\\\\\\u0026nbsp; \\\\\\\\ \\\\simplify{x(y-{m})} \u0026amp;\\\\,= \\\\simplify{{c}-{a}y} \\\\\\\\\\\\\\\\ x\u0026amp;\\\\,= \\\\simplify{({c}-{a}y)/(y-{m})}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eHence, $f^{-1}(y) =\\\\simplify{({c}-{a}y)/(y-{m})}$, and therefore \\\\[ f^{-1}(x) =\\\\simplify{({c}-{a}x)/(x-{m})}.\\\\]\u003c/p\u003e\\n\u003cp\u003e(\u003cstrong\u003eNote:\u003c/strong\u003e\u0026nbsp;The inverse is valid for all values of $x$\u0026nbsp;\u003cem\u003eexcept\u0026nbsp;\u003c/em\u003e$x=\\\\var{m}$, since this would make the denominator equal to 0.)\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Functions\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find resources to help you revise how to find the inverse of functions.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"m\": {\"name\": \"m\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..9)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-9..9 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-8..8)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"abs(m)-abs(c)\u003e0 or abs(m)-abs(c)\u003c0\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"m\", \"c\", \"a\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$f^{-1}(x)=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"({c}-{a}x)/(x-{m})\", \"answerSimplification\": \"fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AL02 Logs - rules 1\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Megan Oliver\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23526/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eSolving $a\\\\log(x)+\\\\log(b)=\\\\log(c)$ for $x$, where $a$, $b$ and $c$ are positive integers.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eSolve for $x$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\var{a}\\\\log(x)+\\\\log(\\\\var{b})=\\\\log(\\\\var{c}). \\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo solve $\\\\var{a}\\\\log(x)+\\\\log(\\\\var{b})=\\\\log(\\\\var{c})$ for $x$, we want to use the following logarithm rules:\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003e$\\\\log(a^n) = n\\\\log(a)$;\u003c/li\u003e\\n\u003cli\u003e$\\\\log(a)+\\\\log(b) = \\\\log(ab)$.\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003eHence,\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\var{a}\\\\log(x)+\\\\log(\\\\var{b}) \u0026amp;\\\\,=\\\\log(\\\\var{c}) \\\\\\\\ \\\\log(x^\\\\var{a})+\\\\log(\\\\var{b}) \u0026amp;\\\\,= \\\\log(\\\\var{c}) \\\\\\\\ \\\\log(\\\\var{b}x^\\\\var{a}) \u0026amp;\\\\,= \\\\log(\\\\var{c}). \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eIf $\\\\log(a)=\\\\log(b)$ then this implies $a=b$. Therefore,\u003c/p\u003e\\n\u003cp\u003e\\\\[\u0026nbsp; \\\\begin{split} \\\\var{b}x^\\\\var{a} \u0026amp;\\\\,=\\\\var{c} \\\\\\\\ x^\\\\var{a} \u0026amp;\\\\,= \\\\simplify[fractionNumbers]{{c/b}} \\\\\\\\ x \u0026amp;\\\\,= \\\\simplify[fractionNumbers]{({c/b})^(1/{a})} \\\\\\\\ x \u0026amp;\\\\,= \\\\var{sol} \\\\text{ (2 d.p.)}\\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Logs\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find rsources to help you revise how the rules of logarithms to help you solve logarithmic equations.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..9)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol\": {\"name\": \"sol\", \"group\": \"Ungrouped variables\", \"definition\": \"precround((c/b)^(1/a),2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..40 except [b,b^(a+1)])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"c\", \"sol\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$x=$ [[0]] (\u003cem\u003eGive you answer to 2 decimal places where necessary)\u003c/em\u003e\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{sol}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"AL03 Logs - Solving equations using logs\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Megan Oliver\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23526/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eSolving an equation of the form $a^x=b$ using logarithms to find $x$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eSolve for $x$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\var{a}^x = \\\\var{b} \\\\,. \\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo solve $\\\\var{a}^x = \\\\var{b}$ for $x$, since $x$ is the exponent we want to make use of the following logarithm rule:\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003e$\\\\log(a^n) = n \\\\log(a)$.\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eBy taking the logarithm of each side and applying the above rule:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split}\\\\var{a}^x \u0026amp;\\\\,= \\\\var{b} \\\\\\\\ \\\\log_{10}(\\\\var{a}^x) \u0026amp; \\\\,= \\\\log_{10}(\\\\var{b})\\\\\\\\ x \\\\log_{10}(\\\\var{a}) \u0026amp;\\\\,= \\\\log_{10}(\\\\var{b}) \\\\\\\\\\\\\\\\ x\u0026amp;\\\\,=\\\\simplify{log({b})/log({a})} \\\\\\\\\\\\\\\\ x \u0026amp;\\\\,= \\\\var{sol} \\\\text{ (2 d.p.)}.\u0026nbsp; \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Logs\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find resources to help you revise how logarithms.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..9)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..9 except [a,a^2,a^3])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol\": {\"name\": \"sol\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(log(b)/log(a),2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"sol\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$x=$ [[0]] (\u003cem\u003eGive you answer to 2 decimal places where necessary)\u003c/em\u003e\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{sol}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD02 Differentiating polynomials 2\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Lauren Desoysa\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21504/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eDifferentiate a polynomial expression involving coefficients and, negative and fractional indices.\u0026nbsp;\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eFind the derivative of $y=\\\\simplify[unitFactor, fractionNumbers]{{a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3}}$.\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eFrom the Table of Derivatives we see that a function of the form \\\\[ f(x)=kx^n \\\\] has a derivative \\\\[ \\\\frac{df}{dx} = knx^{n-1}. \\\\]\u003c/p\u003e\\n\u003cp\u003eAdditionally, the\u0026nbsp;derivative of the sum or difference of two or more functions is equal to the sum or difference of the derivatives of each function: \\\\[ \\\\frac{d}{dx}(f(x)\\\\pm g(x)) = \\\\frac{df}{dx} \\\\pm \\\\frac{dg}{dx}.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e{advice}\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Diff\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\" title=\\\"Link to differentiation resources\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a_1\": {\"name\": \"a_1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b_1\": {\"name\": \"b_1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a_2\": {\"name\": \"a_2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b_2\": {\"name\": \"b_2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..-1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a_3\": {\"name\": \"a_3\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"advice\": {\"name\": \"advice\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a_2\u003e0 and a_3\u003e0,\u0027{solutiona}\u0027,{advice2})\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solutiona\": {\"name\": \"solutiona\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eSo, for the function \\\\\\\\[y=\\\\\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\\\\\] the derivative\u0026nbsp; is \\\\\\\\begin{split}\\\\\\\\frac{dy}{dx} \u0026amp;= (\\\\\\\\var[fractionNumbers]{a_1}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_1})x^{\\\\\\\\var[fractionNumbers]{b_1}-1} +(\\\\\\\\var[fractionNumbers]{a_2}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_2})x^{\\\\\\\\var[fractionNumbers]{b_2}-1} +(\\\\\\\\var[fractionNumbers]{a_3}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_3})x^{\\\\\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\u0026amp;= \\\\\\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\\\\\end{split}\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"solutionb\": {\"name\": \"solutionb\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eSo, for the function \\\\\\\\[y=\\\\\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\\\\\] the derivative\u0026nbsp; is \\\\\\\\begin{split}\\\\\\\\frac{dy}{dx} \u0026amp;= (\\\\\\\\var[fractionNumbers]{a_1}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_1})x^{\\\\\\\\var[fractionNumbers]{b_1}-1} -(\\\\\\\\var[fractionNumbers]{abs(a_2)}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_2})x^{\\\\\\\\var[fractionNumbers]{b_2}-1} +(\\\\\\\\var[fractionNumbers]{a_3}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_3})x^{\\\\\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\u0026amp;= \\\\\\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\\\\\end{split}\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"solutionc\": {\"name\": \"solutionc\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eSo, for the function \\\\\\\\[y=\\\\\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\\\\\] the derivative\u0026nbsp; is \\\\\\\\begin{split}\\\\\\\\frac{dy}{dx} \u0026amp;= (\\\\\\\\var[fractionNumbers]{a_1}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_1})x^{\\\\\\\\var[fractionNumbers]{b_1}-1} +(\\\\\\\\var[fractionNumbers]{a_2}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_2})x^{\\\\\\\\var[fractionNumbers]{b_2}-1} -(\\\\\\\\var[fractionNumbers]{abs(a_3)}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_3})x^{\\\\\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\u0026amp;= \\\\\\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\\\\\end{split}\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"solutiond\": {\"name\": \"solutiond\", \"group\": \"Ungrouped variables\", \"definition\": \"\\\"\u003cp\u003eSo, for the function \\\\\\\\[y=\\\\\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\\\\\] the derivative\u0026nbsp; is \\\\\\\\begin{split}\\\\\\\\frac{dy}{dx} \u0026amp;= (\\\\\\\\var[fractionNumbers]{a_1}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_1})x^{\\\\\\\\var[fractionNumbers]{b_1}-1} -(\\\\\\\\var[fractionNumbers]{abs(a_2)}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_2})x^{\\\\\\\\var[fractionNumbers]{b_2}-1} -(\\\\\\\\var[fractionNumbers]{abs(a_3)}\\\\\\\\times\\\\\\\\var[fractionNumbers]{b_3})x^{\\\\\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\u0026amp;= \\\\\\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\\\\\end{split}\u003c/p\u003e\\\"\", \"description\": \"\", \"templateType\": \"long string\", \"can_override\": false}, \"advice2\": {\"name\": \"advice2\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a_2\u003c0 and a_3\u003e0,\u0027{solutionb}\u0027,{advice3})\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"advice3\": {\"name\": \"advice3\", \"group\": \"Ungrouped variables\", \"definition\": \"if(a_2\u003e0 and a_3\u003c0,\u0027{solutionc}\u0027,\u0027{solutiond}\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b_3\": {\"name\": \"b_3\", \"group\": \"Ungrouped variables\", \"definition\": \"b/c\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"gcd(b,c)=1\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a_1\", \"a_2\", \"a_3\", \"b_1\", \"b_2\", \"b_3\", \"b\", \"c\", \"advice\", \"advice2\", \"advice3\", \"solutiona\", \"solutionb\", \"solutionc\", \"solutiond\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\frac{dy}{dx}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": false, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{a_1*b_1}x^{{b_1}-1}+{a_2*b_2}x^{{b_2}-1}+{a_3*b_3}x^{{b_3}-1}\", \"answerSimplification\": \"fractionNumbers, basic, unitFactor\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CI02 Definite integration\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the definite integral $\\\\int_{n_1}^{n_2}a_1x^{b_1}+a_2x^{b_2}+a_3x^{b_3} dx$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eEvaluate \\\\[ \\\\int_{\\\\var{n_1}}^{\\\\var{n_2}}\\\\simplify[unitFactor, unitPower, fractionNumbers]{{a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3}} \\\\,dx.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIntegrating a function of the form\u0026nbsp;\u0026nbsp;\\\\[ f(x)=x^n \\\\] has the integral \\\\[ \\\\int_a^b x^n dx\u0026nbsp; =\u0026nbsp; \\\\left[\\\\frac{x^{n+1}}{n+1}\\\\right]_a^b,\\\\]\u003c/p\u003e\\n\u003cp\u003eand \\\\[\\\\int_a^b kf(x) dx = k \\\\int_a^b f(x) dx.\\\\]\u003c/p\u003e\\n\u003cp\u003eAdditionally, the\u0026nbsp;integral of the sum or difference of two or more functions is equal to the sum or difference of the\u0026nbsp;integrals of each function: \\\\[ \\\\int(f(x)\\\\pm g(x))\\\\, dx = \\\\int f(x)\\\\, dx\u0026nbsp; \\\\pm \\\\int g(x) \\\\, dx.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eTherefore,\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split}\\\\simplify[unitFactor,unitPower]{defint({a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3},x,{n_1},{n_2})} \u0026amp;\\\\,= \\\\simplify{{a_1}defint(x^{b_1},x,{n_1},{n_2})+{a_2}defint(x^{b_2},x,{n_1},{n_2})+{a_3}defint(x^{b_3},x,{n_1},{n_2})} \\\\\\\\ \u0026amp;\\\\,= \\\\left[\\\\simplify[all,fractionNumbers]{{a_1}x^{b_1+1}/{b_1+1}+{a_2}x^{b_2+1}/{b_2+1}+{a_3}x^{b_3+1}/{b_3+1}}\\\\right]_\\\\var{n_1}^\\\\var{n_2} \\\\\\\\ \u0026amp;\\\\,= \\\\left[\\\\simplify[all,fractionNumbers,!collectNumbers]{{a_1*n_2^(b_1+1)}/{b_1+1}+{a_2*n_2^(b_2+1)}/{b_2+1}+{a_3*n_2^(b_3+1)}/{b_3+1}}\\\\right] -\\\\left[\\\\simplify[all,fractionNumbers,!collectNumbers]{{a_1*n_1^(b_1+1)}/{b_1+1}+{a_2*n_1^(b_2+1)}/{b_2+1}+{a_3*n_1^(b_3+1)}/{b_3+1}}\\\\right]\u0026nbsp;\\\\\\\\ \u0026amp;\\\\,= \\\\simplify[!collectNumbers]{{eval2a}-{eval1a}} \\\\\\\\ \u0026amp;\\\\,=\\\\var{sol1} \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Int\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources on areas under curves which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a_1\": {\"name\": \"a_1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b_1\": {\"name\": \"b_1\", \"group\": \"Ungrouped variables\", \"definition\": \"3\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a_2\": {\"name\": \"a_2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b_2\": {\"name\": \"b_2\", \"group\": \"Ungrouped variables\", \"definition\": \"2\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b_3\": {\"name\": \"b_3\", \"group\": \"Ungrouped variables\", \"definition\": \"1\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"a_3\": {\"name\": \"a_3\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n_1\": {\"name\": \"n_1\", \"group\": \"Ungrouped variables\", \"definition\": \"random(0..2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n_2\": {\"name\": \"n_2\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"eval2\": {\"name\": \"eval2\", \"group\": \"Ungrouped variables\", \"definition\": \"a_1*n_2^(b_1+1)/(b_1+1)+a_2*n_2^(b_2+1)/(b_2+1)+a_3*n_2^(b_3+1)/(b_3+1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"eval1\": {\"name\": \"eval1\", \"group\": \"Ungrouped variables\", \"definition\": \"a_1*n_1^(b_1+1)/(b_1+1)+a_2*n_1^(b_2+1)/(b_2+1)+a_3*n_1^(b_3+1)/(b_3+1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"eval2a\": {\"name\": \"eval2a\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(eval2,3)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"eval1a\": {\"name\": \"eval1a\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(eval1,3)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol\": {\"name\": \"sol\", \"group\": \"Ungrouped variables\", \"definition\": \"eval2-eval1\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"sol1\": {\"name\": \"sol1\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(sol,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"b_1\u003eb_2 and b_2\u003eb_3 and n_2\u003en_1\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a_1\", \"a_2\", \"a_3\", \"b_1\", \"b_2\", \"b_3\", \"n_1\", \"n_2\", \"eval2\", \"eval1\", \"eval2a\", \"eval1a\", \"sol\", \"sol1\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e[[0]]\u0026nbsp;\u003cem\u003e(Give answers to 2 decimal places where necessary)\u003c/em\u003e\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{sol1}\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD08 Finding turning points\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFinding the stationary points of a cubic equation and determining their nature.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eGiven the function \\\\[ \\\\simplify{y={a}x^3+{b}x^2+{c}x+{d}} ,\\\\] find its stationary points and determine their nature.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eTo find the stationary points of the function, we must solve $\\\\tfrac{dy}{dx}=0$ for $x$. For the function $\\\\simplify{y={a}x^3+{b}x^2+{c}x+{d}}$,\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\frac{dy}{dx} = \\\\simplify{{3a}x^2+{2b}x+{c}}. \\\\]\u003c/p\u003e\\n\u003cp\u003eSetting $\\\\frac{dy}{dx}=0$ and solving for $x$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{{3a}x^2+{2b}x+{c}} =0\u0026nbsp; \\\\\\\\ \\\\\\\\ \\\\implies x=\\\\var{solx1dp} \\\\var{x1} \\\\text{ and } x=\\\\var{solx2dp} \\\\var{x2}. \\\\]\u003c/p\u003e\\n\u003cp\u003eHence, the function has two stationary points at $x=\\\\var{solx1dp}$ and $x=\\\\var{solx2dp}$. To find the corresponding $y$-coordinates, we want to plug these values back into the initial equation.\u003c/p\u003e\\n\u003cp\u003eWhen $x=\\\\var{solx1dp}$,\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} y \u0026amp;\\\\,= \\\\simplify[unitFactor,!cancelTerms]{{a}*({solx1dp})^3+{b}*({solx1dp})^2+{c}*({solx1dp})+{d}} \\\\\\\\ \u0026amp;\\\\,=\\\\simplify{{soly1dp}} \\\\var{y1}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eWhen $x=\\\\var{solx2dp}$,\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} y \u0026amp;\\\\,= \\\\simplify[unitFactor,!cancelTerms]{{a}*({solx2dp})^3+{b}*({solx2dp})^2+{c}*({solx2dp})+{d}} \\\\\\\\ \u0026amp;\\\\,=\\\\simplify{{soly2dp}} \\\\var{y2}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, the stationary points of $y=\\\\simplify{{a}x^3+{b}x^2+{c}x+{d}}$ are\u003c/p\u003e\\n\u003cp\u003e\\\\[ (\\\\simplify{{solx1dp}},\\\\, \\\\simplify{{soly1dp}}) \\\\, , \\\\,(\\\\simplify{{solx2dp}},\\\\, \\\\simplify{{soly2dp}}). \\\\]\u003c/p\u003e\\n\u003cp\u003eFinally, we need to determine the nature of the stationary points. To do this we want to calculate the second derivative of the initial function and then evaluate it for each $x$-value of the stationary points.\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003eRecall:\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003eIf $\\\\tfrac{d^2y}{dx^2}\u0026lt;0$ the stationary point is a \u003cstrong\u003emaximum\u003c/strong\u003e;\u003c/li\u003e\\n\u003cli\u003eIf $\\\\tfrac{d^2y}{dx^2}\u0026gt;0$ the stationary point is a\u0026nbsp;\u003cstrong\u003eminimum\u003c/strong\u003e;\u003c/li\u003e\\n\u003cli\u003eIf $\\\\tfrac{d^2y}{dx^2}=0$ the stationary point is a \u003cstrong\u003epoint of inflection\u003c/strong\u003e.\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003eTo calculate\u0026nbsp;$\\\\tfrac{d^2y}{dx^2}$, we want to differentiate\u0026nbsp;$\\\\tfrac{dy}{dx}$ again with respect to $x$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \u0026amp;\\\\frac{dy}{dx} = \\\\simplify{{3a}x^2+{2b}x+{c}}, \\\\\\\\ \\\\\\\\\\\\implies \u0026amp;\\\\frac{d^2y}{dx^2} = \\\\simplify{{6a}x+{2b}}. \\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003eFor $(\\\\simplify{{solx1dp}},\\\\, \\\\simplify{{soly1dp}})$, $\\\\frac{d^2y}{dx^2} = \\\\simplify{{check}}$, so it is a minimum.\u003c/p\u003e\\n\u003cp\u003eFor $(\\\\simplify{{solx2dp}},\\\\, \\\\simplify{{soly2dp}})$, $\\\\frac{d^2y}{dx^2} = \\\\simplify{{check2}}$, so it is a maximum.\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Diff\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..3)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-7..7)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"d\": {\"name\": \"d\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except b)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solx1\": {\"name\": \"solx1\", \"group\": \"Ungrouped variables\", \"definition\": \"(-2b+sqrt((2b)^2-12*a*c))/(6a)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solx2\": {\"name\": \"solx2\", \"group\": \"Ungrouped variables\", \"definition\": \"(-2b-sqrt((2b)^2-12*a*c))/(6a)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"check\": {\"name\": \"check\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(6a*solx1+2b,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"check2\": {\"name\": \"check2\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(6a*solx2+2b,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"soly1\": {\"name\": \"soly1\", \"group\": \"Ungrouped variables\", \"definition\": \"a*(solx1)^3+b*(solx1)^2+c*solx1+d\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"soly2\": {\"name\": \"soly2\", \"group\": \"Ungrouped variables\", \"definition\": \"a*(solx2)^3+b*(solx2)^2+c*solx2+d\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solx1dp\": {\"name\": \"solx1dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(solx1,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"solx2dp\": {\"name\": \"solx2dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(solx2,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"soly1dp\": {\"name\": \"soly1dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(soly1,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"soly2dp\": {\"name\": \"soly2dp\", \"group\": \"Ungrouped variables\", \"definition\": \"precround(soly2,2)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"x1\": {\"name\": \"x1\", \"group\": \"Ungrouped variables\", \"definition\": \"if(round(solx1)=solx1,\u0027\u0027,\u0027(2 d.p.)\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"x2\": {\"name\": \"x2\", \"group\": \"Ungrouped variables\", \"definition\": \"if(round(solx2)=solx2,\u0027\u0027,\u0027(2 d.p.)\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"y1\": {\"name\": \"y1\", \"group\": \"Ungrouped variables\", \"definition\": \"if(round(soly1)=soly1,\u0027\u0027,\u0027(2 d.p.)\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"y2\": {\"name\": \"y2\", \"group\": \"Ungrouped variables\", \"definition\": \"if(round(soly2)=soly2,\u0027\u0027,\u0027(2 d.p.)\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"b^2\u003e3*a*c\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a\", \"b\", \"c\", \"d\", \"solx1\", \"soly1\", \"solx2\", \"soly2\", \"check\", \"check2\", \"solx1dp\", \"solx2dp\", \"soly1dp\", \"soly2dp\", \"x1\", \"x2\", \"y1\", \"y2\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003eThere is a minimum point at ([[0]], [[1]]) and a maximum point at ([[2]] , [[3]]).\u003c/p\u003e\\n\u003cp\u003e\u003cem\u003e(Give the coordinates of the stationary points to 2 decimal places where necessary.)\u003c/em\u003e\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{solx1dp}\", \"showPreview\": false, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{soly1dp}\", \"showPreview\": false, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{solx2dp}\", \"showPreview\": false, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}, {\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{soly2dp}\", \"showPreview\": false, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": []}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD05 Differentiating Trig 1\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFind the derivative of a function of the form $y=a \\\\cos(bx+c)$ using a table of derivatives.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eFind the derivative of $y=\\\\simplify[unitFactor]{{a}cos({b}x+{c})}.$\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eFrom the Table of Derivatives we see that a function of the form \\\\[ f(x)=a \\\\cos(kx+c) \\\\] has a derivative \\\\[-ak \\\\sin (kx+c).\\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, the function\u0026nbsp; \\\\[y=\\\\simplify[unitFactor]{{a}*cos({b}x+{c})}\\\\] has a derivative\\\\[ \\\\begin{split} \\\\frac{dy}{dx} \u0026amp;=-(\\\\var{a}\\\\times \\\\var{b})\\\\sin(\\\\simplify[unitFactor]{{b}x+{c}})\\\\\\\\ \u0026amp;= \\\\simplify[unitFactor]{{-a*b}sin({b}x+{c})}.\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-15..15)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a\", \"b\", \"c\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\frac{dy}{dx}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"-{a*b}sin({b}x+{c})\", \"answerSimplification\": \"fractionNumbers, basic, unitFactor\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD06 Differentiating Trig 2\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFind the derivative of a function of the form $y=a \\\\sin(bx+c)$ using a table of derivatives.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eUsing the Table of Derivatives, calculate the derivative of $y=\\\\simplify[unitFactor]{{a}sin({b}x+{c})}.$\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eFrom the Table of Derivatives we see that a function of the form \\\\[ f(x)=a \\\\sin(kx+c) \\\\] has a derivative \\\\[ak \\\\cos (kx+c).\\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, the function\u0026nbsp; \\\\[y=\\\\simplify[unitFactor]{{a}*sin({b}x+{c})}\\\\] has a derivative\\\\[ \\\\begin{split} \\\\frac{dy}{dx} \u0026amp;=(\\\\var{a}\\\\times \\\\var{b})\\\\cos(\\\\simplify[unitFactor]{{b}x+{c}})\\\\\\\\ \u0026amp;= \\\\simplify[unitFactor]{{a*b}cos({b}x+{c})}.\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-15..15)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a\", \"b\", \"c\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\frac{dy}{dx}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{a*b}cos({b}x+{c})\", \"answerSimplification\": \"fractionNumbers, basic, unitFactor\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD07 Differentiating with Trig 3\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eFind the derivative of a function of the form $y=a \\\\tan(bx+c)$ using a table of derivatives.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eUsing the Table of Derivatives, calculate the derivative of $y=\\\\simplify[unitFactor]{{a}tan({b}x+{c})}.$\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eFrom the Table of Derivatives we see that a function of the form \\\\[ f(x)=a \\\\tan(kx+c) \\\\] has a derivative \\\\[ak \\\\sec^2(kx+c).\\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, the function\u0026nbsp; \\\\[y=\\\\simplify[unitFactor]{{a}*tan({b}x+{c})}\\\\] has a derivative\\\\[ \\\\begin{split} \\\\frac{dy}{dx} \u0026amp;=(\\\\var{a}\\\\times \\\\var{b})\\\\sec^2(\\\\simplify[unitFactor]{{b}x+{c}})\\\\\\\\ \u0026amp;= \\\\simplify[unitFactor]{{a*b}}\\\\sec^2(\\\\simplify[unitFactor]{{b}x+{c}}).\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-15..15)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a\", \"b\", \"c\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\frac{dy}{dx}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"alternatives\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"answer\": \"{a*b}sec^2({b}x+{c})\", \"answerSimplification\": \"fractionNumbers, basic, unitFactor\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"sec\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}], \"answer\": \"{a*b}sec({b}x+{c})^2\", \"answerSimplification\": \"fractionNumbers, basic, unitFactor\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD04 Differentiating with Exponentials\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the derivative of an exponential function of the form $ae^{bx}$, using a table of derivatives.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eCalculate the derivative of $y=\\\\simplify[all]{{a}*e^({b}x)}.$\u003c/p\u003e\", \"advice\": \"\u003cp\u003eFrom the Table of Derivatives we see that a function of the form \\\\[ f(x)=a e^{kx} \\\\] has a derivative \\\\[ak e^{kx}.\\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, the function\u0026nbsp; \\\\[y=\\\\simplify[unitFactor]{{a}*e^({b}x)}\\\\] has a derivative\\\\[ \\\\begin{split} \\\\frac{dy}{dx} \u0026amp;=(\\\\var{a}\\\\times \\\\var{b})e^{\\\\simplify[unitFactor]{{b}x}}\\\\\\\\ \u0026amp;= \\\\simplify[unitFactor]{{a*b}e^({b}x)}.\\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-20..20 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1 .. 20#1)\", \"description\": \"\", \"templateType\": \"randrange\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a\", \"b\", \"c\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\frac{dy}{dx}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{a*b}e^({b}x)\", \"answerSimplification\": \"fractionNumbers, basic, unitFactor\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD09 Chain Rule\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the derivative of a function of the form $\\\\sin(ax^m+bx^n)$ using the chain rule.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eCalculate the derivative of $y=\\\\simplify[all]{sin({a}*x^{n}+{b}*x^{m})}$.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf we have a function of the form $y=f(g(x))$, sometimes described as a function of a function, to calculate its derivative we need to use the chain rule:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\frac{dy}{dx} = \\\\frac{du}{dx} \\\\times \\\\frac{dy}{du}.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eThis can be split up into steps:\u003c/p\u003e\\n\u003cul\u003e\\n\u003cli\u003eLet $u=g(x)$;\u003c/li\u003e\\n\u003cli\u003eRewrite $y$ in terms of $u$, such that $y=f(u)$;\u003c/li\u003e\\n\u003cli\u003eCalculate $\\\\frac{du}{dx}$ and $\\\\frac{dy}{du}$;\u003c/li\u003e\\n\u003cli\u003eWrite $\\\\frac{dy}{dx}$ as a product of\u0026nbsp;$\\\\frac{du}{dx}$ and $\\\\frac{dy}{du}$;\u003c/li\u003e\\n\u003cli\u003eMake sure $\\\\frac{dy}{dx}$ is\u0026nbsp;\u003cem\u003eonly\u003c/em\u003e in terms of $x$. Ensure any $u$ terms have been replaced using the initial substitution.\u003c/li\u003e\\n\u003c/ul\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eFollowing this process, we must first identify $g(x)$. Since the function is of the form $y=f(g(x))$, we are looking for the \u0027inner\u0027 function.\u003c/p\u003e\\n\u003cp\u003eSo, for $y=\\\\simplify[all,fractionNumbers]{sin({a}*x^{n}+{b}*x^{m})}$, \\\\[g(x)=\\\\simplify[all, fractionNumbers, unitFactor]{{a}*x^{n}+{b}*x^{m}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eIf we now set $u=g(x)$, we can rewrite $y$ in terms of $u$ such that $y=f(u)$:\u003c/p\u003e\\n\u003cp\u003e\\\\[y=\\\\simplify[all, fractionNumbers,unitFactor]{sin(u)}.\\\\]\u003c/p\u003e\\n\u003cp\u003eNext, we calculate the two derivatives $\\\\frac{du}{dx}$ and $\\\\frac{dy}{du}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\frac{du}{dx}=\\\\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}}, \\\\quad \\\\frac{dy}{du}=\\\\simplify[all, fractionNumbers, unitFactor]{cos(u)}.\\\\]\u003c/p\u003e\\n\u003cp\u003ePlugging these into the chain rule:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\frac{dy}{dx} \u0026amp;= \\\\frac{du}{dx} \\\\times \\\\frac{dy}{du}, \\\\\\\\\u0026amp;=(\\\\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}}) \\\\times\\\\simplify[all, fractionNumbers, unitFactor]{cos(u)}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eFinally, we need to express $\\\\frac{dy}{dx}$\u0026nbsp;\u003cem\u003eonly\u0026nbsp;\u003c/em\u003ein terms of $x$, so we must replace the $u$ term using the initial substitution $u=\\\\simplify[all, fractionNumbers, unitFactor]{{a}*x^{n}+{b}*x^{m}}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\frac{dy}{dx} =(\\\\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}})\\\\simplify[all, fractionNumbers, unitFactor]{cos({a}*x^{n}+{b}*x^{m})}.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-10..10 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"m\": {\"name\": \"m\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"n\u003em\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"n\", \"m\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\frac{dy}{dx}=$[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"({a*n}*x^{n-1}+{b*m}*x^{m-1})*cos({a}x^{n}+{b}x^{m})\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD10 Product Rule\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the derivative a function of the form $ax^n \\\\sin(bx)$ using the product rule.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eFind the derivative of \\\\[ \\\\simplify{y={a}x^{n} sin({b}x)}. \\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf we have a function of the form $y=u(x)v(x)$, to calculate its derivative we need to use the product rule:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\dfrac{dy}{dx} = u(x) \\\\times \\\\dfrac{dv}{dx} + v(x) \\\\times\\\\dfrac{du}{dx}.\\\\]\u003c/p\u003e\\n\u003cp\u003eThis can be split up into steps:\u003c/p\u003e\\n\u003col\u003e\\n\u003cli\u003eIdentify the functions $u(x)$ and $v(x)$;\u003c/li\u003e\\n\u003cli\u003eCalculate their derivatives $\\\\tfrac{du}{dx}$ and $\\\\tfrac{dv}{dx}$;\u003c/li\u003e\\n\u003cli\u003eSubstitute these into the formula for the product rule to obtain an expression for $\\\\tfrac{dy}{dx}$;\u003c/li\u003e\\n\u003cli\u003eSimplify $\\\\tfrac{dy}{dx}$ where possible.\u003c/li\u003e\\n\u003c/ol\u003e\\n\u003cp\u003eFollowing this process, we must first identify $u(x)$ and $v(x)$.\u003c/p\u003e\\n\u003cp\u003eAs \\\\[ \\\\simplify{y={a}x^{n} sin({b}x)}, \\\\]\u003c/p\u003e\\n\u003cp\u003elet \\\\[ u(x) = \\\\simplify{{a}x^{n}} \\\\quad \\\\text{and} \\\\quad v(x)=\\\\simplify{sin({b}x)}.\\\\]\u003c/p\u003e\\n\u003cp\u003eNext, we need to find the derivatives, $\\\\tfrac{du}{dx}$ and $\\\\tfrac{dv}{dx}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\dfrac{du}{dx} = \\\\simplify{{a*n}x^{n-1}}\\\\quad \\\\text{and} \\\\quad\\\\dfrac{dv}{dx}=\\\\simplify{{b}cos({b}x)}.\\\\]\u003c/p\u003e\\n\u003cp\u003eSubstituting these results into the product rule formula we can obtain an expression for $\\\\tfrac{dy}{dx}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} \u0026amp;\\\\,= \\\\dfrac{du}{dx}\\\\times v(x) + u(x) \\\\times\\\\dfrac{dv}{dx} \\\\\\\\ \u0026amp;\\\\,=\\\\simplify{{a*n}x^{n-1}} \\\\times\\\\simplify{sin({b}x)} +\\\\simplify{{a}x^{n}} \\\\times \\\\simplify{{b}cos({b}x)}.\u0026nbsp; \\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003eSimplifying,\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\dfrac{dy}{dx} = \\\\simplify{{n*a}x^{n-1}sin({b}x) + {a*b}x^{n}cos({b}x)}. \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\"\u003eUse this link to find some resources which will help you revise this topic\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..10)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"n\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\dfrac{dy}{dx}=$[[0]]\u0026nbsp;\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"{n*a}x^{n-1}sin({b}x) + {a*b}x^{n}cos({b}x)\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CD11 Quotient Rule\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the derivative of a function of the form $\\\\frac{ax^n}{bx+c}$ using the quotient rule.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eFind the derivative of \\\\[ \\\\simplify{y={a}x^{n}/({b}x+{c})}. \\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf we have a function of the form $y=\\\\tfrac{u(x)}{v(x)}$, to calculate its derivative we need to use the quotient rule:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\dfrac{dy}{dx} = \\\\dfrac{v(x) \\\\times \\\\frac{du}{dx} - u(x) \\\\times\\\\frac{dv}{dx}}{[v(x)]^2}\\\\,.\\\\]\u003c/p\u003e\\n\u003cp\u003eThis can be split up into steps:\u003c/p\u003e\\n\u003col\u003e\\n\u003cli\u003eIdentify the functions $u(x)$ and $v(x)$;\u003c/li\u003e\\n\u003cli\u003eCalculate their derivatives $\\\\tfrac{du}{dx}$ and $\\\\tfrac{dv}{dx}$;\u003c/li\u003e\\n\u003cli\u003eSubstitute these into the formula for the quotient rule to obtain an expression for $\\\\tfrac{dy}{dx}$;\u003c/li\u003e\\n\u003cli\u003eSimplify $\\\\tfrac{dy}{dx}$ where possible.\u003c/li\u003e\\n\u003c/ol\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eFollowing this process, we must first identify $u(x)$ and $v(x)$.\u003c/p\u003e\\n\u003cp\u003eAs \\\\[ \\\\simplify{y={a}x^{n}/({b}x+{c})}, \\\\]\u003c/p\u003e\\n\u003cp\u003elet \\\\[ u(x) = \\\\simplify{{a}x^{n}} \\\\quad \\\\text{and} \\\\quad v(x)=\\\\simplify{{b}x+{c}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eNext, we need to find the derivatives, $\\\\tfrac{du}{dx}$ and $\\\\tfrac{dv}{dx}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\dfrac{du}{dx} = \\\\simplify{{a*n}x^{n-1}}\\\\quad \\\\text{and} \\\\quad\\\\dfrac{dv}{dx}=\\\\simplify{{b}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eSubstituting these results into the quotient rule formula we can obtain an expression for $\\\\tfrac{dy}{dx}$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} \u0026amp;\\\\,= \\\\dfrac{v(x) \\\\times \\\\frac{du}{dx} - u(x) \\\\times\\\\frac{dv}{dx}}{[v(x)]^2} \\\\\\\\ \\\\\\\\\u0026amp;\\\\,=\\\\dfrac{(\\\\simplify{{b}x+{c}}) \\\\times\\\\simplify{{a*n}x^{n-1}} - \\\\simplify{{a}x^{n}} \\\\times \\\\simplify{{b}}}{\\\\simplify{({b}x+{c})^2}}.\u0026nbsp; \\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003eSimplifying,\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\dfrac{dy}{dx} \u0026amp;\\\\,=\\\\dfrac{(\\\\simplify{{b}x+{c}})\\\\simplify{{a*n}x^{n-1}} - \\\\simplify{{b*a}x^{n}}}{\\\\simplify{({b}x+{c})^2}} \\\\\\\\ \\\\\\\\\u0026amp;\\\\,=\\\\dfrac{\\\\simplify[all,!cancelTerms]{{b*a*n}x^{n}+{c*a*n}x^{n-1} - {b*a}x^{n}}}{\\\\simplify{({b}x+{c})^2}}\\\\\\\\ \\\\\\\\ \u0026amp;\\\\,=\\\\dfrac{\\\\simplify{{b*a*n}x^{n}+{c*a*n}x^{n-1} - {b*a}x^{n}}}{\\\\simplify{({b}x+{c})^2}} \\\\\\\\ \\\\\\\\ \u0026amp;\\\\,=\\\\dfrac{\\\\simplify{{simp}x^{n-1}({(b*a*n-b*a)/simp}x+{c*a*n/simp})}}{\\\\simplify{({b}x+{c})^2}} \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003e\u003c/span\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..6)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"c\": {\"name\": \"c\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-6..6 except [0,b])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"simp\": {\"name\": \"simp\", \"group\": \"Ungrouped variables\", \"definition\": \"gcd(b*a*n-b*a,c*a*n)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"simp\u003e1\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\", \"c\", \"n\", \"simp\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e$\\\\dfrac{dy}{dx}=$[[0]]\u0026nbsp;\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Gap 0\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"answer\": \"({simp}x^{n-1}({(b*a*n-a*b)/simp}x+{c*a*n/simp}))/({b}x+{c})^2\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CI04 Integration - trig identities\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eUsing the various versions of $\\\\cos{2x}$ identity to integrate $\\\\sin^2{x}$ and $\\\\cos^2{x}$.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eIntegrate $f(x)=\\\\var{Func}$.\u003c/p\u003e\", \"advice\": \"\u003cdiv data-jme-visible=\\\"SorC=\u0027sin(x)^2\u0027\\\"\u003e\\n\u003cp\u003eWe can\u0027t integrate $\\\\var{Coeff}\\\\sin^2(x)$ directly so first we have to use the double angle formula $\\\\cos(2x)=1-2\\\\sin^2(x)$. We re-arrange using the double angle formula to give us,\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\var{Coeff}\\\\sin^2(x)=\\\\frac{\\\\var{Coeff}}2-\\\\frac{\\\\var{Coeff}}2\\\\cos(2x).\\\\]\u003c/p\u003e\\n\u003cp\u003eFrom the Table of Integrals we see that a function of the form \\\\[ f(x)= \\\\cos(nx) \\\\] has the integral \\\\[ \\\\int \\\\cos(nx) dx\u0026nbsp; = \\\\frac{1}{n}\\\\sin(nx)+c\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eSo, for the function\u003c/p\u003e\\n\u003cp\u003e\\\\[f(x)=\\\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}cos(2x)},\\\\]\u003c/p\u003e\\n\u003cp\u003ethe integral is\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\int\\\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}cos(2x)} dx \\\\,= \\\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}int(cos(2x),x)} \u0026amp;\\\\,=\\\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}(1/2 sin({2}x))} +c, \\\\\\\\ \u0026amp;\\\\,=\\\\simplify[unitFactor,fractionNumbers]{{-Coeff/4} sin(2x)+c}. \\\\end{split}\u0026nbsp; \\\\]\u003c/p\u003e\\n\u003cp\u003eThe integral of $\\\\frac{\\\\var{Coeff}}2$ is\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\int\\\\frac{\\\\var{Coeff}}2dx=\\\\frac{\\\\var{Coeff}}2x+c,\\\\]\u003c/p\u003e\\n\u003cp\u003eso combining these our final answer is\u003c/p\u003e\\n\u003cp\u003e\\\\[\\\\int\\\\frac{\\\\var{Coeff}}2-\\\\frac{\\\\var{Coeff}}2\\\\cos(2x)dx=\\\\simplify[unitFactor,fractionNumbers]{{Coeff/2}x-{Coeff/4} sin(2x)+c}\\\\]\u003c/p\u003e\\n\u003c/div\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003eWe can\u0027t integrate $\\\\var{Coeff}\\\\cos^2(x)$ directly so first we have to use the double angle formula $\\\\cos(2x)=2\\\\cos^2(x)-1$. We re-arrange using the double angle formula to give us,\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003e\\\\[\\\\var{Coeff}\\\\cos^2(x)=\\\\frac{\\\\var{Coeff}}2+\\\\frac{\\\\var{Coeff}}2\\\\cos(2x).\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003eFrom the Table of Integrals we see that a function of the form \\\\[ f(x)= \\\\cos(nx) \\\\] has the integral \\\\[ \\\\int \\\\cos(nx) dx\u0026nbsp; = \\\\frac{1}{n}\\\\sin(nx)+c\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003eSo, for the function\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003e\\\\[f(x)=\\\\simplify[unitFactor,fractionNumbers]{{Coeff/2}cos(2x)},\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003ethe integral is\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003e\\\\[ \\\\begin{split} \\\\int\\\\simplify[unitFactor,fractionNumbers]{{Coeff/2}cos(2x)} dx \\\\,= \\\\simplify[unitFactor,fractionNumbers]{{Coeff/2}int(cos(2x),x)} \u0026amp;\\\\,=\\\\simplify[unitFactor,fractionNumbers]{{Coeff/2}(1/2 sin({2}x))} +c, \\\\\\\\ \u0026amp;\\\\,=\\\\simplify[unitFactor,fractionNumbers]{{Coeff/4} sin(2x)+c}. \\\\end{split}\u0026nbsp; \\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003eThe integral of $\\\\frac{\\\\var{Coeff}}2$ is\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003e\\\\[\\\\int\\\\frac{\\\\var{Coeff}}2dx=\\\\frac{\\\\var{Coeff}}2x+c,\\\\]\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003eso combining these our final answer is\u003c/span\u003e\u003c/p\u003e\\n\u003cp\u003e\u003cspan data-jme-visible=\\\"SorC=\u0027cos(x)^2\u0027\\\"\u003e\\\\[\\\\int\\\\frac{\\\\var{Coeff}}2+\\\\frac{\\\\var{Coeff}}2\\\\cos(2x)dx=\\\\simplify[unitFactor,fractionNumbers]{{Coeff/2}x+{Coeff/4} sin(2x)+c}\\\\]\u003c/span\u003e\u003cbr/\u003e\u003cbr/\u003e\u003cbr/\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"SorC\": {\"name\": \"SorC\", \"group\": \"Ungrouped variables\", \"definition\": \"random(\u0027sin(x)^2\u0027,\u0027cos(x)^2\u0027)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Coeff\": {\"name\": \"Coeff\", \"group\": \"Ungrouped variables\", \"definition\": \"random(1..9)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"Func\": {\"name\": \"Func\", \"group\": \"Ungrouped variables\", \"definition\": \"simplify((expression(Coeff+\\\"*\\\"+SorC)),[\\\"unitFactor\\\",\\\"trig\\\"])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"OneIfCosMinusOneIfSine\": {\"name\": \"OneIfCosMinusOneIfSine\", \"group\": \"Ungrouped variables\", \"definition\": \"1 if(SorC=\u0027sin(x)^2\u0027,-1,1)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"SorC\", \"Coeff\", \"Func\", \"OneIfCosMinusOneIfSine\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"jme\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"alternatives\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"+k instead of +c\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"answer\": \"1/2{Coeff}*(x+{OneIfCosMinusOneIfSine}/2*sin(2x))+k\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"k\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}, {\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Forgotten constant\", \"marks\": \"0.5\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\u003cp\u003eIt looks like you forgot to include the integration constant. You should always remember the \\\"+C\\\" when doing an indefinite integral.\u003c/p\u003e\", \"useAlternativeFeedback\": false, \"answer\": \"1/2{Coeff}*(x+{OneIfCosMinusOneIfSine}/2*sin(2x))\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"answer\": \"1/2{Coeff}*(x+{OneIfCosMinusOneIfSine}/2*sin(2x))+c\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"c\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CI05 Integration - Substitution\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the integral of a function of the form $\\\\frac{nx^{n-1}}{x^n+a}$ using integration by substitution.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eCalculate \\\\[ \\\\simplify[all]{int(({n}x^{n-1})/(x^{n}+{a}),x)}\\\\]\u003c/p\u003e\\n\u003cp\u003eby using the substitution \\\\[ \\\\simplify[all]{u=x^{n}+{a}}.\\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eSince this integral is of the form \\\\[ \\\\int g\u0027(x)f(g(x))\\\\,dx,\\\\] we can use the method of substitution to calculate the solution.\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003eFirstly, we must make a change of variables from $x$ to $u$, where $u$ is equal to the \u0027inner\u0027 function $g(x)$.\u003c/p\u003e\\n\u003cp\u003eSo, for \\\\[\\\\simplify[fractionNumbers]{int(({n}x^{n-1})/((x^{n}+{a})),x)}\\\\]\u003c/p\u003e\\n\u003cp\u003elet $\\\\color{red}{u=\\\\simplify[fractionNumbers]{x^{n}+{a}}}.$\u003c/p\u003e\\n\u003cp\u003eNow, we need to calculate the differential, $du$, where \\\\[ du = \\\\left(\\\\frac{du}{dx}\\\\right)dx. \\\\]\u003c/p\u003e\\n\u003cp\u003eDifferentiating $u$ with respect to $x$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\frac{du}{dx}= \\\\simplify[fractionNumbers]{{n}x^{n-1}}.\\\\]\u003c/p\u003e\\n\u003cp\u003eTherefore, \\\\[ \\\\color{blue}{du = \\\\simplify[fractionNumbers]{{n}x^{n-1}}\\\\, dx}.\\\\]\u003c/p\u003e\\n\u003cp\u003eWe can now rewrite the original integral in terms of $u$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\int \\\\frac{\\\\color{blue}{\\\\simplify{{n}x^{n-1}}}}{\\\\color{red}{\\\\simplify{x^{n}+{a}}}}\\\\color{blue}{\\\\text{d}x} =\u0026nbsp;\\\\int \\\\frac{1}{\\\\color{red}{u}}\\\\color{blue}{\\\\text{d}u}.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003cstrong\u003e(Note: It is important to see that both the function we are integrating, and the variable we are integrating with respect to, has changed.)\u003c/strong\u003e\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify[fractionNumbers]{int(1/u,u) = ln(abs(u)) + c}.\\\\]\u003c/p\u003e\\n\u003cp\u003eFinally, we must rewrite our solution back in terms of the original variable $x$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify[fractionNumbers]{ln(abs(u)) + c = ln(abs(x^{n}+{a}))\u0026nbsp;+ c}.\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Int\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(-5..5 except 0)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"n\": {\"name\": \"n\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..6)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": \"100\"}, \"ungrouped_variables\": [\"a\", \"n\"], \"variable_groups\": [{\"name\": \"Unnamed group\", \"variables\": []}], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Correct answer\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, 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This can be done in NUMBAS by using \\\"abs(*function*)\\\".\u003c/p\u003e\", \"useAlternativeFeedback\": false, \"answer\": \"ln(x^{n}+{a})+c\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"c\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}, {\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Alternative using \\\"+k\\\"\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"answer\": \"ln(abs(x^{n}+{a})) + k\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"k\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}, {\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Alternative using brackets and \\\"+k\\\"\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\u003cp\u003eTechnically we should use the absolute value symbols for the logs. This can be done in NUMBAS by using \\\"abs(*function*)\\\".\u003c/p\u003e\", \"useAlternativeFeedback\": false, \"answer\": \"ln(x^{n}+{a})+k\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"k\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}, {\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Forgotten constant\", \"marks\": \"0.5\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\u003cp\u003eIt looks like you forgot to include the integration constant. You should always remember the \\\"+C\\\" when doing an indefinite integral.\u003c/p\u003e\", \"useAlternativeFeedback\": false, \"answer\": \"ln(abs(x^{n}+{a}))\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"answer\": \"ln(abs(x^{n}+{a}))+c\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": \"0.01\", \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"c\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"CI06 Integration - Parts\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Ben McGovern\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4872/\"}, {\"name\": \"Andrew Neate\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21832/\"}, {\"name\": \"Will Morgan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21933/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculating the integral of a function of the form $ax^2 \\\\cos(bx)$ using integration by parts.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eCalculate the integral \\\\[ \\\\simplify{int({a}x^2 cos({b}x),x)}\\\\]\u003c/p\u003e\", \"advice\": \"\u003cp\u003eIf we have a function of $x$ which is the product of two functions of $x$, to integrate such a function it is often necessary to use Integration by Parts. The formula for Integration by Parts is:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\int u(x) \\\\frac{dv}{dx} dx = u(x)v(x) - \\\\int v(x) \\\\frac{du}{dx} dx.\\\\]\u003c/p\u003e\\n\u003cp\u003eUsing this method can be broken down into steps:\u003c/p\u003e\\n\u003col\u003e\\n\u003cli\u003eIdentify $u(x)$ and $\\\\tfrac{dv}{dx}$ (The function you pick for each is important, in general you want $u(x)$ to become simpler when differentiating it, and you must be able to integrate $\\\\tfrac{dv}{dx}$ to find $v(x)$);\u003c/li\u003e\\n\u003cli\u003eCalculate $\\\\tfrac{du}{dx}$ and $v(x)$;\u003c/li\u003e\\n\u003cli\u003ePut the functions $u(x)$, $v(x)$, and their derivatives into the Integration by Parts formula;\u003c/li\u003e\\n\u003cli\u003eCalculate the integral $\\\\int v(x) \\\\tfrac{du}{dx} dx$ (This may require you to use Integration by Parts again, this is OK!);\u003c/li\u003e\\n\u003cli\u003eSimplify your answer where possible and don\u0027t forget to add the constant of integration.\u003c/li\u003e\\n\u003c/ol\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003eFor the integral\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{int({a}x^2 cos({b}x),x)},\\\\]\u003c/p\u003e\\n\u003cp\u003ewe must first identify $u(x)$ and $\\\\tfrac{dv}{dx}$. In this case, let \\\\[ u(x)=\\\\simplify{{a}x^2},\\\\quad \\\\frac{dv}{dx}= \\\\simplify{cos({b}x)}. \\\\]\u003c/p\u003e\\n\u003cp\u003eNext, we need to calculate $\\\\tfrac{du}{dx}$ and $v(x)$:\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} u(x) = \\\\var{a}x^2 \\\\quad \u0026amp;\\\\implies \\\\frac{du}{dx} = \\\\simplify{{2a}x}; \\\\\\\\ \\\\frac{dv}{dx} = \\\\cos(\\\\var{b}x) \u0026amp;\\\\implies v(x) = \\\\simplify[fractionNumbers]{1/{b} sin({b}x)}. \\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003ePlugging these 4 terms into the integration by parts formula:\u003c/p\u003e\\n\u003cp\u003e\\\\[\u0026nbsp; \\\\begin{split} \\\\simplify{int({a}x^2 cos({b}x),x)} \u0026amp;\\\\,= \\\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) - int({2a/b}x sin({b}x),x)},\u0026nbsp; \\\\\\\\ \\\\\\\\ \u0026amp;\\\\,= \\\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) -{2a/b}int(x sin({b}x),x)}.\\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003eSince the integral on the right-hand side is still the product of two functions of $x$, we need to use integration by parts again.\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003eSo, for\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\simplify{int(x sin({b}x),x)}, \\\\]\u003c/p\u003e\\n\u003cp\u003e\u0026nbsp;Let $u=x$ and $\\\\tfrac{dv}{dx} = \\\\sin(\\\\var{b}x)$. Therefore, $\\\\tfrac{du}{dx}=1$ and $v(x)=\\\\simplify{-1/{b} cos({b}x)}$.\u003c/p\u003e\\n\u003cp\u003eHence,\u003c/p\u003e\\n\u003cp\u003e\\\\[ \\\\begin{split} \\\\simplify{int(x sin({b}x),x)} \u0026amp;\\\\,= \\\\simplify{-1/{b}x cos({b}x)- int(-1/{b} cos({b}x),x)} \\\\\\\\ \\\\\\\\ \u0026amp;\\\\,= \\\\simplify{-1/{b}x cos({b}x)+1/{b^2}sin({b}x)}. \\\\end{split}\\\\]\u003c/p\u003e\\n\u003cp\u003ePlugging this back into the original calculation:\u003c/p\u003e\\n\u003cp\u003e\\\\[\u0026nbsp; \\\\begin{split} \\\\simplify{int({a}x^2 cos({b}x),x)} \u0026amp;\\\\,= \\\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) -{2a/b}int(x cos({b}x),x)} \\\\\\\\ \\\\\\\\ \u0026amp;\\\\,=\u0026nbsp;\\\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) -{2a/b}[-1/{b}x cos({b}x)+1/{b^2}sin({b}x)]} \\\\\\\\ \\\\\\\\ \u0026amp;\\\\,=\\\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) +{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)} + c.\\\\end{split} \\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Int\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"a\": {\"name\": \"a\", \"group\": \"Ungrouped variables\", \"definition\": \"random(2..7)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"b\": {\"name\": \"b\", \"group\": \"Ungrouped variables\", \"definition\": \"random(3..5)\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"gcd(a,b)=1 and b\u003ea\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"a\", \"b\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"gapfill\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003e[[0]]\u003c/p\u003e\", \"gaps\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Correct Answer\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"alternatives\": [{\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Alt constant +k\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"answer\": \"{a/b}x^2 sin({b}x)+{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)+k\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"k\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}, {\"type\": \"jme\", \"useCustomName\": true, \"customName\": \"Forgotten constant\", \"marks\": \"0.5\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\u003cp\u003eIt looks like you forgot to include the integration constant. You should always remember the \\\"+C\\\" when doing an indefinite integral.\u003c/p\u003e\", \"useAlternativeFeedback\": false, \"answer\": \"{a/b}x^2 sin({b}x)+{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)\", \"answerSimplification\": \"all,!collectLikeFractions,fractionNumbers\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"x\", \"value\": \"\"}]}], \"answer\": \"{a/b}x^2 sin({b}x)+{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)+c\", \"answerSimplification\": \"fractionNumbers, basic\", \"showPreview\": true, \"checkingType\": \"absdiff\", \"checkingAccuracy\": 0.001, \"failureRate\": 1, \"vsetRangePoints\": 5, \"vsetRange\": [0, 1], \"checkVariableNames\": false, \"singleLetterVariables\": false, \"allowUnknownFunctions\": true, \"implicitFunctionComposition\": false, \"caseSensitive\": false, \"valuegenerators\": [{\"name\": \"c\", \"value\": \"\"}, {\"name\": \"x\", \"value\": \"\"}]}], \"sortAnswers\": false}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"SM07 Identify measures of spread/location\", \"extensions\": [\"jsxgraph\"], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Gareth Woods\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/978/\"}, {\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Lauren Desoysa\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21504/\"}, {\"name\": \"Michael Pan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23528/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eIdentifying measures of spread or location (average)\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International\"}, \"statement\": \"\u003cp\u003eMatch each of the following with what they measure.\u003c/p\u003e\", \"advice\": \"\u003cp\u003eThe mean is a measure of location or central tendancy. It is calcuated by summing all of the data values and dividing by the number of values.\u003c/p\u003e\\n\u003cp\u003eThe median is a measure of location or central tendancy. It is the middle value of an ordered data set.\u003c/p\u003e\\n\u003cp\u003eThe inter-quartile range is a measure of spread. The interquartile range is the difference between upper and lower quartiles.The lower quartile, or first quartile (Q1), is the value under which 25% of data points are found when they are arranged in increasing order. The upper quartile, or third quartile (Q3), is the value under which 75% of data points are found when arranged in increasing order. The inter-quartile range therefore gives us an idea of the middle 50% of the ordered data set.\u003c/p\u003e\\n\u003cp\u003eThe standard deviation is a measure of spread. It measures the dispersion of a data set relative to its mean.\u0026nbsp;\u003c/p\u003e\\n\u003cp\u003eThe variance is a measure spread because it is the square of the standard deviation.\u003c/p\u003e\\n\u003cp\u003eA p-value the probability that a particular statistical measure, such as the mean or standard deviation, of an assumed probability distribution will be greater than or equal to (or less than or equal to in some instances) observed results. A p-value is used to determine statistical significance, not measures of spread or location.\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/stats-resources/first-steps#mean,-median-and-mode\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {\"std\": [\"all\", \"fractionNumbers\"]}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"function dragpoint_board() {\\n  var scope = question.scope;\\n\\n  JXG.Options.text.display = \u0027internal\u0027;\\n  \\n  var yo0 = scope.variables.yo0.value;\\n  var yo1 = scope.variables.yo1.value;\\n  var yo2 = scope.variables.yo2.value;\\n  var yo3 = 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board.create(\u0027point\u0027, [1.5,0],{name:\u0027Morgan\u0027,fixed:true,size:0,color:\u0027black\u0027,face:\u0027diamond\u0027, label:{offset:[-20,-8]}});\\n  var pop1 = board.create(\u0027point\u0027,[4.5,0],{name:\u0027Strome\u0027,fixed:true,size:0,color:\u0027black\u0027,face:\u0027diamond\u0027, label:{offset:[-20,-8]}});\\n  var pop2 = board.create(\u0027point\u0027,[7.5,0],{name:\u0027Bentley\u0027,fixed:true,size:0,color:\u0027black\u0027, face:\u0027diamond\u0027, label:{offset:[-15,-8]}});\\n  var pop3 = board.create(\u0027point\u0027,[10.5,0],{name:\u0027Sand\u0027,fixed:true,size:0,color:\u0027black\u0027, face:\u0027diamond\u0027, label:{offset:[-15,-8]}});\\n  var pop4 = board.create(\u0027point\u0027,[13.5,0],{name:\u0027Karchen\u0027,fixed:true,size:0,color:\u0027black\u0027, face:\u0027diamond\u0027, label:{offset:[-15,-8]}});\\n\\n  var leg1 = board.create(\u0027point\u0027,[12,75],{name:\u0027last year\u0027,fixed:true,size:6,color:\u0027#DA2228\u0027, face:\u0027square\u0027, label:{offset:[9,0]}});\\n  var leg2 = board.create(\u0027point\u0027,[12,72],{name:\u0027this year\u0027,fixed:true,size:6,color:\u0027#6F1B75\u0027, face:\u0027square\u0027, label:{offset:[9,0]}});\\n  \\n  \\n//  var chart = board.createElement(\u0027chart\u0027, dataArr, \\n  //                              {chartStyle:\u0027bar\u0027, fillOpacity:1, width:1,\\n    //     colorArray:[\u0027#8E1B77\u0027,\u0027#8E1B77\u0027,\u0027Red\u0027,\u0027Red\u0027,\u0027blue\u0027,\u0027red\u0027,\u0027blue\u0027,\u0027red\u0027,\u0027red\u0027,\u0027blue\u0027, \u0027red\u0027,\u0027blue\u0027,\u0027red\u0027,\u0027red\u0027], shadow:false});\\n \\n//var chart = board.createElement(\u0027chart\u0027, dataArr, \\n  //                              {chartStyle:\u0027bar\u0027, width:1,fillOpacity:1, fillColor:\u0027red\u0027, shadow:false});  \\n  \\n  \\n    var a = board.create(\u0027chart\u0027, [[1,2,3],[yo0,yo5,0]], {chartStyle:\u0027bar\u0027,colors:[\u0027#DA2228\u0027,\u0027#6F1B75\u0027,\u0027#6F1B75\u0027],width:1,fillOpacity:1});\\n    var b = board.create(\u0027chart\u0027, [[4,5,6],[yo1,yo6,0]], {chartStyle:\u0027bar\u0027,width:1,colors:[\u0027#DA2228\u0027,\u0027#6F1B75\u0027,\u0027#6F1B75\u0027],fillOpacity:1});\\n    var c = board.create(\u0027chart\u0027, [[7,8,9],[yo2,yo7,0]], {chartStyle:\u0027bar\u0027,width:1,colors:[\u0027#DA2228\u0027,\u0027#6F1B75\u0027,\u0027#6F1B75\u0027],fillOpacity:1});\\n    var d = board.create(\u0027chart\u0027, [[10,11,12],[yo3,yo8,0]], {chartStyle:\u0027bar\u0027,width:1,colors:[\u0027#DA2228\u0027,\u0027#6F1B75\u0027,\u0027#6F1B75\u0027],fillOpacity:1});\\n    var e = board.create(\u0027chart\u0027, [[13,14],[yo4,yo9]], {chartStyle:\u0027bar\u0027,width:1,colors:[\u0027#DA2228\u0027,\u0027#6F1B75\u0027],fillOpacity:1});\\n  \\n    board.unsuspendUpdate();\\n  \\n  var txt1 = board.create(\u0027text\u0027,[-0.3,30, \u0027Investment \\u00a3(m)\u0027], 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Correlation\", \"extensions\": [\"jsxgraph\"], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Richard Miles\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/882/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Lauren Desoysa\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/21504/\"}, {\"name\": \"Upuli Wickramaarachchi\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23527/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eTests understanding of scatter plots and related concepts.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\u003cp\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eThe scatter plot below shows the relationship between an employee\u0026rsquo;s height in centimetres and how long it takes them to walk to work in minutes.\u003c/span\u003e\u003c/p\u003e\\n\u003ctable\u003e\\n\u003ctbody\u003e\\n\u003ctr\u003e\\n\u003ctd\u003etime (mins)\u003c/td\u003e\\n\u003ctd\u003e{drawgraph()}\u003c/td\u003e\\n\u003c/tr\u003e\\n\u003ctr\u003e\\n\u003ctd\u003e\u003c/td\u003e\\n\u003ctd style=\\\"text-align: center;\\\"\u003eheight (cm)\u003c/td\u003e\\n\u003c/tr\u003e\\n\u003c/tbody\u003e\\n\u003c/table\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\", \"advice\": \"\u003cp\u003eThe graph shows that there is a positive correlation between a person\u0027s height and how long it takes them to walk to work.\u003c/p\u003e\\n\u003cp\u003eA postive correlation is a relationship between two variables where both variables move in the same diection.\u003c/p\u003e\\n\u003cp\u003eThis tells us that as a person\u0027s height increases, the time it takes to walk to work increases.\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/stats-resources/next-steps#Introduction%20to%20correlation\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic\u003c/a\u003e\u003c/p\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"slope\": {\"name\": \"slope\", \"group\": \"Regression variables\", \"definition\": \"(6*sumxy-sumx*sumy)/(6*sumxx-(sumx)^2)\", \"description\": \"\u003cp\u003es\u003c/p\u003e\", \"templateType\": \"anything\", \"can_override\": false}, \"timemax\": {\"name\": \"timemax\", \"group\": \"Calculation variables\", \"definition\": \"max([p1y,p2y,p3y,p4y,p5y,p6y])\", \"description\": \"\", \"templateType\": \"anything\", \"can_override\": false}, \"minx\": {\"name\": \"minx\", \"group\": \"Graph Limits\", 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Numbas.jme.unwrapValue(scope.variables.regy1);\\n      var regy2 = Numbas.jme.unwrapValue(scope.variables.regy2);\\n\\n      var p1x = Numbas.jme.unwrapValue(scope.variables.p1x);\\n      var p1y = Numbas.jme.unwrapValue(scope.variables.p1y);\\n      var p2x = Numbas.jme.unwrapValue(scope.variables.p2x);\\n      var p2y= Numbas.jme.unwrapValue(scope.variables.p2y);\\n      var p3x = Numbas.jme.unwrapValue(scope.variables.p3x);\\n      var p3y= Numbas.jme.unwrapValue(scope.variables.p3y);\\n      var p4x = Numbas.jme.unwrapValue(scope.variables.p4x);\\n      var p4y= Numbas.jme.unwrapValue(scope.variables.p4y);\\n      var p5x = Numbas.jme.unwrapValue(scope.variables.p5x);\\n      var p5y= Numbas.jme.unwrapValue(scope.variables.p5y);\\n      var p6x = Numbas.jme.unwrapValue(scope.variables.p6x);\\n      var p6y= Numbas.jme.unwrapValue(scope.variables.p6y);\\n      \\n      var div = Numbas.extensions.jsxgraph.makeBoard(\u0027400px\u0027,\u0027400px\u0027,\\n      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\"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"1_n_2\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 0, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003eMark the statement that best \u003cstrong\u003edescribes\u003c/strong\u003e what this scatter plot shows.\u003c/p\u003e\", \"minMarks\": 0, \"maxMarks\": 0, \"shuffleChoices\": true, \"displayType\": \"radiogroup\", \"displayColumns\": 0, \"showCellAnswerState\": true, \"choices\": [\"\u003cp\u003eIn general, there is a positive correlation between a person\u0027s height and how long it takes them to walk to work.\u003c/p\u003e\", \"\u003cp\u003eIn general, there is a negative correlation between a person\u0027s height and how long it takes them to walk to work.\u003c/p\u003e\", \"\u003cp\u003eIn general, there is a no correlation between a person\u0027s height and how long it takes them to walk to work.\u003c/p\u003e\"], \"matrix\": [\"1\", 0, 0], \"distractors\": [\"\", \"\", \"\"]}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"SP01 Probability - \\\"sample space\\\"\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Ruth Hand\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/3228/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}], \"tags\": [], \"metadata\": {\"description\": \"\u003cp\u003eCalculate probability of selecting coloured counters from a bag.\u003c/p\u003e\", \"licence\": \"None specified\"}, \"statement\": \"\u003cp\u003eA bag contains:\u003cbr/\u003e\u003cbr/\u003e $\\\\var{srn}$ small, red tokens,\u0026nbsp;\u003cbr/\u003e$\\\\var{sbn}$ small,\u0026nbsp;blue\u0026nbsp;tokens,\u0026nbsp;\u003cbr/\u003e$\\\\var{brn}$ large, red\u0026nbsp;tokens, and\u0026nbsp;\u003cbr/\u003e$\\\\var{bbn}$ large, blue tokens.\u003c/p\u003e\", \"advice\": \"\u003ch3\u003epart a)\u003c/h3\u003e\\n\u003cp\u003eA probability is a fraction. \u0026nbsp;You can give your answer as a fraction, decimal or percentage as these are all equivalent.\u003cbr/\u003e\u003cbr/\u003eThe formula for probability is:\u003c/p\u003e\\n\u003cp\u003e\\\\[ P(A) = \\\\frac{\\\\text{number of possibilities for A}}{\\\\text{number of total possible outcomes}} \\\\]\u003c/p\u003e\\n\u003cp\u003eFor this question the total possible outcomes are $\\\\var{srn}+\\\\var{sbn}+\\\\var{brn}+\\\\var{bbn} = \\\\var{total}$.\u003cbr/\u003e\u003cbr/\u003eTherefore\u003c/p\u003e\\n\u003cp\u003e\\\\[ P(\\\\text{A large red token}) = \\\\frac{\\\\var{brn}}{\\\\var{total}} = \\\\var[fractionnumbers]{brn/total}\\\\]\u003c/p\u003e\\n\u003ch3\u003epart b)\u003c/h3\u003e\\n\u003cp\u003eFor this question we need to know the total number of small tokens, i.e.\u0026nbsp;$\\\\var{srn}+\\\\var{sbn} = \\\\var{srn+sbn}$.\u003cbr/\u003e\u003cbr/\u003eTherefore\u003c/p\u003e\\n\u003cp\u003e\\\\[ P(\\\\text{A small token}) = \\\\frac{\\\\var{srn+sbn}}{\\\\var{total}} = \\\\var[fractionnumbers]{(srn+sbn)/total}\\\\]\u003c/p\u003e\\n\u003cp\u003e\u003c/p\u003e\\n\u003cp\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/stats-resources/probability#equally-likely-outcomes\\\"\u003e\u003cspan style=\\\"font-weight: 400;\\\"\u003eUse this link to find some resources which will help you revise this 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random.\u003cbr/\u003e\u003cbr/\u003eWhat is the probability that it is a small token?\u003cbr/\u003e\u003cbr/\u003e\u003cem\u003eGive your answer as a fraction, or a decimal correct to 2dp.\u003c/em\u003e\u003c/p\u003e\", \"alternatives\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": \"1\", \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"alternativeFeedbackMessage\": \"\", \"useAlternativeFeedback\": false, \"minValue\": \"ans2\", \"maxValue\": \"ans2\", \"correctAnswerFraction\": false, \"allowFractions\": false, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"minValue\": \"{{srn}+{sbn}}/{total}\", \"maxValue\": \"{{srn}+{sbn}}/{total}\", \"correctAnswerFraction\": true, \"allowFractions\": true, \"mustBeReduced\": false, \"mustBeReducedPC\": 0, \"displayAnswer\": \"\", \"showFractionHint\": true, \"notationStyles\": [\"plain\", \"en\", \"si-en\"], \"correctAnswerStyle\": \"plain\"}], \"partsMode\": \"all\", \"maxMarks\": 0, \"objectives\": [], \"penalties\": [], \"objectiveVisibility\": \"always\", \"penaltyVisibility\": \"always\"}, {\"name\": \"SP02 Intuitive Probability\", \"extensions\": [], \"custom_part_types\": [], \"resources\": [], \"navigation\": {\"allowregen\": true, \"showfrontpage\": false, \"preventleave\": false, \"typeendtoleave\": false}, \"contributors\": [{\"name\": \"Christian Lawson-Perfect\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/7/\"}, {\"name\": \"Mash Sheffield\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/4679/\"}, {\"name\": \"Upuli Wickramaarachchi\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23527/\"}, {\"name\": \"Michael Pan\", \"profile_url\": \"https://numbas.mathcentre.ac.uk/accounts/profile/23528/\"}], \"tags\": [\"taxonomy\"], \"metadata\": {\"description\": \"\u003cp\u003ePredicting the probability of an unbiased coin landing on heads based on the results of previous throws.\u003c/p\u003e\", \"licence\": \"Creative Commons Attribution 4.0 International\"}, \"statement\": \"\", \"advice\": \"\u003cp\u003eWhen we flip an unbiased\u0026nbsp;coin there are two possible events that we could measure: the coin lands on heads or\u0026nbsp;the coin lands on tails.\u003c/p\u003e\\n\u003cp\u003eEach toss of the coin is independent; if we flip a coin once and it lands\u0026nbsp;on heads then the next time we flip the coin it is still equally likely to land on either heads or tails.\u003c/p\u003e\\n\u003cp\u003eIt doesn\u0027t matter what the coin landed on previously as this\u0026nbsp;outcome does not affect the\u0026nbsp;outcome of the next flip of the coin.\u003c/p\u003e\\n\u003cp\u003eEven when we flip\u0026nbsp;an unbiased coin\u0026nbsp;$\\\\var{no_flips}$ times and it lands on\u0026nbsp;heads each time; the next time we flip the coin, it is still equally likely to land on either heads or tails.\u003c/p\u003e\\n\u003cp\u003eSo the probability that the coin lands on heads the next time that the coin is flipped is still $\\\\displaystyle\\\\frac{1}{2}$.\u003c/p\u003e\\n\u003ch4 style=\\\"text-align: center;\\\"\u003e\u003ca href=\\\"https://www.sheffield.ac.uk/mash/stats-resources/probability#equally-likely-outcomes\\\" target=\\\"_blank\\\" rel=\\\"noopener\\\"\u003eUse this link to find some resources which will help you revise this topic.\u003c/a\u003e\u003c/h4\u003e\", \"rulesets\": {}, \"builtin_constants\": {\"e\": true, \"pi,\\u03c0\": true, \"i\": true, \"j\": false}, \"constants\": [], \"variables\": {\"no_flips\": {\"name\": \"no_flips\", \"group\": \"Ungrouped variables\", \"definition\": \"random(6..9)\", \"description\": \"\u003cp\u003eNumber of flips of the coin\u003c/p\u003e\", \"templateType\": \"anything\", \"can_override\": false}}, \"variablesTest\": {\"condition\": \"\", \"maxRuns\": 100}, \"ungrouped_variables\": [\"no_flips\"], \"variable_groups\": [], \"functions\": {}, \"preamble\": {\"js\": \"\", \"css\": \"\"}, \"parts\": [{\"type\": \"numberentry\", \"useCustomName\": false, \"customName\": \"\", \"marks\": 1, \"scripts\": {}, \"customMarkingAlgorithm\": \"\", \"extendBaseMarkingAlgorithm\": true, \"unitTests\": [], \"showCorrectAnswer\": true, \"showFeedbackIcon\": true, \"variableReplacements\": [], \"variableReplacementStrategy\": \"originalfirst\", \"nextParts\": [], \"suggestGoingBack\": false, \"adaptiveMarkingPenalty\": 0, \"exploreObjective\": null, \"prompt\": \"\u003cp\u003eAn unbiased coin is flipped $\\\\var{no_flips}$ times. 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included or not.\u003c/p\u003e\", \"help_url\": \"\", \"input_widget\": \"string\", \"input_options\": {\"correctAnswer\": \"join(\\n  if(settings[\\\"correctAnswerFractions\\\"],\\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\\\"correctAnswer\\\"])\\n  ,\\n    settings[\\\"correctAnswer\\\"]\\n  ),\\n  settings[\\\"separator\\\"] + \\\" \\\"\\n)\", \"hint\": {\"static\": false, \"value\": \"if(settings[\\\"show_input_hint\\\"],\\n  \\\"Enter a list of numbers separated by \u003ccode\u003e{settings[\u0027separator\u0027]}\u003c/code\u003e.\\\",\\n  \\\"\\\"\\n)\"}, \"allowEmpty\": {\"static\": true, \"value\": true}}, \"can_be_gap\": true, \"can_be_step\": true, \"marking_script\": \"bits:\\nlet(b,filter(x\u003c\u003e\\\"\\\",x,split(studentAnswer,settings[\\\"separator\\\"])),\\n    if(isSet,list(set(b)),b)\\n)\\n\\nexpected_numbers:\\nlet(l,settings[\\\"correctAnswer\\\"] as \\\"list\\\",\\n    if(isSet,list(set(l)),l)\\n)\\n\\nvalid_numbers:\\nif(all(map(not 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The order does not matter.\", \"input_type\": \"code\", \"default_value\": \"\", \"evaluate\": true}, {\"name\": \"allowFractions\", \"label\": \"Allow the student to enter fractions?\", \"help_url\": \"\", \"hint\": \"\", \"input_type\": \"checkbox\", \"default_value\": false}, {\"name\": \"correctAnswerFractions\", \"label\": \"Display the correct answers as fractions?\", \"help_url\": \"\", \"hint\": \"\", \"input_type\": \"checkbox\", \"default_value\": false}, {\"name\": \"isSet\", \"label\": \"Is the answer a set?\", \"help_url\": \"\", \"hint\": \"If ticked, the number of times an element occurs doesn\u0027t matter, only whether it\u0027s included at all.\", \"input_type\": \"checkbox\", \"default_value\": false}, {\"name\": \"show_input_hint\", \"label\": \"Show the input hint?\", \"help_url\": \"\", \"hint\": \"\", \"input_type\": \"checkbox\", \"default_value\": true}, {\"name\": \"separator\", \"label\": \"Separator\", \"help_url\": \"\", \"hint\": \"The substring that should separate items in the student\u0027s list\", \"input_type\": \"string\", \"default_value\": \",\", \"subvars\": false}], \"public_availability\": \"always\", \"published\": true, \"extensions\": []}], \"resources\": []}",
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    <!-- ko if: numQuestions() > 1 -->
    <label class="sr-only" for="exam-progress" data-localise="exam.progress"></label>
    <progress id="exam-progress" data-bind="attr: {max: numQuestions(), value: questionsAttempted(), title: questionsAttemptedDisplay}, text: questionsAttemptedDisplay"></progress>
    <!-- /ko -->

    <menu id="questionList" class="nav navmenu-nav" role="tablist" aria-orientation="vertical" data-bind="tablist: true">
        <li class="introduction" data-bind="visible: exam.hasIntro, css: {active: $root.exam().infoPage()=='introduction'}">
            <a href="#" role="tab" data-bind="click: () => showInfoPage('introduction'), attr: {'aria-current': $root.exam().infoPage()=='introduction' ? 'step' : false, tabindex: $root.exam().infoPage()=='introduction' ? 0 : -1, 'data-name': R('control.show introduction')}" data-localise="control.show introduction"></a>
       </li>
        <!-- ko foreach: question_groups -->
            <!-- ko if: $root.exam().showQuestionGroupNames -->
            <li class="question-group-header" data-bind="visible: show_name">
                <h2 data-bind="text: name, typeset: name"></h2>
                <span class="score" data-bind="html: feedback.message"></span>
            </li>
            <!-- /ko -->
            <!-- ko foreach: questions -->
            <li class="questionSelector" data-bind="visible: visible, css: {active: $root.exam().currentQuestion()==$data}">
                <a href="#" role="tab" data-bind="click: Numbas.controls.makeQuestionJumper(question.number, question.exam), attr: {'aria-current': isCurrentQuestion() ? 'step' : false, tabindex: isCurrentQuestion() ? 0 : -1, 'data-number': question.number+1, 'data-name': displayName}">
                    <div class="question-info">
                        <span class="feedback-icon" data-bind="css: scoreFeedback.iconClass, attr: scoreFeedback.iconAttr" aria-hidden="true"></span>
                        <span class="name" data-bind="text: displayName"></span>
                        <span class="sr-only" data-bind="text: scoreFeedback.iconAttr().title"></span>
                        <span class="score" data-bind="html: scoreFeedback.message"></span>
                    </div>
                    <div class="messages">
                        <div class="answered-state" data-bind="">
                            <span data-bind="text: scoreFeedback.answeredString"></span>
                        </div>
                        <div data-bind="visible: isDirty" class="dirty-message">
                            <span class="warning-icon icon-exclamation-sign"></span>
                            <span data-localise="question.selector.unsubmitted changes"></span>
                        </div>
                    </div>
                </a>
            </li>
            <!-- /ko -->
        <!-- /ko -->
        <!-- total exam score display -->
        <li class="examScore" data-bind="visible: showActualMark">
            <span data-localise="control.total"></span> 
            <span id="examScore" data-bind="html: examScoreDisplay"></span>
        </li>
    </menu>

    <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>

    <menu class="nav navmenu-nav menuButtons">
        <li><button class="btn center-block thin-button display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button></li>
        <li data-bind="visible: allowPause && mode()=='normal'"><button class="btn center-block thin-button" id="pauseBtn" data-localise="control.pause" data-bind="click: pause"></button></li>
        <li data-bind="visible: mode()=='normal'"><button class="btn danger center-block" id="endBtn" data-localise="control.end exam" data-bind="click: endExam"></button></li>
        <li data-bind="visible: mode()=='review'"><button class="btn center-block" data-localise="control.back to results" data-bind="click: () => showInfoPage('result')"></button></li>
    </menu>

</nav>

<!-- top menu shown on narrow screens -->
<nav id="topNav" class="navbar">
    <!-- ko if: numQuestions() > 1 -->
    <div class="prev"><button type="button" class="btn" data-bind="click: reverse, attr: {disabled: !canReverse()}"><span class="sr-only" data-localise="control.previous"></span>&larr;</button></div>
    <div class="next"><button type="button" class="btn" data-bind="click: advance, attr: {disabled: !canAdvance()}"><span class="sr-only" data-localise="control.next"></span>&rarr;</span></button></div>
    <!-- /ko -->
    <div class="end"><button class="btn danger" data-localise="control.end exam" data-bind="click: endExam, visible: mode()=='normal' && !canAdvance()"></button></div>
    <div class="titleBits" data-bind="with: currentQuestion">
        <span class="review-header" data-bind="visible: $root.exam().mode()=='review'" data-localise="exam.review header"></span>
        <span id="questionNameDisplay" data-bind="text: displayName"></span>
    </div>

    <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>

    <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
        <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <rect x="0" y="0" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="4" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="8" width="9" height="1" fill="currentColor"/>
</svg>
    </button>
</nav>
<!-- /ko -->

<!-- ko if: exam.settings.navigateMode=='menu' -->
    <nav id="topNav" class="navbar">
    <button type="button" class="btn primary" data-bind="visible: mode()=='normal', click: showMenu">&larr; <span data-localise="control.back to menu"></span></button>
    <button type="button" class="btn primary" data-bind="visible: mode()=='review', click: () => showInfoPage('result')">&larr; <span data-localise="control.back to results"></span></button>
    <div class="titleBits" data-bind="with: currentQuestion">
        <span class="review-header" data-bind="visible: $root.exam().mode()=='review'" data-localise="exam.review header"></span>
        <span id="questionNameDisplay" data-bind="latex: displayName"></span>
    </div>
    <button class="btn thin-button display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button>
</nav>
<!-- /ko -->

<!-- ko if: exam.settings.navigateMode=='diagnostic' -->
    <!-- side menu, slides out on narrow screens -->
<nav id="sideNav">

    <header>
        <div class="navmenu-brand">
            <img class="logo center-block" src="resources/numbas-logo.svg" alt="Numbas"/>
        </div>
        <h1 class="exam-name navmenu-brand" data-bind="text: exam.settings.name, typeset: exam.settings.name"></h1>
        <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
            <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <line x1="1" y1="1" x2="8" y2="8" stroke="currentColor"/>
    <line x1="1" y1="8" x2="8" y2="1" stroke="currentColor"/>
</svg>
        </button>
    </header>

    <dl id="learning-objective-list" class="nav navemnu-nav">
        <!-- ko foreach: diagnostic_progress -->
        <dt class="name" data-bind="latex: name"></dt>
        <dd data-bind="text: Math.floor(100*progress)+'%'"></dd>
        <!-- /ko -->
    </dl>


    <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>

    <menu class="nav navmenu-nav menuButtons">
        <li><button class="btn center-block thin-button display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button></li>
        <li data-bind="visible: allowPause && mode()=='normal'"><button class="btn center-block thin-button" id="pauseBtn" data-localise="control.pause" data-bind="click: pause"></button></li>
        <li data-bind="visible: mode()=='normal'"><button class="btn danger center-block" id="endBtn" data-localise="control.end exam" data-bind="click: endExam"></button></li>
        <li data-bind="visible: mode()=='review'"><button class="btn center-block" data-localise="control.back to results" data-bind="click: () => showInfoPage('result')"></button></li>
    </menu>

</nav>

<!-- top menu shown on narrow screens -->
<nav id="topNav" class="navbar">
    <div class="nav">
        <div class="titleBits" data-bind="with: currentQuestion">
            <span class="review-header" data-bind="visible: $root.exam().mode()=='review'" data-localise="exam.review header"></span>
            <span id="questionNameDisplay" data-bind="text: displayName"></span>
        </div>

        <div class="stopWatch" data-bind="visible: showTime">
    <span class="" data-localise="timing.time remaining"></span>
    <time class="time" data-bind="duration: timeRemaining"></time>
</div>
    </div>

    <button type="button" class="navbar-toggle btn" aria-controls="navMenu">
        <svg viewBox="0 0 9 9">
    <title data-localise="control.toggle navigation menu"></title>
    <rect x="0" y="0" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="4" width="9" height="1" fill="currentColor"/>
    <rect x="0" y="8" width="9" height="1" fill="currentColor"/>
</svg>
    </button>
</nav>
<!-- /ko -->

            <!-- main content area - info pages or questions go in here -->
            <main class="mainDisplay" role="main" data-bind="attr: {'aria-labelledby': main_labelledby}">
                <div id="infoDisplay" class="main-content" data-bind="visible: infoPage">
    <div id="frontpage" data-bind="visible: infoPage()=='frontpage', if: infoPage()=='frontpage'">
        <h1 class="break-words" id="infopage-frontpage-header" data-bind="html: exam.settings.name"></h1>
<table class="summary">
    <tbody>
    <!-- ko if: exam.settings.navigateMode != 'diagnostic' -->
    <tr>
        <th data-localise="exam.number of questions"></th>
        <td class="rtd" data-bind="html: numAssessedQuestions"></td>
    </tr>
    <tr data-bind="visible: exam.settings.showTotalMark">
        <th data-localise="exam.marks available"></th>
        <td class="rtd" data-bind="niceNumber: marks"></td>
    </tr>
    <!-- /ko -->
    <tr data-bind="visible: exam.settings.percentPass>0">
        <th data-localise="exam.pass percentage"></th>
        <td class="rtd" data-bind="text: percentPass"></td>
    </tr>
    <tr data-bind="visible: duration()>0">
        <th data-localise="exam.time allowed"></th>
        <td class="rtd"><time data-bind="duration: duration"></time></td>
    </tr>
    <tr data-bind="visible: exam.settings.showStudentName && exam.student_name && !needsStudentName">
        <th data-localise="exam.student name"></th>
        <td><span data-bind="text: exam.student_name"></span><span data-bind="visible: exam.student_id"> (<span data-bind="text: exam.student_id"></span>)</span></td>
    </tr>
    </tbody>
</table>

<div id="intro" data-bind="visible: exam.hasIntro, latex: exam.introMessage" localise-data-jme-context-description="exam.introduction"></div>

<div class="alert danger" data-bind="visible: showAttemptDownloadSecurityWarning">
    <p data-localise="exam.attempt download security warning"></p>
</div>
<div id="standalone-warning" class="alert info" data-bind="visible: !Numbas.storage.lmsConnected && !allowAttemptDownload" data-localise="frontpage.scorm.lms not connected"></div>

<form data-bind="submit: beginExam" class="info-buttons">
    <button class="btn display-options force-original-color" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button>
    <div class="student-name" data-bind="visible: needsStudentName">
        <label for="begin-exam-student-name" data-localise="exam.enter your name"></label>
        <input id="begin-exam-student-name" name="begin-exam-student-name" type="text" data-bind="textInput: student_name, autosize: student_name"/>
    </div>
    <div class="password" data-bind="visible: needsPassword">
        <label for="begin-exam-password" data-localise="exam.enter password"></label>
        <input id="begin-exam-password" autocomplete="off" name="begin-exam-password" type="text" data-bind="textInput: passwordHandler.value, autosize: passwordHandler.value, css: {'has-errors': passwordHandler.value().length && !passwordHandler.valid(), 'success': passwordHandler.valid()}"/>
        <span class="password-feedback feedback-icon" data-bind="css: passwordHandler.feedback().iconClass, attr: {title: passwordHandler.feedback().title}"></span>
    </div>
    <button class="btn" id="startBtn" type="submit" data-bind="disable: !canBegin(), css: passwordHandler.feedback().buttonClass" data-localise="frontpage.start"></button>
</form>
    </div>
    <div id="menu" data-bind="visible: infoPage()=='menu', if: infoPage()=='menu'">
        <h1 id="infopage-menu-header" class="break-words" data-bind="html: exam.settings.name"></h1>

<div class="intro" class="alert" data-bind="visible: exam.hasIntro, latex: exam.introMessage" localise-data-jme-context-description="exam.introduction"></div>

<nav id="question-chooser">
    <table class="table questionSelector">
        <colgroup>
            <col class="question">
            <col class="score">
            <col class="icon">
        </colgroup>
        <tbody>
        <!-- ko foreach: question_groups -->
            <!-- ko if: $parent.showQuestionGroupNames -->
            <tr>
                <th data-bind="visible: $parent.question_groups.length>1">
                    <span data-bind="latex: name"></span>
                </th>
                <th class="messages">
                    <span class="score" data-bind="html: feedback.plainMessage"></span>
                </th>
                <th class="icon">
                    <span class="feedback-icon" data-bind="css: feedback.iconClass, attr: feedback.iconAttr" aria-hidden="true"></span>
                </th>
            </tr>
            <!-- /ko -->
            <!-- ko foreach: questions -->
            <tr class="" role="button" data-bind="click: Numbas.controls.makeQuestionJumper(question.number, question.exam), attr: {'aria-labelledby': question.path+'-menu-name'}">
                <td class="info">
                    <button class="btn link" type="button" class="name" data-bind="latex: displayName, attr: {id: question.path+'-menu-name'}"></button>
                </td>
                <td class="messages">
                    <span class="sr-only" data-bind="text: scoreFeedback.iconAttr().title"></span>
                    <span class="score" data-bind="html: scoreFeedback.plainMessage"></span>
                    <div class="answered-state" data-bind="">
                        <span data-bind="text: scoreFeedback.attemptedString"></span>
                    </div>
                    <div data-bind="visible: isDirty" class="dirty-message">
                        <span class="warning-icon icon-exclamation-sign"></span>
                        <span data-localise="question.selector.unsubmitted changes"></span>
                    </div>
                </td>
                <td class="icon">
                    <span class="feedback-icon" data-bind="css: scoreFeedback.iconClass, attr: scoreFeedback.iconAttr" aria-hidden="true"></span>
                </td>
            </tr>
            <!-- /ko -->
        <!-- /ko -->
        </tbody>
    </table>
</nav>

<menu class="nav navmenu-nav menuButtons">
    <li><button class="btn thin-button" type="button" data-bind="click: $root.showStyleModal" data-localise="control.style options"></button></li>
    <li><button class="btn danger thin-button" id="endBtn" data-localise="control.end exam" data-bind="click: endExam"></button></li>
</menu>
    </div>
    <div id="introduction" data-bind="visible: infoPage()=='introduction', if: infoPage()=='introduction'">
        <h1 id="infopage-introduction-header" data-localise="exam.introduction"></h1>
<div data-bind="latex: exam.introMessage" localise-data-jme-context-description="exam.introduction"></div>
    </div>
    <div id="suspend" data-bind="visible: infoPage()=='paused' || infoPage()=='resumed', if: infoPage()=='paused' || infoPage()=='resumed'">
        <h1 id="infopage-suspend-header">
    <span data-bind="visible: infoPage()=='paused'" data-localise="suspend.paused header" ></span>
    <span data-bind="visible: infoPage()=='resumed'" data-localise="suspend.resumed header" ></span>
</h1>
<p data-localise="suspend.exam suspended"></p>
<p data-localise="suspend.you can resume"></p>
<button class="btn primary" id="resumeBtn" data-bind="click: resumeExam" data-localise="suspend.resume"></button>
    </div>
    <div id="results" data-bind="visible: infoPage()=='result', if: infoPage()=='result'">
        <header>
    <h1 class="exam-name text-center break-words" data-bind="latex: exam.settings.name"></h1>
</header>

<!-- ko if: exam.settings.navigateMode=='diagnostic' -->
    <table id="diagnostic-breakdown">
    <colgroup>
        <col>
        <col>
    </colgroup>
    <thead>
        <th data-localise="result.learning objective"></th>
        <!-- ko if: showActualMark -->
        <th data-localise="result.question score"></th>
        <!-- /ko -->
    </thead>
    <tbody>
        <!-- ko foreach: diagnostic_progress -->
            <tr data-bind="css: {total: $index()==$parent.diagnostic_progress().length-1}">
                <td class="name" data-bind="latex: name"></td>
                <!-- ko if: $parent.showActualMark -->
                <td class="score" data-bind="text: Math.floor(100*credit)+'%'"></td>
                <!-- /ko -->
            </tr>
        <!-- /ko -->
    </tbody>
</table>
<!-- /ko -->

<!-- ko if: exam.settings.navigateMode=='sequence' || exam.settings.navigateMode=='menu' -->
<p data-localise="result.click a question to review"></p>
<table id="question-breakdown">
    <colgroup>
        <col>
        <col span="3" class="score">
        <col>
    </colgroup>
    <thead>
        <tr>
        <!-- ko if: exam.settings.navigateMode=='sequence' -->
            <th data-localise="result.question number"></th>
        <!-- /ko -->
        <!-- ko if: exam.settings.navigateMode=='menu' -->
            <th data-localise="question"></th>
        <!-- /ko -->
        <th data-bind="text: resultsScoreColumnHeader"></th>
        </tr>
    </thead>
    <tbody>
        <!-- ko foreach: question_groups -->
            <!-- ko if: $root.exam().showQuestionGroupNames -->
            <tr class="question-group-header" data-bind="visible: show_name">
                <th data-bind="text: name, typeset: name"></th>
                <th class="score-cell" data-bind="html: feedback.plainMessage"></th>
            </tr>
            <!-- /ko -->
            <!-- ko foreach: questions -->
            <tr class="question" data-bind="click: review, attr: {title: R('result.question review title')}">
                <td class="name"><a href="#" data-bind="latex: displayName"></a></td>
                <td class="score-cell" data-bind="html: scoreFeedback.plainMessage"></td>
            </tr>
            <!-- /ko -->
        <!-- /ko -->
        <!-- ko if: scoreFeedback.showActualMark() || scoreFeedback.showTotalMark() -->
        <tr class="total">
            <th scope="row" data-localise="Total"></th>
            <td class="score-cell">
                <span class="score" data-bind="html: scoreFeedback.plainMessage"></span>
                <!-- ko if: scoreFeedback.showActualMark() && scoreFeedback.showTotalMark() -->(<span data-bind="text: percentScore"></span>%)<!-- /ko -->
            </td>
        </tr>
        <!-- /ko -->
    </tbody>
</table>

<!-- /ko -->

<div id="end-message" class="alert" data-bind="visible: exam.has_end_message, latex: exam.end_message" localise-data-jme-context-description="exam.end_message"></div>

<!-- ko if: scoreFeedback.showActualMark -->
<div id="feedback" class="alert results-feedback" data-bind="visible: feedbackMessage()!==null, latex: feedbackMessage" localise-data-jme-context-description="exam.feedback"></div>
<!-- /ko -->

<div id="performance-summary">
    <h2 data-localise="result.performance summary"></h2>

    <table class="summary">
        <tr>
            <th data-localise="exam.exam name"></th>
            <td class="break-words" data-bind="html: exam.settings.name"></td>
        </tr>
        <tr>
            <th data-localise="exam.random seed"></th>
            <td class="break-words" data-bind="html: exam.seed"></td>
        </tr>
        <tr data-bind="visible: exam.settings.showStudentName && exam.student_name">
            <th data-localise="exam.student name"></th>
            <td class="break-words"><span data-bind="text: exam.student_name"></span><span data-bind="visible: exam.student_id"> (<span data-bind="text: exam.student_id"></span>)</span></td>
        </tr>
        <tr>
            <th data-localise="result.exam start"></th>
            <td class="break-words"><time data-bind="datetime: startTime"></time></td>
        </tr>
        <tr>
            <th data-localise="result.exam stop"></th>
            <td class="break-words"><time data-bind="datetime: endTime"></time></td>
        </tr>
        <tr>
            <th data-localise="result.time spent"></th>
            <td class="break-words"><time data-bind="duration: timeSpent"></time></td>
        </tr>
    </table>
</div>



<div class="info-buttons">
    <button class="btn info" data-bind="visible: allowPrinting, click: function(){ window.print(); }, text: printLabel"></button>
    <button class="btn info" data-bind="visible: allowAttemptDownload, click: download_attempt_data" data-localise="result.download exam object"></button>
</div>

<p id="finished-text" class="text-center" data-localise="end.exam has finished"></p>
    </div>

    
    <footer class="copyright-footer">
    Created using <a target="_blank" href="https://www.numbas.org.uk">Numbas</a> v9.1, developed by <a target="_blank" href="https://www.newcastle.ac.uk">Newcastle University</a>.
</footer>
    
</div>

                <div id="questionContainer" class="print-visible" data-bind="visible: currentQuestion, css: {review: mode()=='review'}">
    <div class="contentArea ">
        <div id="diagnostic-feedback" class="diagnostic-feedback" data-bind="visible: diagnostic_feedback, latex: diagnostic_feedback"></div>

        <div id="questionDisplay" data-bind="foreach: questions">
            <div class="question-wrapper print-visible" data-bind="promise: html_promise, visible: isCurrentQuestion">
            </div>
        </div>

        <footer class="copyright-footer">
    Created using <a target="_blank" href="https://www.numbas.org.uk">Numbas</a> v9.1, developed by <a target="_blank" href="https://www.newcastle.ac.uk">Newcastle University</a>.
</footer>
    </div>

</div>
            </main>
        </div>
    </exam-container>
</template>
        <div id="no-stylesheet">
            <p data-localise="page.no stylesheet loaded"></p>
        </div>

        <div id="with-stylesheet">
<div id="die" hidden class="force-default-style">
    <h1 data-localise="die.numbas failed">Numbas failed</h1>
    <p data-localise="die.sorry"></p>
    <div class="error" tabindex="1">
        <div class="message"></div>
        <div class="stack"></div>
    </div>
</div>
<div id="loading">
    <img class="logo center-block" src="resources/numbas-logo.svg" alt="Numbas"/>
    <progress min="0" max="100" value="0"></progress>
    <span class="loading-text" data-localise="page.loading">Loading...</span>
</div>

            
            <numbas-exam storage="scorm">
                <script type="application/numbas-exam">// Numbas version: finer_feedback_settings
{"name": "Skills Audit for Maths and Stats - Physics programmes (PHY129)", "metadata": {"description": "<p>Draft Skills Audit for Physics Students.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], [], []], "questions": [{"name": "NU02 Convert Units (m/s and km/h)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Unit conversion between two compound units.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A Cheetah runs at a speed of {speedkm} kilometres per hour (km/h). What is the Cheetah's speed in metres per second (m/s)?</p>\n<p>Give your ansswer to 2 decimal places where appropriate.</p>", "advice": "<p>There are a number of ways to work out the conversion. Here are a couple of suggestions.</p>\n<p><strong>METHOD 1</strong></p>\n<p>Since there are $1000$m in $1$km we first multiply by $1000$ to get the speed in metres per hour:</p>\n<p>\\begin{equation} 1000*\\var{speedkm} = \\var{step1}\\end{equation}</p>\n<p>Then we divide by $3600$ since that is the number of seconds in an hour to get the speed in metres per second:</p>\n<p>\\begin{equation} \\frac{\\var{step1}}{3600} = \\var{speedms} \\end{equation}</p>\n<p>Finally we round off to 2 decimal places as required, $\\var{roundanswer}$m/s.</p>\n<p><strong>METHOD 2</strong></p>\n<p>We can actually do all of the above in one step of working by using a single conversion factor. Since there are $1000$m in a km and $3600$ seconds in an hour, we can calaculate the conversion factor:</p>\n<p>\\begin{equation} \\frac{3600}{1000} = 3.6 \\end{equation}</p>\n<p>and then simply divide by that conversion factor:</p>\n<p>\\begin{equation} \\frac{\\var{speedkm}}{3.6} = \\var{speedms} \\end{equation}</p>\n<p>finally rounding off as before, $\\var{roundanswer}$m/s.</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Conversions\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"speedkm": {"name": "speedkm", "group": "Ungrouped variables", "definition": "random(70 .. 120#10)", "description": "", "templateType": "randrange", "can_override": false}, "speedms": {"name": "speedms", "group": "Ungrouped variables", "definition": "1000*speedkm/3600", "description": "", "templateType": "anything", "can_override": false}, "step1": {"name": "step1", "group": "Ungrouped variables", "definition": "1000*speedkm", "description": "", "templateType": "anything", "can_override": false}, "roundanswer": {"name": "roundanswer", "group": "Ungrouped variables", "definition": "dpformat(speedms,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["speedkm", "speedms", "step1", "roundanswer"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]m/s</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "answer", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "speedms", "maxValue": "speedms", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "speedms", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "NF03 Rounding SF (decimal)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Oliver Spenceley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23557/"}], "tags": ["rounding"], "metadata": {"description": "<p>Round numbers to&nbsp;a given number of&nbsp;significant figures.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>The&nbsp;first&nbsp;thing to do&nbsp;when we are rounding numbers is to identify the last digit we are keeping.</p>\n<p>When you're asked to <strong>round</strong> your answer to a number of significant figures, you need to decide whether to keep the last&nbsp;digit same (rounding down) or increase it by 1 (rounding up). If the following digit is less than 5 we round down and we round up&nbsp;when the next digit is 5 or more.</p>\n<p>To write it down in steps:</p>\n<ol>\n<ol>\n<ol>\n<ol>\n<li style=\"text-align: left;\">Identify the last digit we need to keep.</li>\n<li style=\"text-align: left;\">Look at the following digit.</li>\n<li style=\"text-align: left;\">If it's 5 or more, increase the previous digit by one.</li>\n<li class=\"c-list__ordered-item\" style=\"text-align: left;\">If it's 4 or less, keep the previous digit the same.</li>\n<li class=\"c-list__ordered-item\" style=\"text-align: left;\">Fill any spaces to the right of the digit with zeros if needed.</li>\n</ol>\n</ol>\n</ol>\n</ol>\n<p class=\"c-list__ordered-item\">It is important to keep in mind that if the digit we are increasing is 9, it becomes zero and we increase the previous digit instead. If this digit is 9 as well, we&nbsp;move along&nbsp;to the left side until we find a digit less than 9.</p>\n<p class=\"c-list__ordered-item\">The last digit we need to keep will depend on how many zeros there are. &nbsp;We don't consider leading zeros to be significant, <br/>i.e.&nbsp;0.03 and 0.3 both have 1 significant figure (but 0.30 has two significant figures, since the second zero isn't a 'leading' zero).</p>\n<p><em>i)</em></p>\n<p style=\"padding-left: 30px;\">We round $\\var{e1}$ to 1 significant figure. The first non-zero digit is $\\var{edig[4]}$, followed by $\\var{edig[3]}$. This is <span data-jme-visible=\"edig[3] &lt; 5\">lower than 5 so we round down</span><span data-jme-visible=\"edig[3] &gt; 4\">more than 5 so we round up</span> to get $\\var{sigformat(e1,1)}$.</p>\n<p><em>ii)</em></p>\n<p style=\"padding-left: 30px;\">We round $\\var{e1}$ to&nbsp;{sf} significant figures. The first non-zero digit is $\\var{edig[4]}$. The second following digit is $\\var{edig[3]}$<span data-jme-visible=\"sf &gt; 2\">, the third following digit is $\\var{edig[2]}$</span><span data-jme-visible=\"sf = 4\"> and the fourth following digit is $\\var{edig[1]}$</span>. The&nbsp;digit&nbsp;following the last digit we are keeping is <span data-jme-visible=\"sf = 2\">$\\var{edig[2]}$</span><span data-jme-visible=\"sf = 3\">$\\var{edig[1]}$</span><span data-jme-visible=\"sf = 4\">$\\var{edig[0]}$</span>, so we round to&nbsp;get $\\var{sigformat(e1, sf)}$. These are our {sf} significant figures.&nbsp;</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#number\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>\n<p style=\"padding-left: 30px;\"></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"edig": {"name": "edig", "group": "Ungrouped variables", "definition": "repeat(random(1..9), 5)", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "n_from_digits(ddig)", "description": "<p>Random integer.</p>", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "n_from_digits(edig)*10^(random(-6,-7,-8))", "description": "<p>Random number with&nbsp;7 decimal places.</p>", "templateType": "anything", "can_override": false}, "ddig": {"name": "ddig", "group": "Ungrouped variables", "definition": "repeat(random(1..9), 6)", "description": "", "templateType": "anything", "can_override": false}, "sf": {"name": "sf", "group": "Ungrouped variables", "definition": "3", "description": "<p>Number of significant figures to round.</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["sf", "ddig", "edig", "d1", "e1"], "variable_groups": [], "functions": {"n_from_digits": {"parameters": [["digits", "list"]], "type": "number", "language": "jme", "definition": "if(\n  len(digits)=0,\n    0,\n    digits[0]+10*n_from_digits(digits[1..len(digits)])\n)"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Round $\\var{e1}$</p>\n<p style=\"padding-left: 60px;\"><em>iii)</em> $\\var{e1}$ rounded&nbsp;to 1 significant figure is:&nbsp; [[0]]</p>\n<p style=\"padding-left: 60px;\"><em>iv)</em> $\\var{e1}$ rounded&nbsp;to {sf} significant figures is:&nbsp; [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "siground(e1, 1)", "maxValue": "siground(e1, 1)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "siground(e1, sf)", "maxValue": "siground(e1, sf)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "NF07 Percentage change (decrease then increase)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Compound percentage change.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The value of a car is initially {StartingPrice}. If the value decreases by {dec}%, and then increases by {inc}%, what is the final value?</p>\n<p>Give your answer correct to two decimal places.</p>", "advice": "<p>There is a {dec}% decrease in price. This means that price after the decrease will be {100-dec}% of the old price.</p>\n<p>\\[\\frac{\\var{100-dec}}{100} \\times \\var{StartingPrice} = \\var{(100-dec)/100*StartingPrice}\\]</p>\n<p>Then there is a {inc}% increase in price. This means the final price will be {100+inc}% of the price after the decrease.</p>\n<p><span style=\"font-weight: 400;\">\\[\\frac{\\var{100+inc}}{100} \\times \\var{(100-dec)/100*StartingPrice} = &pound;\\var{dpformat((100+inc)/100*(100-dec)/100*StartingPrice,2)}\\]</span></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Percentages\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"dec": {"name": "dec", "group": "Ungrouped variables", "definition": "random(1..50)", "description": "", "templateType": "anything", "can_override": false}, "inc": {"name": "inc", "group": "Ungrouped variables", "definition": "random(1..50)", "description": "", "templateType": "anything", "can_override": false}, "FinalPrice": {"name": "FinalPrice", "group": "Ungrouped variables", "definition": "StartingPrice*(1-dec/100)*(1+inc/100)", "description": "", "templateType": "anything", "can_override": false}, "StartingPrice": {"name": "StartingPrice", "group": "Ungrouped variables", "definition": "random(600..8000 # 10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["dec", "inc", "FinalPrice", "StartingPrice"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p></p>\n<p>&pound;[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "FinalPrice", "maxValue": "FinalPrice", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "NF24 Dividing amounts in ratios", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Dividing amounts in ratios</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The ratio of ethanol to water is {a}:{b} for an experiment. If I have {volWater}ml of water, how much ethanol do I need?</p>", "advice": "<p>If there is a ratio of {a}:{b} for ethanol:water then that means for every {b}ml of water we need {a}ml of ethanol.</p>\n<p>In our experiment there is {volwater}ml of water so to find the amount of ethanol we divide by {b} and then multiply by {a}.</p>\n<p>\\[\\var{volwater}\\text{ml}\\times\\frac{\\var{a}}{\\var{b}}=\\var{volwater*a/b}\\text{ml}\\]<br/><br/><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Ratio\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..12 except a)", "description": "", "templateType": "anything", "can_override": false}, "volwater": {"name": "volwater", "group": "Ungrouped variables", "definition": "UsedforVol-mod(UsedforVol,b)", "description": "", "templateType": "anything", "can_override": false}, "UsedforVol": {"name": "UsedforVol", "group": "Ungrouped variables", "definition": "random(70..1000)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "volwater", "UsedforVol"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]ml</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "volwater/b*a", "maxValue": "volwater/b*a", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "NS03 Standard Form (Calculations)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Calculations involving Standard form.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<p>To divide two numbers in standard form we can calculate the division of each part of the standard form number separately. In general we have,</p>\n<p>\\[\\frac{x\\times10^j}{y\\times10^k}=\\frac xy\\times\\frac{10^j}{10^k}=\\frac xy\\times 10^{j-k}\\]</p>\n<p></p>\n<p>In this question we therefore have,</p>\n<p>\\[\\frac{\\var{a}\\times10^{\\var{n}}}{\\var{b}\\times10^{\\var{m}}}=\\frac{\\var{a}}{\\var{b}}\\times\\frac{10^{\\var{n}}}{10^{\\var{m}}}=\\var{aDivBRound}\\times10^\\var{n-m}.\\]<br/><br/></p>\n<div data-jme-visible=\"IsADivBLessOne\">\n<p>Since {aDivBRound} is less than 1 then our answer isn't in standard form. In this case we need to reduce the exponent by 1 so the final answer is</p>\n<p>\\[\\var{MantAnsRound}\\times10^{\\var{ExponentAns}}.\\]</p>\n</div>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Standard\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9.9 # 0.1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..9.9 # 0.1)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "IsADivBLessOne": {"name": "IsADivBLessOne", "group": "Ungrouped variables", "definition": "a/b<1", "description": "", "templateType": "anything", "can_override": false}, "ExponentAns": {"name": "ExponentAns", "group": "Ungrouped variables", "definition": "if(IsADivBLessOne,n-m-1,n-m)", "description": "", "templateType": "anything", "can_override": false}, "MantAns": {"name": "MantAns", "group": "Ungrouped variables", "definition": "if(IsADivBLessOne, a/b*10, a/b)", "description": "", "templateType": "anything", "can_override": false}, "aDivBRound": {"name": "aDivBRound", "group": "Ungrouped variables", "definition": "precround(a/b,2)", "description": "", "templateType": "anything", "can_override": false}, "MantAnsRound": {"name": "MantAnsRound", "group": "Ungrouped variables", "definition": "precround(MantAns,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n", "m", "IsADivBLessOne", "ExponentAns", "MantAns", "aDivBRound", "MantAnsRound"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>For the equation</p>\n<p>\\[\\frac{\\var{a}\\times10^{\\var{n}}}{\\var{b}\\times10^{\\var{m}}}=a\\times10^n\\]</p>\n<p>find the values of $a$ and $n$ which keep the answer in standard form.</p>\n<p>Give $a$ to two decimal places.</p>\n<p>$a=$[[0]]<br/><br/>$n=$[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "MantAns", "maxValue": "MantAns", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ExponentAns", "maxValue": "ExponentAns", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AC08 Collecting terms (higher powers)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Simple exercise in collecting terms in different powers of \\(x\\)</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Simplify the following expression by combining \"like\" terms.</p>", "advice": "<p>First we expand the minus sign in the bracket.</p>\n<p>\\[\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4-({f}x+{e}x^3)}=\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4+{-f}x+{-e}x^3}\\]</p>\n<p>The idea is to collect together and combine any terms that are the same kind of term so:</p>\n<p>$\\var{b}x$ and $\\var{-f}x$ both have an $x$ term. We can combine them to get $\\var{b-f}x$</p>\n<p>We can combine $\\var{a}x^4$ and $\\var{d}x^4$ to get $\\var{a+d}x^4$.</p>\n<p>We combine $\\var{c}x^3$ and $\\var{-e}x^3$ to get $\\var{c-e}x^3$. So our answer is:</p>\n<p>$\\simplify{{a+d}x^4+{c+e}x^3+{b+f}}$</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Collect\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(-5..10 except 0 except b)", "description": "", "templateType": "anything", "can_override": false}, "e": {"name": "e", "group": "Ungrouped variables", "definition": "random(-5..10 except 0 except c)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "f", "e"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify[!collectNumbers]{{a}x^4+{b}x+{c}x^3+{d}x^4-({f}x+{e}x^3)}$</p>", "answer": "({a}+{d})x^4+({c}-{e})x^3+({b}-{f})x", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`+-$n`?*x^4+`+-$n`?*x^3+`+-$n`?*x", "partialCredit": 0, "message": "", "nameToCompare": "", "warningTime": "submission"}, "valuegenerators": [{"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AC09 Multiply algebraic terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Filling in the blanks from the answer to a simplified expression involving indices.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the missing factor in the following statement:</p>\n<p>\\[ \\var{3*a}x^{\\var{b}}y^{\\var{c}} = \\var{a}x^{\\var{d}}(?)\\]</p>", "advice": "<p><span style=\"font-weight: 400;\">We can divide the left handside of the expression by the factor given on the right hand side of the expression to work out the missing factor:</span></p>\n<p><span style=\"font-weight: 400;\">\\[\\begin{split}<br/>\\var{3*a}x^{\\var{b}}y^{\\var{c}}&amp;=\\var{a}x^{\\var{d}}(?)\\\\<br/>\\Rightarrow \\frac{\\var{3*a}x^{\\var{b}}y^{\\var{c}}}{\\var{a}x^{\\var{d}}}&amp;=(?)\\\\<br/>\\Rightarrow 3x^{\\var{b}-\\var{d}}y^{\\var{c}} &amp;=(?),<br/>\\end{split}\\]</span></p>\n<p><span style=\"font-weight: 400;\">which after simplifying gives the answer:</span></p>\n<p><span style=\"font-weight: 400;\">\\[(?) = 3x^{\\var{b-d}}y^{\\var{c}}\\]</span></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Collect\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "Random(-3..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "Random(-5..8 except [b,0])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "Random(-3..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "Random(2..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b", "d", "c", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "3x^{b-d}y^{c}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AC13 Expand Double Brackets (Hard)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Poppy Jeffries", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21275/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Expand two brackets involving powers of $x$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Expand the brackets and simplify</p>", "advice": "<p>To expand the brackets $\\simplify{({a[1]}x^{b[1]}+{a[2]}x^{b[2]})({a[3]}x^{b[3]}+{c[1]}x^{b[4]})}$ We first multiply all the terms in the left bracket by all the terms in the right bracket. This gives us</p>\n<p>\\[\\var{a[1]}x^\\var{b[1]}\\times\\var{a[3]}x^\\var{b[3]}+\\var{a[1]}x^\\var{b[1]}\\times\\var{c[1]}x^\\var{b[4]}+\\var{a[2]}x^\\var{b[2]}\\times\\var{a[3]}x^\\var{b[3]}+\\var{a[2]}x^\\var{b[2]}\\times\\var{c[1]}x^\\var{b[4]}\\]</p>\n<p>We can then simplify to give us the final answer of</p>\n<p>$\\simplify{{a[1]*a[3]}*x^{b[1]+b[3]}+{a[1]*c[1]}*x^{b[1]+b[4]}+{a[2]*a[3]}*x^{b[2]+b[3]}+{a[2]*c[1]}*x^{b[2]+b[4]}}.$</p>\n<p><br/><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/maths-all#Factorise\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "repeat(random(-5..10 except [0]),5\n)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random([-5,-9/2,-4,7/2,-3,-5/2,-2,-3/2,-1,-1/2,1/2,1,3/2,2,5/2,3,7/2,4,9/2,5]),5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "shuffle(1..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify{({a[1]}x^{b[1]}+{a[2]}x^{b[2]})({a[3]}x^{b[3]}+{c[1]}x^{b[4]})}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a[1]*a[3]}*x^{b[1]+b[3]}+{a[1]*c[1]}*x^{b[1]+b[4]}+{a[2]*a[3]}*x^{b[2]+b[3]}+{a[2]*c[1]}*x^{b[2]+b[4]}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AS03 Simultaneous Equations (2 linear)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": ["Category: Simultaneous equations"], "metadata": {"description": "<p>Solving a pair of linear simultaneous equations, giving answers as integers or fractions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the simultaneous equations for x and y, giving your answers as integers or fractions, but not decimals.</p>\n<p>\\[ \\begin{split}&nbsp;\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &amp;\\,=\\var{c} \\\\ \\simplify[!noLeadingminus,unitFactor]{{a1}x +{b1}y}&nbsp; &amp;\\,=\\var{c1} \\end{split}\\]</p>", "advice": "<p>\\[\\begin{split}\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &amp;\\,=\\var{c} \\qquad\\qquad&amp;(1)\\\\ \\simplify[!noLeadingminus,unitFactor]{{a1}x +{b1}y}&nbsp; &amp;\\,=\\var{c1} \\qquad\\qquad&amp;(2)\\end{split}\\]</p>\n<p>{advice1}</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Rearranging\"><span style=\"font-weight: 400;\"><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sim\">Use this link to find some resources which will help you revise this topic.</a></span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2..8 except [0,1])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-8..8 except [0,1,a])", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-5..8 except [0,1])", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2..10 except [round(a1*b/a),b,0,1])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-7..7 except 0)", "description": "", "templateType": "anything", "can_override": false}, "aorsb": {"name": "aorsb", "group": "Ungrouped variables", "definition": "if(b*abs(b1)=b1*abs(b),'subtract','add')", "description": "", "templateType": "anything", "can_override": false}, "torfb": {"name": "torfb", "group": "Ungrouped variables", "definition": "if(b*abs(b1)=b1*abs(b),'from','to')", "description": "", "templateType": "anything", "can_override": false}, "sgn": {"name": "sgn", "group": "Ungrouped variables", "definition": "if(b*abs(b1)=b1*abs(b),-1,1)", "description": "", "templateType": "anything", "can_override": false}, "xn": {"name": "xn", "group": "Ungrouped variables", "definition": "c*abs(b1)+sgn*c1*abs(b)", "description": "", "templateType": "anything", "can_override": false}, "xd": {"name": "xd", "group": "Ungrouped variables", "definition": "a*abs(b1)+sgn*a1*abs(b)", "description": "", "templateType": "anything", "can_override": false}, "xsimp": {"name": "xsimp", "group": "Ungrouped variables", "definition": "xn/xd", "description": "", "templateType": "anything", "can_override": false}, "samex": {"name": "samex", "group": "Ungrouped variables", "definition": "\"<p>For these equations, it is easiest to get a solution for $y$ first, due to the $x$-terms having {eqoroppa} coefficients.</p>\\n<p>If we {aorsa} equation (2) {torfa} equation (1) this eliminates the $x$-terms leaving us with one equation in terms of $y$:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({b}+{sgna*(b1)})y} &amp;\\\\,=&nbsp;\\\\simplify[!collectNumbers, !noLeadingminus]{{c}+{sgna*(c1)}}\\\\\\\\ \\\\simplify{{b+sgna*(b1)}y} &amp;\\\\,= \\\\simplify{{c+sgna*(c1)}} \\\\\\\\ y &amp;\\\\,= \\\\simplify[all, fractionNumbers]{{c+sgna*(c1)}/{b+sgna*(b1)}} \\\\end{split} \\\\]</p>\\n<p></p>\\n<p>To obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\var{a}x + \\\\var{b} \\\\times \\\\simplify[all, fractionNumbers]{{c+sgna*(c1)}/{b+sgna*(b1)}} &amp;\\\\,= \\\\var{c} \\\\\\\\ \\\\var{a}x &amp;\\\\,= \\\\simplify[all, !collectNumbers, !noLeadingminus]{{c} - {c*b+b*sgna*(c1)}/{b+sgna*(b1)}} \\\\\\\\ x &amp;\\\\,= \\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "eqoroppb": {"name": "eqoroppb", "group": "Ungrouped variables", "definition": "if(abs(b)*b1=abs(b1)*b,'equal','equal and opposite')", "description": "", "templateType": "anything", "can_override": false}, "eqoroppa": {"name": "eqoroppa", "group": "Ungrouped variables", "definition": "if(abs(a)*a1=abs(a1)*a,'equal','equal and opposite')", "description": "", "templateType": "anything", "can_override": false}, "samey": {"name": "samey", "group": "Ungrouped variables", "definition": "\"<p>For these equations, it is easiest to get a solution for $x$ first, due to the $y$-terms having {eqoroppb} coefficients.</p>\\n<p>If we {aorsb} equation (2) {torfb} equation (1) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({a}+{sgn*(a1)})x} &amp;\\\\,=&nbsp;\\\\simplify[!collectNumbers, !noLeadingminus]{{c}+{sgn*(c1)}}\\\\\\\\ \\\\simplify{{a+sgn*(a1)}x} &amp;\\\\,= \\\\simplify{{c+sgn*(c1)}} \\\\\\\\ x &amp;\\\\,= \\\\simplify[all, fractionNumbers]{{c+sgn*(c1)}/{a+sgn*(a1)}} \\\\end{split} \\\\]</p>\\n<p></p>\\n<p>To obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\var{a} \\\\times\\\\simplify[fractionNumbers]{{c+sgn*(c1)}/{a+sgn*(a1)}} + \\\\var{b}y &amp;\\\\,= \\\\var{c} \\\\\\\\ \\\\var{b}y &amp;\\\\,= \\\\simplify[!collectNumbers, !noLeadingminus]{{c} - {c*a+a*sgn*(c1)}/{a+sgn*(a1)}} \\\\\\\\ y &amp;\\\\,= \\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "lcmb": {"name": "lcmb", "group": "Ungrouped variables", "definition": "\"<p>To get a solution for $x$, if we multiply equation (2) by $\\\\simplify{{abs(b/b1)}}$ we will have two equations with {eqoroppb} $y$-coefficients:</p>\\n<p>\\\\[ \\\\begin{split}&nbsp;\\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &amp;\\\\,=\\\\var{c} \\\\qquad\\\\qquad&amp;(3)\\\\\\\\ \\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(b/b1)}x +{b1*abs(b/b1)}y}&nbsp; &amp;\\\\,=\\\\var{c1*abs(b/b1)} \\\\qquad\\\\qquad&amp;(4)\\\\end{split}\\\\]</p>\\n<p>If we {aorsb} equation (4) {torfb} equation (3) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({a}+{sgn*(a1*abs(b/b1))})x} &amp;\\\\,=&nbsp;\\\\simplify[all, !collectNumbers, !noLeadingminus]{{c}+{sgn*(c1*abs(b/b1))}}\\\\\\\\ \\\\simplify{{a+sgn*(a1*abs(b/b1))}x} &amp;\\\\,= \\\\simplify{{c+sgn*(c1*abs(b/b1))}} \\\\\\\\ x &amp;\\\\,= \\\\simplify[all,fractionNumbers]{{c+sgn*(c1*abs(b/b1))}/{a+sgn*(a1*abs(b/b1))}}. \\\\end{split} \\\\]</p>\\n<p></p>\\n<p>To obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\var{a}\\\\times\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c+sgn*c1*abs(b/b1)}/{(a)+sgn*a1*abs(b/b1)}) + {b}y} &amp;\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{b}y} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(a*c)+a*sgn*c1*abs(b/b1)}/{(a)+sgn*a1*abs(b/b1)})} \\\\\\\\ \\\\simplify{{b}y} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(a*c+a*sgn*c1*abs(b/b1))/(a+sgn*a1*abs(b/b1))}} \\\\\\\\ y &amp;\\\\,=\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "lcmb1": {"name": "lcmb1", "group": "Ungrouped variables", "definition": "\"<p>To get a solution for $x$, if we multiply equation (1) by $\\\\simplify{{abs(b1/b)}}$ we will have two equations with {eqoroppb} $y$-coefficients:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a*abs(b1/b)}x +{b*abs(b1/b)}y}&nbsp; &amp;\\\\,=\\\\var{c*abs(b1/b)} \\\\qquad\\\\qquad&amp;(3) \\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1}x+{b1}y} &amp;\\\\,=\\\\var{c1} \\\\qquad\\\\qquad&amp;(4)\\\\\\\\ \\\\end{split} \\\\]</p>\\n<p>If we {aorsb} equation (4) {torfb} equation (3) this eliminates the $y$-terms, leaving us with one equation in terms of $x$:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({(a*abs(b1/b))}+{sgn*a1})x} &amp;\\\\,=&nbsp;\\\\simplify[!collectNumbers, !noLeadingminus]{{(c*abs(b1/b))}+{sgn*c1}}\\\\\\\\ \\\\simplify{{(a*abs(b1/b))+sgn*a1}x} &amp;\\\\,= \\\\simplify{{(c*abs(b1/b))+sgn*c1}} \\\\\\\\ x &amp;\\\\,= \\\\simplify[all, fractionNumbers]{{(c*abs(b1/b))+sgn*c1}/{(a*abs(b1/b))+sgn*a1}}. \\\\end{split} \\\\]</p>\\n<p></p>\\n<p>To obtain a solution for $y$ we can substitute this $x$-value into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\var{a}\\\\times\\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({(c*abs(b1/b))+sgn*c1}/{(a*abs(b1/b))+sgn*a1}) + {b}y} &amp;\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{b}y} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(a*c*abs(b1/b))+a*sgn*c1}/{(a*abs(b1/b))+sgn*a1})} \\\\\\\\ \\\\simplify{{b}y} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(a*c*abs(b1/b)+a*sgn*c1)/(a*abs(b1/b)+sgn*a1)}} \\\\\\\\ y &amp;\\\\,=\\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "full": {"name": "full", "group": "Ungrouped variables", "definition": "\"<p>To get a solution for $x$, if we multiply equation (1) by $\\\\var{abs(b1)}$ and equation (2) by $\\\\var{abs(b)}$, we will have two equations with {eqoroppb} $y$-coefficients:</p>\\n<p>\\\\[ \\\\begin{split}&nbsp;\\\\simplify[!noLeadingminus,unitFactor]{{a*abs(b1)}x+{b*abs(b1)}y} &amp;\\\\,=\\\\var{c*abs(b1)} \\\\qquad\\\\qquad&amp;(3)\\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(b)}x +{b1*abs(b)}y}&nbsp; &amp;\\\\,=\\\\var{c1*abs(b)} \\\\qquad\\\\qquad&amp;(4) \\\\end{split}\\\\]</p>\\n<p>Now, {aorsb} equation (4) {torfb} equation (3) to eliminate the $y$ terms:</p>\\n<p>\\\\[ \\\\begin{split} (\\\\simplify[!collectNumbers]{{a*abs(b1)} +{sgn*a1*abs(b)}}) x &amp;\\\\,= \\\\simplify[!collectNumbers]{{c*abs(b1)}+{sgn*c1*abs(b)}}&nbsp;\\\\\\\\&nbsp;&nbsp;\\\\simplify{{a*abs(b1)+sgn*a1*abs(b)}} x &amp;\\\\,= \\\\simplify{{c*abs(b1)+sgn*c1*abs(b)}} .\\\\end{split} \\\\]</p>\\n<p>So the solution for $x$ is \\\\[ x=\\\\simplify{{c*abs(b1)+sgn*c1*abs(b)}/{a*abs(b1)+sgn*a1*abs(b)}}.\\\\]</p>\\n<p>To obtain a solution for $y$ we can substitute this value of $x$ into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[noLeadingminus,fractionNumbers,unitFactor]{{a}&nbsp; {xsimp} + {b}y} &amp;\\\\,=\\\\var{c} \\\\\\\\ \\\\var{b}y &amp;\\\\,= \\\\simplify[!collectNumbers,fractionNumbers]{{c}-{a*xsimp}} \\\\\\\\\\\\var{b}y &amp;\\\\,= \\\\simplify[fractionNumbers]{{c-a*xsimp}} \\\\\\\\y &amp;\\\\,= \\\\simplify[fractionNumbers]{{(c-a*xsimp)/b}} \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "aorsa": {"name": "aorsa", "group": "Ungrouped variables", "definition": "if(a*abs(a1)=abs(a)*a1,'subtract','add')", "description": "", "templateType": "anything", "can_override": false}, "torfa": {"name": "torfa", "group": "Ungrouped variables", "definition": "if(a*abs(a1)=abs(a)*a1,'from','to')", "description": "", "templateType": "anything", "can_override": false}, "sgna": {"name": "sgna", "group": "Ungrouped variables", "definition": "if(a*abs(a1)=abs(a)*a1,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "lcma": {"name": "lcma", "group": "Ungrouped variables", "definition": "\"<p>To get a solution for $y$, if we multiply equation (2) by $\\\\simplify{{abs(a/a1)}}$ we will have two equations with {eqoroppa} $x$-coefficients:</p>\\n<p>\\\\[ \\\\begin{split}&nbsp;\\\\simplify[!noLeadingminus,unitFactor]{{a}x+{b}y} &amp;\\\\,=\\\\var{c} \\\\qquad\\\\qquad&amp;(3)\\\\\\\\ \\\\simplify[!noLeadingminus,unitFactor]{{a1*abs(a/a1)}x +{b1*abs(a/a1)}y}&nbsp; &amp;\\\\,=\\\\var{c1*abs(a/a1)} \\\\qquad\\\\qquad&amp;(4)\\\\end{split}\\\\]</p>\\n<p>If we {aorsa} equation (4) {torfa} equation (3) this eliminates the $x$-terms, leaving us with one equation in terms of $y$:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({b}+{sgna*(b1*abs(a/a1))})y} &amp;\\\\,=&nbsp;\\\\simplify[all, !collectNumbers, !noLeadingminus]{{c}+{sgna*(c1*abs(a/a1))}}\\\\\\\\ \\\\simplify{{b+sgna*(b1*abs(a/a1))}y} &amp;\\\\,= \\\\simplify{{c+sgna*(c1*abs(a/a1))}} \\\\\\\\ y &amp;\\\\,= \\\\simplify[all,fractionNumbers]{{c+sgna*(c1*abs(a/a1))}/{b+sgna*(b1*abs(a/a1))}}. \\\\end{split} \\\\]</p>\\n<p></p>\\n<p>To obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{{a}x + {b}}\\\\times \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c+sgna*c1*abs(a/a1)}/{(b)+sgna*b1*abs(a/a1)})} &amp;\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{a}x} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(b*c)+b*sgna*c1*abs(a/a1)}/{(b)+sgna*b1*abs(a/a1)})} \\\\\\\\ \\\\simplify{{a}x} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(b*c+b*sgna*c1*abs(a/a1))/(b+sgna*b1*abs(a/a1))}} \\\\\\\\ x &amp;\\\\,=\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "lcma1": {"name": "lcma1", "group": "Ungrouped variables", "definition": "\"<p>To get a solution for $y$, if we multiply equation (1) by $\\\\simplify{{abs(a1/a)}}$ we will have two equations with {eqoroppa} $x$-coefficients:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!noLeadingminus,unitFactor]{{a*abs(a1/a)}x +{b*abs(a1/a)}y}&nbsp; &amp;\\\\,=\\\\var{c*abs(a1/a)} \\\\qquad\\\\qquad&amp;(3) \\\\\\\\\\\\simplify[!noLeadingminus,unitFactor]{{a1}x+{b1}y} &amp;\\\\,=\\\\var{c1} \\\\qquad\\\\qquad&amp;(4) \\\\end{split}\\\\]</p>\\n<p>If we {aorsa} equation (4) {torfa} equation (3) this eliminates the $x$-terms, leaving us with one equation in terms of $y$:</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[!collectNumbers, !noLeadingminus]{({(b*abs(a1/a))}+{sgna*b1})y} &amp;\\\\,=&nbsp;\\\\simplify[!collectNumbers, !noLeadingminus]{{(c*abs(a1/a))}+{sgna*c1}}\\\\\\\\ \\\\simplify{{(b*abs(a1/a))+sgna*b1}y} &amp;\\\\,= \\\\simplify{{(c*abs(a1/a))+sgna*c1}} \\\\\\\\ y &amp;\\\\,= \\\\simplify[all, fractionNumbers]{{(c*abs(a1/a))+sgna*c1}/{(b*abs(a1/a))+sgna*b1}}. \\\\end{split} \\\\]</p>\\n<p></p>\\n<p>To obtain a solution for $x$ we can substitute this $y$-value into either of our initial equations. Using equation (1), we obtain</p>\\n<p>\\\\[ \\\\begin{split} \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{{a}x + {b}}\\\\times \\\\simplify[all, !noLeadingminus, !expandBrackets, fractionNumbers]{({c*abs(a1/a)+sgna*c1}/{(b*abs(a1/a))+sgna*b1})} &amp;\\\\,= \\\\var{c} \\\\\\\\ \\\\simplify{{a}x} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c} -({(b*c*abs(a1/a))+b*sgna*c1}/{(b*abs(a1/a))+sgna*b1})} \\\\\\\\ \\\\simplify{{a}x} &amp;\\\\,= \\\\simplify[all, !noLeadingminus, fractionNumbers]{{c -(b*c*abs(a1/a)+b*sgna*c1)/(b*abs(a1/a)+sgna*b1)}} \\\\\\\\ x &amp;\\\\,=\\\\simplify[fractionNumbers]{{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice1": {"name": "advice1", "group": "Ungrouped variables", "definition": "if(abs(b)=abs(b1), {samey},if(abs(a)=abs(a1),{samex},if(lcm(abs(b),abs(b1))=abs(b),{lcmb},if(lcm(abs(b),abs(b1))=abs(b1),{lcmb1},if(lcm(abs(a),abs(a1))=abs(a),{lcma},if(lcm(abs(a),abs(a1))=abs(a1),{lcma1},{full}))))))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(b-b1)>1 and\nabs(a-a1)>1 and\ngcd(a,c)=1 and\ngcd(a1,c1)=1", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "a1", "b1", "c", "c1", "aorsa", "torfa", "aorsb", "torfb", "sgna", "sgn", "xn", "xd", "xsimp", "eqoroppa", "eqoroppb", "advice1", "samey", "samex", "lcmb", "lcmb1", "lcma", "lcma1", "full"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$x=$ [[0]]</p>\n<p>$y=$ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(c*abs(b1)+sgn*c1*abs(b))/(a*abs(b1)+sgn*a1*abs(b))}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(c-a*xsimp)/b}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AS04 Simultaneous Equations (one non-linear)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Solving a pair of simultaneous equations of the form $a_1x+y=c_1$ and $a_2x^2+b_2xy=c_2$ by forming a quadratic equation.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following simultaneous equations:</p>\n<p>\\[ \\begin{split} \\simplify{{a1}x+y} &amp;\\,= \\var{c1} \\\\ \\simplify{{a2}x^2+{b2}x*y} &amp;\\,= \\var{c2} \\end{split} \\]</p>\n<p></p>\n<p><em>Give your answers to 2 decimal places where necessary.</em></p>", "advice": "<p>To solve a pair of simultaneous equations of this type we want to rearrange the linear equation such that $y$ is the subject, which we can then substitute into the equation with the quadratic $x$-term. This will result in a quadratic equation in terms of $x$ only.</p>\n<p>For the equations&nbsp;</p>\n<p>\\[ \\begin{split} \\simplify{{a1}x+y} &amp;\\,= \\var{c1} \\qquad \\qquad &amp;(1) \\\\\\simplify{{a2}x^2+{b2}x*y} &amp;\\,= \\var{c2} \\qquad \\qquad &amp;(2) \\end{split} \\]</p>\n<p>we can rearrange equation (1)&nbsp;to make $y$ the subject:</p>\n<p>\\[ y = \\simplify{{c1}-{a1}x}. \\qquad\\qquad (3)\\]</p>\n<p>Substituting this into equation (2):</p>\n<p>\\[ \\begin{split}\\simplify{{a2}x^2+{b2}x({c1}-{a1}x)} &amp;\\,=\\var{c2} \\\\ \\simplify[!cancelTerms,unitFactor]{{a2}x^2+{b2*c1}x-{b2*a1}x^2} &amp;\\,=\\var{c2}. \\end{split} \\]</p>\n<p>Collecting similar terms:</p>\n<p>\\[ \\simplify{({a2}-{b2*a1})x^2+{b2*c1}x-{c2}} =0. \\qquad\\qquad (4) \\]</p>\n<p>Using the quadratic formula, we find two solutions for $x$:</p>\n<p>{check}</p>\n<p>Therefore, the 2 pairs of solutions for these simultaneous equations are</p>\n<p>\\[ (x_1,y_1) = (\\var{x1dp},\\var{y1dp}) \\] and&nbsp;\\[ (x_2,y_2) = (\\var{x2dp},\\var{y2dp}). \\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Rearranging\"><span style=\"font-weight: 400;\"></span></a><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sim\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(-5..-1)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "solx1": {"name": "solx1", "group": "Ungrouped variables", "definition": "(-b2*c1-sqrt((b2^2*c1^2)+4(a2-a1*b2)*c2))/(2(a2-b2*a1))", "description": "", "templateType": "anything", "can_override": false}, "solx2": {"name": "solx2", "group": "Ungrouped variables", "definition": "(-b2*c1+sqrt((b2^2*c1^2)+4(a2-a1*b2)*c2))/(2(a2-b2*a1))", "description": "", "templateType": "anything", "can_override": false}, "soly1": {"name": "soly1", "group": "Ungrouped variables", "definition": "c1-a1*solx1", "description": "", "templateType": "anything", "can_override": false}, "soly2": {"name": "soly2", "group": "Ungrouped variables", "definition": "c1-a1*solx2", "description": "", "templateType": "anything", "can_override": false}, "x2dp": {"name": "x2dp", "group": "Ungrouped variables", "definition": "precround(solx2,2)", "description": "", "templateType": "anything", "can_override": false}, "y1dp": {"name": "y1dp", "group": "Ungrouped variables", "definition": "precround(soly1,2)", "description": "", "templateType": "anything", "can_override": false}, "y2dp": {"name": "y2dp", "group": "Ungrouped variables", "definition": "precround(soly2,2)", "description": "", "templateType": "anything", "can_override": false}, "x1dp": {"name": "x1dp", "group": "Ungrouped variables", "definition": "precround(solx1,2)", "description": "", "templateType": "anything", "can_override": false}, "solutions1": {"name": "solutions1", "group": "Ungrouped variables", "definition": "matrix([x1dp,y1dp])", "description": "", "templateType": "anything", "can_override": false}, "solutions2": {"name": "solutions2", "group": "Ungrouped variables", "definition": "matrix([x2dp,y2dp])", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(x1dp=round(x1dp) and x2dp=round(x2dp),'{text}', if(x1dp=round(x1dp),'{text1}',if(x2dp=round(x2dp),'{text2}','{text3}')))", "description": "", "templateType": "anything", "can_override": false}, "text1": {"name": "text1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp} \\\\, \\\\text{ (2 d.p.)} \\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\, \\\\text{(2 d.p.)} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "text2": {"name": "text2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\, \\\\text{ (2 d.p.)}\\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp} \\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\, \\\\text{(2 d.p.)} \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "text3": {"name": "text3", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\,&nbsp; \\\\text{ (2 d.p.)}\\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp} \\\\, \\\\text{(2 d.p.)} \\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\, \\\\text{(2 d.p.)} \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\,&nbsp;\\\\text{(2 d.p.)} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "text": {"name": "text", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ x_1 = \\\\var{x1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad x_2=\\\\var{x2dp}\\\\]</p>\\n<p>To find the corresponding $y$-values, we can plug these solutions for $x$ back into equation (3), which gives:</p>\\n<p>\\\\[ y_1 = \\\\var{y1dp} \\\\,&nbsp; \\\\quad \\\\text{and} \\\\quad y_2=\\\\var{y2dp} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}}, "variablesTest": {"condition": "(a2-a1*b2)>0 and (b2^2*c1^2+4(a2-a1*b2)*c2)>0 and gcd(a2,b2)=1", "maxRuns": 100}, "ungrouped_variables": ["a1", "c1", "a2", "b2", "c2", "solx1", "solx2", "soly1", "soly2", "x1dp", "y1dp", "x2dp", "y2dp", "solutions1", "solutions2", "check", "text", "text1", "text2", "text3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$(x_1,y_1)=$[[0]]</p>\n<p>$(x_2,y_2)=$[[1]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "correctAnswer": "solutions2", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "correctAnswer": "solutions1", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "correctAnswer": "solutions1", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "correctAnswer": "solutions2", "correctAnswerFractions": false, "numRows": 1, "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AS07 Quadratics - solve", "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "<p>Solving a quadratic equation via factorisation (or otherwise) with the $x^2$-term having a coefficient of 1.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the following quadratic equation by factorisation or otherwise:</p>\n<p>\\[ \\simplify[unitFactor]{x^2+{b}x+{c}=0} \\]</p>", "advice": "<p>To solve a quadratic equation of the form \\[ x^2+bx+c=0\\] by factorisation, we want to factorise the equation into the form \\[(x+p)(x+q)=0,\\] where $p+q=b$ and $p \\times q = c$.&nbsp;</p>\n<p>Hence, for the equation \\[\\simplify{x^2+{b}x+{c}=0}, \\]</p>\n<p>this can be factorised to \\[\\simplify{(x+{p})(x+{q})=0}.\\] This equation is satisfied when either \\[\\simplify{x+{p}=0} \\quad \\text{or} \\quad \\simplify{x+{q}=0}, \\] which implies the solutions to this quadratic equation are \\[ \\simplify{x={-p}} \\quad \\text{and} \\quad \\simplify{x={-q}} .\\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\" target=\"_blank\" rel=\"noopener\">Use this link to find resources to help you revise how to solve quadratic equations by factorising the expression.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "{p+q}", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "{p*q}", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(-10..10 except [0,p])", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "[-p,-q]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(p+q)>0", "maxRuns": 100}, "ungrouped_variables": ["b", "c", "p", "q", "sol"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$x= $[[0]]</p>", "gaps": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "{sol}", "allowFractions": false, "correctAnswerFractions": false, "isSet": false, "show_input_hint": true, "separator": ","}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AS08 Completing the square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": ["complete the square", "completing the square", "taxonomy"], "metadata": {"description": "<p>Rearrange expressions in the form $ax^2+bx+c$ to $a(x+b)^2+c$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>We can rewrite quadratic equations given in the form $ax^2+bx+c$ as a&nbsp;square plus another term - this is called \"completing the square\".</p>\n<p>This can be useful when it isn't obvious how to fully factorise a quadratic equation.</p>\n<p>Rewrite the following expressions in the form \\[(x+b)^2-c\\]</p>", "advice": "<p>Completing the square works by noticing that</p>\n<p>\\[ (x+a)^2 = x^2 + 2ax + a^2 \\]</p>\n<p>So when we see an expression of the form $x^2 + 2ax$, we can rewrite it as $(x+a)^2-a^2$.</p>\n<p></p>\n<p>Replace $x^2+\\var{evens2}x$ with $(x+\\var{evens2/2})^2 - \\var{evens2/2}^2$. Remember&nbsp;to keep the $\\var{evens2-evens1}$ term on the end!</p>\n<p>\\begin{align}<br/>\\simplify[basic]{ x^2 + {evens2}x + {evens2-evens1}} &nbsp;&amp;= \\simplify[basic]{ (x+{evens2/2})^2 - {evens2/2}^2 + {evens2-evens1} } \\\\<br/>&amp;= \\simplify[basic]{ (x+{evens2/2})^2 + {evens2-evens1 - evens2^2/4} }<br/>\\end{align}</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Quadratics\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>\n<h4><em></em></h4>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"multiall2": {"name": "multiall2", "group": "Ungrouped variables", "definition": "all2*random(2..10 #2)", "description": "", "templateType": "anything", "can_override": false}, "odds3": {"name": "odds3", "group": "Odds and Evens", "definition": "random(11..30 #2 except odds odds2)", "description": "", "templateType": "anything", "can_override": false}, "evens3": {"name": "evens3", "group": "Odds and Evens", "definition": "random(2..30 #2 except evens1 evens2)", "description": "", "templateType": "anything", "can_override": false}, "evens2": {"name": "evens2", "group": "Odds and Evens", "definition": "random(10..30 #2 except evens1)", "description": "", "templateType": "anything", "can_override": false}, "multiall": {"name": "multiall", "group": "Ungrouped variables", "definition": "all*random(2..10#2)", "description": "", "templateType": "anything", "can_override": false}, "evens1": {"name": "evens1", "group": "Odds and Evens", "definition": "random(10..30 #2)", "description": "", "templateType": "anything", "can_override": false}, "all2": {"name": "all2", "group": "Ungrouped variables", "definition": "random(2..6 except all)", "description": "", "templateType": "anything", "can_override": false}, "odds2": {"name": "odds2", "group": "Odds and Evens", "definition": "random(11..30 #2 except odds)", "description": "", "templateType": "anything", "can_override": false}, "big": {"name": "big", "group": "Ungrouped variables", "definition": "random(30..50)", "description": "", "templateType": "anything", "can_override": false}, "all": {"name": "all", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "sml": {"name": "sml", "group": "Ungrouped variables", "definition": "random(2..6#2)", "description": "", "templateType": "anything", "can_override": false}, "odds": {"name": "odds", "group": "Odds and Evens", "definition": "random(11..30 #2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["all", "all2", "multiall", "big", "sml", "multiall2"], "variable_groups": [{"name": "Odds and Evens", "variables": ["evens1", "evens2", "evens3", "odds", "odds2", "odds3"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\simplify {x^2+ {evens2}x +{evens2-evens1}} =$&nbsp;[[0]]<em><br/></em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(x+{evens2/2})^2+{-(evens2/2)^2+evens2-evens1}", "answerSimplification": "basic, fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["(x", ")^2"], "showStrings": false, "partialCredit": 0, "message": "<p>It doesn't look like you've completed the square.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AC18 Algebraic Fractions - addition (harder)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Simplify the sum of two algebraic fractions where spotting factorising of both numerators and denominators can reduce the work massively.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Write the following as a single fraction $\\frac{\\var{num1}}{\\var{den1}}+\\frac{\\var{num2}}{\\var{den2}}$ simplifying as much as possible. Your answer should be in the form $\\frac{\\alpha\\var{v}+\\beta}{\\delta\\var{v}^2-\\gamma}.$</p>", "advice": "<p>To write the following as a single fraction $\\frac{\\var{num1}}{\\var{den1}}+\\frac{\\var{num2}}{\\var{den2}}$ first factorise as much as possible and look for any cancellations:</p>\n<p>\\[\\begin{split}<br/>&amp;\\frac{\\var{a}\\times\\var{b}}{\\var{den1fact}} + \\frac{\\var{num2}}{\\var{den2fact}}\\\\<br/>&amp; = \\frac{\\var{b}}{\\var{den1simp}} + \\frac{1}{\\var{f1c}}.<br/>\\end{split}\\]</p>\n<p>Then get a common denominator for the two fractions and combine into a single fraction:</p>\n<p>\\[\\begin{split}<br/>&amp;\\frac{\\var{b}}{\\var{den1simp}} + \\frac{\\var{f1}}{\\var{den1simp}}\\\\<br/>&amp; = \\frac{\\var{b}+\\var{f1}}{\\var{den1simp}}\\\\<br/>&amp; = \\var{ans}.<br/>\\end{split}\\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#AlgFrac\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Set up", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Set up", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "v": {"name": "v", "group": "Set up", "definition": "random(\"a\",\"b\",\"c\",\"d\",\"f\",\"g\",\"h\",\"k\",\"m\",\"n\",\"p\",\"q\",\"r\",\"s\",\"t\",\"u\",\"v\",\"w\",\"x\",\"y\",\"z\")", "description": "", "templateType": "anything", "can_override": false}, "cf1": {"name": "cf1", "group": "Set up", "definition": "repeat(random(2..4),2)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Set up", "definition": "simplify(cf1[0]+\"*\"+v+\"+\"+cf1[1],\"all\")", "description": "", "templateType": "anything", "can_override": false}, "f1c": {"name": "f1c", "group": "Set up", "definition": "simplify(cf1[0]+\"*\"+v+\"-\"+cf1[1],\"all\")", "description": "", "templateType": "anything", "can_override": false}, "cf2": {"name": "cf2", "group": "Set up", "definition": "repeat(random(2..5),2)", "description": "", "templateType": "anything", "can_override": false}, "f2": {"name": "f2", "group": "Set up", "definition": "simplify(cf2[0]+\"*\"+v+\"+\"+cf2[1],\"all\")", "description": "", "templateType": "anything", "can_override": false}, "den1fact": {"name": "den1fact", "group": "Advice", "definition": "simplify(a+\"*\"+\"(\"+string(f1)+\")*(\"+string(f1c)+\")\",\"all\")", "description": "", "templateType": "anything", "can_override": false}, "num1": {"name": "num1", "group": "Question", "definition": "a*b", "description": "", "templateType": "anything", "can_override": false}, "den2": {"name": "den2", "group": "Question", "definition": "simplify(den2fact,[\"expandBrackets\",\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "num2": {"name": "num2", "group": "Question", "definition": "simplify(f2,\"all\")", "description": "", "templateType": "anything", "can_override": false}, "den1": {"name": "den1", "group": "Question", "definition": "simplify(den1fact,[\"expandBrackets\",\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "den2fact": {"name": "den2fact", "group": "Advice", "definition": "simplify(expression(\"(\"+string(f1c)+\")*(\"+string(f2)+\")\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "ansn": {"name": "ansn", "group": "Question", "definition": "simplify(string(f1) + \"+\" + b,\"all\")", "description": "", "templateType": "anything", "can_override": false}, "ansd": {"name": "ansd", "group": "Question", "definition": "simplify(expression(\"(\"+string(f1)+\")\"+\"*\"+ \"(\"+string(f1c)+\")\"),[\"expandBrackets\",\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Question", "definition": "simplify(expression(\"(\"+string(ansn)+\")\"+\"/\"+\"(\"+string(ansd)+\")\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "den1simp": {"name": "den1simp", "group": "Advice", "definition": "simplify(\"(\"+string(f1)+\")*(\"+string(f1c)+\")\",\"all\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "f1<>f2 AND f1c<>f2 AND cf1[0]<>cf1[1] AND cf2[0]<>cf2[1]", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Set up", "variables": ["a", "b", "v", "cf1", "f1", "f1c", "cf2", "f2"]}, {"name": "Question", "variables": ["num1", "den1", "num2", "den2", "ansn", "ansd", "ans"]}, {"name": "Advice", "variables": ["den1fact", "den2fact", "den1simp"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "?`+/?`+", "partialCredit": 0, "message": "", "nameToCompare": "", "warningTime": "submission"}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AC19 Cancelling algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}, {"name": "Anna Strzelecka", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2945/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>A question to practice simplifying fractions with the use of factorisation (for binomial and quadratic expressions).</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Simplify the following algebraic expression.<em></em></p>", "advice": "<p>\\[\\frac{{\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}}{{\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}}\\]</p>\n<p>In this question there is a quadratic expression which needs to be factorised into the products of binomials in both the numerator and denominator.</p>\n<p>\\[\\frac{({\\simplify{n+{e1}}})({\\simplify{n+{e2}}})}{({\\simplify{n+{e1}}})({\\simplify{n+{e3}}})}\\]</p>\n<p>The repeated binomials in the numerator and denominator cancel, leaving:</p>\n<p>\\[\\frac{({\\simplify{n+{e2}}})}{({\\simplify{n+{e3}}})}\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#AlgFrac\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"e2": {"name": "e2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "e1": {"name": "e1", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "e3": {"name": "e3", "group": "Ungrouped variables", "definition": "random(-5..5 except 0 except e2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["e1", "e2", "e3"], "variable_groups": [], "functions": {"": {"parameters": [], "type": "number", "language": "jme", "definition": ""}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\[\\frac{\\simplify{(n^2+({e1}+{e2})n+{e1}{e2})}}{\\simplify{(n^2+({e1}+{e3})n+{e1}{e3})}}\\]</p>", "answer": "(n+{e2})/(n+{e3})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["^2", "^"], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "n", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AC22 Partial Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Oliver Spenceley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23557/"}], "tags": [], "metadata": {"description": "<p>Rewrite the expression $\\frac{mx^2+nx+k}{(x+a)(x^2+bx+c)}$ as partial fractions in the form $\\frac{A}{x+a}+\\frac{Bx+C}{x^2+bx+c}$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as partial fractions:</p>\n<p>\\[ \\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))}. \\]</p>\n<p></p>", "advice": "<p>To express \\[ \\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} \\] as partial fractions, we want to set this equal to the sum of two fractions with denominators $\\simplify{x+{a}}$ and $\\simplify{x^2+{b}x+{c}}$. Since we have a linear factor and a quadratic factor, this tells us that the form of the partial fractions will be</p>\n<p>\\[&nbsp;\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\simplify{A/(x+{a}) + (B*x+C)/(x^2+{b}x+{c})},\\]</p>\n<p>where $A$, $B$, and $C$ are constants.</p>\n<p>To find the values of $A$, $B$, and $C$, we want to first multiply this equation by the denominator of the left-hand side. This gives</p>\n<p>\\[ \\simplify{{m}x^2+{n}x+{k}=A(x^2+{b}x+{c})+B*x(x+{a}) + C(x+{a})}.\\]</p>\n<p><strong>(Note: To find $A$, $B$, and $C$, we will use a combination of choosing suitable values of $x$ to eliminate terms, and equating coefficients. It can be solved by <em>only</em> equating coefficients, but this is a more efficient process.)</strong></p>\n<p></p>\n<p>To find $A$, we can eliminate $B$ and $C$ by setting $x=\\var{-a}$:</p>\n<p>\\[ \\simplify{{m*a^2-n*a+k}=A{(a^2-b*a+c)}} \\implies A=\\simplify[fractionNumbers]{{Asol}}.\\]</p>\n<p>To find $C$, we can eliminate $B$ by setting $x=0$ and substituting in the result of $A$:</p>\n<p>\\[ \\simplify{{k}={c}A+{a}C} \\implies C=\\simplify[all,fractionNumbers]{({k}-{c}A)/{a}}.\\]</p>\n<p>Hence,</p>\n<p>\\[ C = \\simplify[fractionNumbers]{{Csol}}.\\]</p>\n<p>Finally, by equating coefficients of the $x^2$-terms we can find $B$:</p>\n<p>\\[ (x^2): \\quad \\var{m} = \\simplify{A+B} \\implies B=\\var{m}-A. \\]</p>\n<p>Therefore, \\[ B=\\simplify[fractionNumbers]{{Bsol}}, \\]</p>\n<p>and</p>\n<p>{check}</p>\n<p><strong><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/directory#P\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></strong></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "pairs[index][1]", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "if(k=1,random(-1,1)*random([1,3,4,5]),if (k=2,random(-1,1)*random([1,2,4,5]),if(k=3,random(-1,1)*random([1,2,3,5]),if(k=5,random(-1,1)*random([1,2,3,4,5,7]),random(-1,1)*random([1,2,3,4,5,7])))))", "description": "", "templateType": "anything", "can_override": false}, "Asol": {"name": "Asol", "group": "Ungrouped variables", "definition": "(m*a^2-n*a+k)/(a^2-b*a+c)", "description": "", "templateType": "anything", "can_override": false}, "Bsol": {"name": "Bsol", "group": "Ungrouped variables", "definition": "(m*c-m*b*a+n*a-k)/(a^2-b*a+c)", "description": "", "templateType": "anything", "can_override": false}, "Csol": {"name": "Csol", "group": "Ungrouped variables", "definition": "(k*(a-b)-m*a*c+n*c)/(a^2-a*b+c)", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(Asol=round(Asol) and Bsol=round(Bsol),'{sol1}',if(simp2=1,'{sol2}','{sol3}'))", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\simplify{{Asol}/(x+{a})+({Bsol}x+{Csol})/(x^2+{b}x+{c})}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "sol2": {"name": "sol2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\simplify[all,fractionNumbers]{{m*a^2-n*a+k}/({a^2-a*b+c}(x+{a}))+({m*c-m*b*a+n*a-k}x+{k*(a-b)-m*a*c+n*c})/({a^2-a*b+c}(x^2+{b}x+{c}))}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "pairs[index][0]", "description": "", "templateType": "anything", "can_override": false}, "simp1": {"name": "simp1", "group": "Ungrouped variables", "definition": "gcd(k*(a-b)-m*a*c+n*c,m*c-m*b*a+n*a-k)", "description": "", "templateType": "anything", "can_override": false}, "simp2": {"name": "simp2", "group": "Ungrouped variables", "definition": "gcd(simp1,a^2-a*b+c)", "description": "", "templateType": "anything", "can_override": false}, "sol3": {"name": "sol3", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify{({m}x^2+{n}x+{k})/((x+{a})(x^2+{b}x+{c}))} = \\\\simplify[all,fractionNumbers]{{m*a^2-n*a+k}/({a^2-a*b+c}(x+{a}))+({(m*c-m*b*a+n*a-k)/simp2}x+{(k*(a-b)-m*a*c+n*c)/simp2})/({(a^2-a*b+c)/simp2}(x^2+{b}x+{c}))}.\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random([1,2,3,5,7])", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "pairs": {"name": "pairs", "group": "Ungrouped variables", "definition": "[[1,random(-1,1)*random([1,3,4,5])],[2,random(-1,1)*random([1,2,4,5])],[3,random(-1,1)*random([1,2,3,5])],[5,random(-1,1)*random([1,2,3,4,5,7])],[7,random(-1,1)*random([1,2,3,4,5,7])]]", "description": "", "templateType": "anything", "can_override": false}, "index": {"name": "index", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a^2-a*b+c>0 or a^2-a*b+c<0", "maxRuns": 100}, "ungrouped_variables": ["a", "pairs", "index", "b", "c", "m", "k", "n", "Asol", "Bsol", "Csol", "check", "sol1", "sol2", "sol3", "simp1", "simp2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(m*a^2-n*a+k)}/({a^2-a*b+c}(x+{a}))+({(m*c-m*b*a+n*a-k)/simp2}x+{(k*(a-b)-m*a*c+n*c)/simp2})/({(a^2-a*b+c)/simp2}(x^2+{b}x+{c}))", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`! (((`+-$n`?*x^2+`+-$n`?*x+`+-$n)/((x+`+-$n)(x^2+`+-$n*x+`+-$n))))", "partialCredit": 0, "message": "", "nameToCompare": "", "warningTime": "submission"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AF01 Sigma Notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "<p>Basic calculation from a sum given in Sigma notation.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate:</p>\n<p>\\[\\displaystyle{\\Sigma_{n=1}^3} \\var{b}n.\\]</p>\n<p></p>", "advice": "<p>The sigma notation $\\displaystyle<span class=\"enlighter-k0\">\\sum_</span><span class=\"enlighter-g1\">{</span><span class=\"enlighter-text\">n=</span><span class=\"enlighter-n1\">1</span><span class=\"enlighter-g1\">}</span><span class=\"enlighter-text\">^</span><span class=\"enlighter-g1\">{</span><span class=\"enlighter-k0\">3</span><span class=\"enlighter-g1\">}\\var{b}n$ is asking us to find the </span>sum of the first three terms of the sequence $\\var{b}n$<span class=\"enlighter-g1\">.&nbsp;</span></p>\n<p>\\[\\begin{split}\\Sigma_{n=1}^3 \\var{b}n &amp;\\, = (\\var{b}\\times 1) + (\\var{b}\\times 2) + (\\var{b}\\times 3) \\\\ &amp;\\, = \\var{b1} + \\var{b2} + \\var{b3} \\\\ &amp;\\, = \\var{sum}.\\end{split}\\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Sigma\" target=\"_blank\" rel=\"noopener\">Use this link to find resources to help you revise sigma notation.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "b*1", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "b*2", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "b*3", "description": "", "templateType": "anything", "can_override": false}, "sum": {"name": "sum", "group": "Ungrouped variables", "definition": "b1+b2+b3", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["b", "b1", "b2", "b3", "sum"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{sum}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "GA09 sec/cosec/cot", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Match the graphs to the functions. No randomisation. Multiple choice.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<p><span style=\"font-weight: 400;\">This is about knowledge of graphs. Generally with trigonometric graphs it is best to start with making sure you know and understand the graphs of the functionts $\\sin(x)$, $\\cos(x)$ and $\\tan(x)$. From there you can use knowledge of where they are zero to work out the position of the asymptotes in the graphs of $\\sec(x)$, $\\text{cosec}(x)$ and $\\cot(x)$. However, you still need really to be able to recall the shape of each graph for some purposes and be confident about where the zeros and turning points are.</span></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#TrigFun\"><span style=\"font-weight: 400;\">Use this link to find some resources to help you familiarise yourself with these graphs.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Match the graph to its function.</p>", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\sec(x)$", "$\\text{cosec}(x)$", "$\\cot(x)$"], "matrix": [["1", "0", 0], [0, "1", 0], ["0", 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["{geogebra_applet('https://www.geogebra.org/m/h9d8hzna')}", "{geogebra_applet('https://www.geogebra.org/m/kqnrbjzy')}", "{geogebra_applet('https://www.geogebra.org/m/xm44vcwe')}"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "GA08 Trigonometric Identities 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Oliver Spenceley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23557/"}], "tags": [], "metadata": {"description": "<p>Rewriting a trigonometric expression of the form $A\\cos(\\theta)\\pm B\\sin(\\theta)$ to either $R\\sin(\\theta+\\alpha)$ or $R\\cos(\\theta+\\alpha)$ by calculating $R$ and $\\alpha$.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If</p>\n<p>{question}</p>\n<p>find the values for $R$ and $\\alpha$, given $R&gt;0$ and $0&lt;\\alpha&lt;\\frac{\\pi}{2}$.</p>", "advice": "<p></p>\n<p>{answer}</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Trig\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(1..5 except A)", "description": "", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "sqrt(A^2+B^2)", "description": "", "templateType": "anything", "can_override": false}, "Rround": {"name": "Rround", "group": "Ungrouped variables", "definition": "precround(R,2)", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "arctan(B/A)", "description": "", "templateType": "anything", "can_override": false}, "Rsol": {"name": "Rsol", "group": "Ungrouped variables", "definition": "if(R=round(R),'{Rsol1}','{Rsol2}')", "description": "", "templateType": "anything", "can_override": false}, "Rsol1": {"name": "Rsol1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split} R^2\\\\cos^2(\\\\alpha) + R^2 \\\\sin^2(\\\\alpha) &amp;\\\\,= \\\\var{A}^2+\\\\var{B}^2 \\\\\\\\ R^2 (\\\\cos^2(\\\\alpha) +\\\\sin^2(\\\\alpha)) &amp;\\\\,= \\\\var{A^2+B^2} \\\\\\\\ R &amp;\\\\,= \\\\var{R}. \\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "Rsol2": {"name": "Rsol2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\begin{split} R^2\\\\cos^2(\\\\alpha) + R^2 \\\\sin^2(\\\\alpha) &amp;\\\\,= \\\\var{A}^2+\\\\var{B}^2 \\\\\\\\ R^2 (\\\\cos^2(\\\\alpha) +\\\\sin^2(\\\\alpha)) &amp;\\\\,= \\\\var{A^2+B^2} \\\\\\\\ R &amp;\\\\,=&nbsp;\\\\sqrt{\\\\var{A^2+B^2}}\\\\\\\\ &amp;\\\\,=\\\\var{Rround} \\\\text{ (2 d.p.)}.&nbsp;\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "alpharound": {"name": "alpharound", "group": "Ungrouped variables", "definition": "precround(alpha,2)", "description": "", "templateType": "anything", "can_override": false}, "question": {"name": "question", "group": "Ungrouped variables", "definition": "if(Q=1,'{q1}','{q2}')", "description": "", "templateType": "anything", "can_override": false}, "Q": {"name": "Q", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "templateType": "anything", "can_override": false}, "sign": {"name": "sign", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify[unitFactor]{{A}sin(theta)+{sign*B}cos(theta)} = \\\\simplify[unitFactor]{R sin (theta+{sign}*alpha)},\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "\"<p>\\\\[ \\\\simplify[unitFactor]{{A}cos(theta)-{sign*B}sin(theta)} = \\\\simplify[unitFactor]{R cos (theta+{sign}*alpha)},\\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "\"<p>To find $R$ and $\\\\alpha$ we want to first rewrite our equation using the double-angle formula, $\\\\simplify[unitFactor]{sin(a+{sign}*b)=sin(a)cos(b)+{sign}*sin(b)cos(a)}$:</p>\\n<p>\\\\[ \\\\begin{split}\\\\simplify[unitFactor]{{A}sin(theta)+{sign*B}cos(theta)} &amp;\\\\,= \\\\simplify{R sin(theta+{sign}*alpha)} \\\\\\\\ &amp;\\\\,= \\\\simplify{R(sin(theta)cos(alpha) + {sign}*sin(alpha)cos(theta))} \\\\\\\\ &amp;\\\\,= \\\\simplify{Rsin(theta)cos(alpha) + {sign}*R sin(alpha)cos(theta)}. \\\\end{split} \\\\]</p>\\n<p>By comparing the coefficients of $\\\\sin(\\\\theta)$ and $\\\\cos(\\\\theta)$, we find that</p>\\n<p>\\\\[ R\\\\cos(\\\\alpha) = \\\\var{A},\\\\quad \\\\text{and} \\\\quad R\\\\sin(\\\\alpha) = \\\\var{B}. \\\\]</p>\\n<p>To calculate $R$, we want to square these results and add them together, allowing us to make use of $\\\\sin^2(\\\\alpha)+\\\\cos^2(\\\\alpha) = 1$:</p>\\n<p>{Rsol}</p>\\n<p>Similarly, to find $\\\\alpha$ we can divide $R\\\\sin(\\\\alpha) = \\\\var{B}$ by $R\\\\cos(\\\\alpha) = \\\\var{A}$, and use the identity $\\\\tan(\\\\alpha) = \\\\frac{\\\\sin(\\\\alpha)}{\\\\cos(\\\\alpha)}$:</p>\\n<p>\\\\[ \\\\frac{R\\\\sin(\\\\alpha)}{R\\\\cos(\\\\alpha)} = \\\\frac{\\\\var{B}}{\\\\var{A}} \\\\implies \\\\tan(\\\\alpha) = \\\\simplify[fractionNumbers]{{B/A}}.\\\\]</p>\\n<p>Therefore, \\\\[ \\\\begin{split} \\\\alpha &amp;\\\\,= \\\\tan^{-1}\\\\left(\\\\simplify[fractionNumbers]{{B/A}}\\\\right) \\\\\\\\ &amp;\\\\,= \\\\var{alpharound} \\\\text{ (2 d.p.)}. \\\\end{split} \\\\]</p>\\n<p></p>\"", "description": "", "templateType": "long string", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "\"<p>To find $R$ and $\\\\alpha$ we want to first rewrite our equation using the double-angle formula, $\\\\simplify{cos(a+{sign}*b)=cos(a)cos(b)-{sign}*sin(a)sin(b)}$:</p>\\n<p>\\\\[ \\\\begin{split}\\\\simplify[unitFactor]{{A}cos(theta)-{sign*B}sin(theta)} &amp;\\\\,= \\\\simplify[unitFactor]{R cos (theta + {sign}*alpha)} \\\\\\\\ &amp;\\\\,= \\\\simplify{R(cos(theta)cos(alpha) - {sign}*sin(theta)sin(alpha))} \\\\\\\\ &amp;\\\\,= \\\\simplify{Rcos(theta)cos(alpha) - {sign}*R sin(theta)sin(alpha)}. \\\\end{split} \\\\]</p>\\n<p>By comparing the coefficients of $\\\\cos(\\\\theta)$ and $\\\\sin(\\\\theta)$, we find that</p>\\n<p>\\\\[ R\\\\cos(\\\\alpha) = \\\\var{A},\\\\quad \\\\text{and} \\\\quad R\\\\sin(\\\\alpha) = \\\\var{B}. \\\\]</p>\\n<p>To calculate $R$, we want to square these results and add them together, allowing us to make use of $\\\\sin^2(\\\\alpha)+\\\\cos^2(\\\\alpha) = 1$:</p>\\n<p>{Rsol}</p>\\n<p>Similarly, to find $\\\\alpha$ we can divide $R\\\\sin(\\\\alpha) = \\\\var{B}$ by $R\\\\cos(\\\\alpha) = \\\\var{A}$, and use the identity $\\\\tan(\\\\alpha) = \\\\frac{\\\\sin(\\\\alpha)}{\\\\cos(\\\\alpha)}$:</p>\\n<p>\\\\[ \\\\frac{R\\\\sin(\\\\alpha)}{R\\\\cos(\\\\alpha)} = \\\\frac{\\\\var{B}}{\\\\var{A}} \\\\implies \\\\tan(\\\\alpha) = \\\\simplify[fractionNumbers]{{B/A}}.\\\\]</p>\\n<p>Therefore, \\\\[ \\\\begin{split} \\\\alpha &amp;\\\\,= \\\\tan^{-1}\\\\left(\\\\simplify[fractionNumbers]{{B/A}}\\\\right) \\\\\\\\ &amp;\\\\,= \\\\var{alpharound} \\\\text{ (2 d.p.)}. \\\\end{split} \\\\]</p>\\n<p></p>\"", "description": "", "templateType": "long string", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "if(Q=1,'{a1}','{a2}')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["Q", "A", "B", "sign", "R", "Rround", "alpha", "alpharound", "Rsol", "Rsol1", "Rsol2", "question", "q1", "q2", "answer", "a1", "a2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$R=$[[0]]</p>\n<p>$\\alpha=$[[1]]</p>\n<p><em>(Give your answers to 2 decimal places where necessary.)</em></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{Rround}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{alpharound}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AF04 Graphs of trig functions (sin, cos, tan)", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Match the relevant graph (sin, cos, tan) with its equation.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "<p>This is about core knowledge of graphs. You should know the shapes of the fundamental trig graphs, if you don't familiarize yourself with them from the resources linked below. In this setting the $x$-axis is given with a scale in radians but you might also find some where it is given in degrees. You should also be aware of the difference between those two different units of angles.</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#TrigFun\"><span style=\"font-weight: 400;\">Use this link to find some resources to help you familiarise yourself with these graphs.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Match the graph to its function.</p>", "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["$\\sin(x)$", "$\\cos(x)$", "$\\tan(x)$"], "matrix": [["1", 0, 0], [0, "1", 0], [0, 0, "1"]], "layout": {"type": "all", "expression": ""}, "answers": ["{geogebra_applet('https://www.geogebra.org/m/ntqvuwqr')}", "{geogebra_applet('https://www.geogebra.org/m/fsqmnhsc')}", "{geogebra_applet('https://www.geogebra.org/m/yg6f9eqz')}"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AF05 Function notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "<p>Evaluating a linear function for a given value of $x$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given $f(x)=\\simplify{{m}x+{c}}$, find $f(\\var{n})$.</p>", "advice": "<p>If $f(x)=\\simplify{{m}x+{c}}$, to find $f(\\var{n})$ we need to evaluate $f(x)$ when $x=\\var{n}$:</p>\n<p>\\[ \\begin{split} f(\\var{n}) &amp;\\,= \\simplify[alwaysTimes]{{m}({n})+{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{m*n}+{c}} \\\\ &amp;\\,= \\simplify{{m*n+c}}. \\end{split} \\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Functions\" target=\"_blank\" rel=\"noopener\">Use this link to find resources to help you revise function notation.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,m])", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,1])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "c", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$f(\\var{n})=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{m*n+c}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AF07 Inverse functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "<p>Finding the inverse of a function of the form $f(x)=\\frac{mx+c}{x+a},\\,x\\neq-a$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>If $f(x)=\\simplify{({m}x+{c})/(x+{a})},\\,x\\neq \\simplify{{-a}}$, find the inverse function, $f^{-1}(x)$.</p>", "advice": "<p>To find $f^{-1}x$, it can help to first set $f(x)$ to a different variable, which we will call $y$:</p>\n<p>\\[ y = f(x) = \\simplify{({m}x+{c})/(x+{a})}\\]</p>\n<p>Since the function $f(x)$ takes us from $x$ to $y$, the inverse function $f^{-1}$ will take us from $y$ to $x$. So to obtain $f^{-1}$, we want to rearrange $y=\\simplify{({m}x+{c})/(x+{a})}$ so that it is $x$ as a function of $y$:</p>\n<p>\\[ \\begin{split} y &amp;\\,= \\simplify{({m}x+{c})/(x+{a})} \\\\\\\\ \\simplify{(x+{a})y} &amp;\\,= \\simplify{{m}x+{c}} \\\\\\\\ \\simplify{x*y+{a}y} &amp;\\,= \\simplify{{m}x+{c}} \\\\\\\\ \\simplify{x*y - {m}x} &amp;\\,= \\simplify{{c}- {a}y}&nbsp; &nbsp;\\\\&nbsp; \\\\ \\simplify{x(y-{m})} &amp;\\,= \\simplify{{c}-{a}y} \\\\\\\\ x&amp;\\,= \\simplify{({c}-{a}y)/(y-{m})}. \\end{split} \\]</p>\n<p>Hence, $f^{-1}(y) =\\simplify{({c}-{a}y)/(y-{m})}$, and therefore \\[ f^{-1}(x) =\\simplify{({c}-{a}x)/(x-{m})}.\\]</p>\n<p>(<strong>Note:</strong>&nbsp;The inverse is valid for all values of $x$&nbsp;<em>except&nbsp;</em>$x=\\var{m}$, since this would make the denominator equal to 0.)</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Functions\" target=\"_blank\" rel=\"noopener\">Use this link to find resources to help you revise how to find the inverse of functions.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-8..8)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "abs(m)-abs(c)>0 or abs(m)-abs(c)<0", "maxRuns": 100}, "ungrouped_variables": ["m", "c", "a"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$f^{-1}(x)=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({c}-{a}x)/(x-{m})", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AL02 Logs - rules 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "<p>Solving $a\\log(x)+\\log(b)=\\log(c)$ for $x$, where $a$, $b$ and $c$ are positive integers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve for $x$:</p>\n<p>\\[ \\var{a}\\log(x)+\\log(\\var{b})=\\log(\\var{c}). \\]</p>", "advice": "<p>To solve $\\var{a}\\log(x)+\\log(\\var{b})=\\log(\\var{c})$ for $x$, we want to use the following logarithm rules:</p>\n<ul>\n<li>$\\log(a^n) = n\\log(a)$;</li>\n<li>$\\log(a)+\\log(b) = \\log(ab)$.</li>\n</ul>\n<p>Hence,&nbsp;</p>\n<p>\\[ \\begin{split} \\var{a}\\log(x)+\\log(\\var{b}) &amp;\\,=\\log(\\var{c}) \\\\ \\log(x^\\var{a})+\\log(\\var{b}) &amp;\\,= \\log(\\var{c}) \\\\ \\log(\\var{b}x^\\var{a}) &amp;\\,= \\log(\\var{c}). \\end{split} \\]</p>\n<p>If $\\log(a)=\\log(b)$ then this implies $a=b$. Therefore,</p>\n<p>\\[&nbsp; \\begin{split} \\var{b}x^\\var{a} &amp;\\,=\\var{c} \\\\ x^\\var{a} &amp;\\,= \\simplify[fractionNumbers]{{c/b}} \\\\ x &amp;\\,= \\simplify[fractionNumbers]{({c/b})^(1/{a})} \\\\ x &amp;\\,= \\var{sol} \\text{ (2 d.p.)}\\end{split} \\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Logs\" target=\"_blank\" rel=\"noopener\">Use this link to find rsources to help you revise how the rules of logarithms to help you solve logarithmic equations.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "precround((c/b)^(1/a),2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..40 except [b,b^(a+1)])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "sol"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$x=$ [[0]] (<em>Give you answer to 2 decimal places where necessary)</em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{sol}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "AL03 Logs - Solving equations using logs", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Megan Oliver", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23526/"}], "tags": [], "metadata": {"description": "<p>Solving an equation of the form $a^x=b$ using logarithms to find $x$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve for $x$:</p>\n<p>\\[ \\var{a}^x = \\var{b} \\,. \\]</p>", "advice": "<p>To solve $\\var{a}^x = \\var{b}$ for $x$, since $x$ is the exponent we want to make use of the following logarithm rule:</p>\n<ul>\n<li>$\\log(a^n) = n \\log(a)$.</li>\n</ul>\n<p></p>\n<p>By taking the logarithm of each side and applying the above rule:</p>\n<p>\\[ \\begin{split}\\var{a}^x &amp;\\,= \\var{b} \\\\ \\log_{10}(\\var{a}^x) &amp; \\,= \\log_{10}(\\var{b})\\\\ x \\log_{10}(\\var{a}) &amp;\\,= \\log_{10}(\\var{b}) \\\\\\\\ x&amp;\\,=\\simplify{log({b})/log({a})} \\\\\\\\ x &amp;\\,= \\var{sol} \\text{ (2 d.p.)}.&nbsp; \\end{split} \\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Logs\" target=\"_blank\" rel=\"noopener\">Use this link to find resources to help you revise how logarithms.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9 except [a,a^2,a^3])", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "precround(log(b)/log(a),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "sol"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$x=$ [[0]] (<em>Give you answer to 2 decimal places where necessary)</em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{sol}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD02 Differentiating polynomials 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Differentiate a polynomial expression involving coefficients and, negative and fractional indices.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of $y=\\simplify[unitFactor, fractionNumbers]{{a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3}}$.</p>\n<p></p>\n<p></p>", "advice": "<p>From the Table of Derivatives we see that a function of the form \\[ f(x)=kx^n \\] has a derivative \\[ \\frac{df}{dx} = knx^{n-1}. \\]</p>\n<p>Additionally, the&nbsp;derivative of the sum or difference of two or more functions is equal to the sum or difference of the derivatives of each function: \\[ \\frac{d}{dx}(f(x)\\pm g(x)) = \\frac{df}{dx} \\pm \\frac{dg}{dx}.\\]</p>\n<p></p>\n<p>{advice}</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Diff\" target=\"_blank\" rel=\"noopener\" title=\"Link to differentiation resources\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a_1": {"name": "a_1", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "b_1": {"name": "b_1", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "a_2": {"name": "a_2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b_2": {"name": "b_2", "group": "Ungrouped variables", "definition": "random(-10..-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "a_3": {"name": "a_3", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "advice": {"name": "advice", "group": "Ungrouped variables", "definition": "if(a_2>0 and a_3>0,'{solutiona}',{advice2})", "description": "", "templateType": "anything", "can_override": false}, "solutiona": {"name": "solutiona", "group": "Ungrouped variables", "definition": "\"<p>So, for the function \\\\[y=\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\] the derivative&nbsp; is \\\\begin{split}\\\\frac{dy}{dx} &amp;= (\\\\var[fractionNumbers]{a_1}\\\\times\\\\var[fractionNumbers]{b_1})x^{\\\\var[fractionNumbers]{b_1}-1} +(\\\\var[fractionNumbers]{a_2}\\\\times\\\\var[fractionNumbers]{b_2})x^{\\\\var[fractionNumbers]{b_2}-1} +(\\\\var[fractionNumbers]{a_3}\\\\times\\\\var[fractionNumbers]{b_3})x^{\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\ \\\\\\\\&amp;= \\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\end{split}</p>\"", "description": "", "templateType": "long string", "can_override": false}, "solutionb": {"name": "solutionb", "group": "Ungrouped variables", "definition": "\"<p>So, for the function \\\\[y=\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\] the derivative&nbsp; is \\\\begin{split}\\\\frac{dy}{dx} &amp;= (\\\\var[fractionNumbers]{a_1}\\\\times\\\\var[fractionNumbers]{b_1})x^{\\\\var[fractionNumbers]{b_1}-1} -(\\\\var[fractionNumbers]{abs(a_2)}\\\\times\\\\var[fractionNumbers]{b_2})x^{\\\\var[fractionNumbers]{b_2}-1} +(\\\\var[fractionNumbers]{a_3}\\\\times\\\\var[fractionNumbers]{b_3})x^{\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\ \\\\\\\\&amp;= \\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\end{split}</p>\"", "description": "", "templateType": "long string", "can_override": false}, "solutionc": {"name": "solutionc", "group": "Ungrouped variables", "definition": "\"<p>So, for the function \\\\[y=\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\] the derivative&nbsp; is \\\\begin{split}\\\\frac{dy}{dx} &amp;= (\\\\var[fractionNumbers]{a_1}\\\\times\\\\var[fractionNumbers]{b_1})x^{\\\\var[fractionNumbers]{b_1}-1} +(\\\\var[fractionNumbers]{a_2}\\\\times\\\\var[fractionNumbers]{b_2})x^{\\\\var[fractionNumbers]{b_2}-1} -(\\\\var[fractionNumbers]{abs(a_3)}\\\\times\\\\var[fractionNumbers]{b_3})x^{\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\ \\\\\\\\&amp;= \\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\end{split}</p>\"", "description": "", "templateType": "long string", "can_override": false}, "solutiond": {"name": "solutiond", "group": "Ungrouped variables", "definition": "\"<p>So, for the function \\\\[y=\\\\simplify[all, fractionNumbers]{{a_1}x^{b_1}+{a_2}x^{b_2}+{a_3}x^{b_3}} \\\\] the derivative&nbsp; is \\\\begin{split}\\\\frac{dy}{dx} &amp;= (\\\\var[fractionNumbers]{a_1}\\\\times\\\\var[fractionNumbers]{b_1})x^{\\\\var[fractionNumbers]{b_1}-1} -(\\\\var[fractionNumbers]{abs(a_2)}\\\\times\\\\var[fractionNumbers]{b_2})x^{\\\\var[fractionNumbers]{b_2}-1} -(\\\\var[fractionNumbers]{abs(a_3)}\\\\times\\\\var[fractionNumbers]{b_3})x^{\\\\var[fractionNumbers]{b_3}-1},\\\\\\\\ \\\\\\\\&amp;= \\\\simplify[all, fractionNumbers]{{a_1*b_1}x^{b_1-1} +{a_2*b_2}x^{b_2-1} +{a_3*b_3}x^{b_3-1}}.\\\\end{split}</p>\"", "description": "", "templateType": "long string", "can_override": false}, "advice2": {"name": "advice2", "group": "Ungrouped variables", "definition": "if(a_2<0 and a_3>0,'{solutionb}',{advice3})", "description": "", "templateType": "anything", "can_override": false}, "advice3": {"name": "advice3", "group": "Ungrouped variables", "definition": "if(a_2>0 and a_3<0,'{solutionc}','{solutiond}')", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "b_3": {"name": "b_3", "group": "Ungrouped variables", "definition": "b/c", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(b,c)=1", "maxRuns": "100"}, "ungrouped_variables": ["a_1", "a_2", "a_3", "b_1", "b_2", "b_3", "b", "c", "advice", "advice2", "advice3", "solutiona", "solutionb", "solutionc", "solutiond"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a_1*b_1}x^{{b_1}-1}+{a_2*b_2}x^{{b_2}-1}+{a_3*b_3}x^{{b_3}-1}", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CI02 Definite integration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Calculating the definite integral $\\int_{n_1}^{n_2}a_1x^{b_1}+a_2x^{b_2}+a_3x^{b_3} dx$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Evaluate \\[ \\int_{\\var{n_1}}^{\\var{n_2}}\\simplify[unitFactor, unitPower, fractionNumbers]{{a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3}} \\,dx.\\]</p>\n<p></p>", "advice": "<p>Integrating a function of the form&nbsp;&nbsp;\\[ f(x)=x^n \\] has the integral \\[ \\int_a^b x^n dx&nbsp; =&nbsp; \\left[\\frac{x^{n+1}}{n+1}\\right]_a^b,\\]</p>\n<p>and \\[\\int_a^b kf(x) dx = k \\int_a^b f(x) dx.\\]</p>\n<p>Additionally, the&nbsp;integral of the sum or difference of two or more functions is equal to the sum or difference of the&nbsp;integrals of each function: \\[ \\int(f(x)\\pm g(x))\\, dx = \\int f(x)\\, dx&nbsp; \\pm \\int g(x) \\, dx.\\]</p>\n<p></p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split}\\simplify[unitFactor,unitPower]{defint({a_1}*x^{b_1}+{a_2}*x^{b_2}+{a_3}*x^{b_3},x,{n_1},{n_2})} &amp;\\,= \\simplify{{a_1}defint(x^{b_1},x,{n_1},{n_2})+{a_2}defint(x^{b_2},x,{n_1},{n_2})+{a_3}defint(x^{b_3},x,{n_1},{n_2})} \\\\ &amp;\\,= \\left[\\simplify[all,fractionNumbers]{{a_1}x^{b_1+1}/{b_1+1}+{a_2}x^{b_2+1}/{b_2+1}+{a_3}x^{b_3+1}/{b_3+1}}\\right]_\\var{n_1}^\\var{n_2} \\\\ &amp;\\,= \\left[\\simplify[all,fractionNumbers,!collectNumbers]{{a_1*n_2^(b_1+1)}/{b_1+1}+{a_2*n_2^(b_2+1)}/{b_2+1}+{a_3*n_2^(b_3+1)}/{b_3+1}}\\right] -\\left[\\simplify[all,fractionNumbers,!collectNumbers]{{a_1*n_1^(b_1+1)}/{b_1+1}+{a_2*n_1^(b_2+1)}/{b_2+1}+{a_3*n_1^(b_3+1)}/{b_3+1}}\\right]&nbsp;\\\\ &amp;\\,= \\simplify[!collectNumbers]{{eval2a}-{eval1a}} \\\\ &amp;\\,=\\var{sol1} \\end{split} \\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Int\"><span style=\"font-weight: 400;\">Use this link to find some resources on areas under curves which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a_1": {"name": "a_1", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b_1": {"name": "b_1", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything", "can_override": false}, "a_2": {"name": "a_2", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b_2": {"name": "b_2", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "anything", "can_override": false}, "b_3": {"name": "b_3", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything", "can_override": false}, "a_3": {"name": "a_3", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n_1": {"name": "n_1", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "n_2": {"name": "n_2", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "eval2": {"name": "eval2", "group": "Ungrouped variables", "definition": "a_1*n_2^(b_1+1)/(b_1+1)+a_2*n_2^(b_2+1)/(b_2+1)+a_3*n_2^(b_3+1)/(b_3+1)", "description": "", "templateType": "anything", "can_override": false}, "eval1": {"name": "eval1", "group": "Ungrouped variables", "definition": "a_1*n_1^(b_1+1)/(b_1+1)+a_2*n_1^(b_2+1)/(b_2+1)+a_3*n_1^(b_3+1)/(b_3+1)", "description": "", "templateType": "anything", "can_override": false}, "eval2a": {"name": "eval2a", "group": "Ungrouped variables", "definition": "precround(eval2,3)", "description": "", "templateType": "anything", "can_override": false}, "eval1a": {"name": "eval1a", "group": "Ungrouped variables", "definition": "precround(eval1,3)", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "eval2-eval1", "description": "", "templateType": "anything", "can_override": false}, "sol1": {"name": "sol1", "group": "Ungrouped variables", "definition": "precround(sol,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "b_1>b_2 and b_2>b_3 and n_2>n_1", "maxRuns": "100"}, "ungrouped_variables": ["a_1", "a_2", "a_3", "b_1", "b_2", "b_3", "n_1", "n_2", "eval2", "eval1", "eval2a", "eval1a", "sol", "sol1"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]&nbsp;<em>(Give answers to 2 decimal places where necessary)</em></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{sol1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD08 Finding turning points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Finding the stationary points of a cubic equation and determining their nature.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given the function \\[ \\simplify{y={a}x^3+{b}x^2+{c}x+{d}} ,\\] find its stationary points and determine their nature.</p>", "advice": "<p>To find the stationary points of the function, we must solve $\\tfrac{dy}{dx}=0$ for $x$. For the function $\\simplify{y={a}x^3+{b}x^2+{c}x+{d}}$,&nbsp;</p>\n<p>\\[ \\frac{dy}{dx} = \\simplify{{3a}x^2+{2b}x+{c}}. \\]</p>\n<p>Setting $\\frac{dy}{dx}=0$ and solving for $x$:</p>\n<p>\\[ \\simplify{{3a}x^2+{2b}x+{c}} =0&nbsp; \\\\ \\\\ \\implies x=\\var{solx1dp} \\var{x1} \\text{ and } x=\\var{solx2dp} \\var{x2}. \\]</p>\n<p>Hence, the function has two stationary points at $x=\\var{solx1dp}$ and $x=\\var{solx2dp}$. To find the corresponding $y$-coordinates, we want to plug these values back into the initial equation.</p>\n<p>When $x=\\var{solx1dp}$,</p>\n<p>\\[ \\begin{split} y &amp;\\,= \\simplify[unitFactor,!cancelTerms]{{a}*({solx1dp})^3+{b}*({solx1dp})^2+{c}*({solx1dp})+{d}} \\\\ &amp;\\,=\\simplify{{soly1dp}} \\var{y1}. \\end{split} \\]</p>\n<p>When $x=\\var{solx2dp}$,&nbsp;</p>\n<p>\\[ \\begin{split} y &amp;\\,= \\simplify[unitFactor,!cancelTerms]{{a}*({solx2dp})^3+{b}*({solx2dp})^2+{c}*({solx2dp})+{d}} \\\\ &amp;\\,=\\simplify{{soly2dp}} \\var{y2}. \\end{split} \\]</p>\n<p>Therefore, the stationary points of $y=\\simplify{{a}x^3+{b}x^2+{c}x+{d}}$ are</p>\n<p>\\[ (\\simplify{{solx1dp}},\\, \\simplify{{soly1dp}}) \\, , \\,(\\simplify{{solx2dp}},\\, \\simplify{{soly2dp}}). \\]</p>\n<p>Finally, we need to determine the nature of the stationary points. To do this we want to calculate the second derivative of the initial function and then evaluate it for each $x$-value of the stationary points.&nbsp;</p>\n<p>Recall:</p>\n<ul>\n<li>If $\\tfrac{d^2y}{dx^2}&lt;0$ the stationary point is a <strong>maximum</strong>;</li>\n<li>If $\\tfrac{d^2y}{dx^2}&gt;0$ the stationary point is a&nbsp;<strong>minimum</strong>;</li>\n<li>If $\\tfrac{d^2y}{dx^2}=0$ the stationary point is a <strong>point of inflection</strong>.</li>\n</ul>\n<p>To calculate&nbsp;$\\tfrac{d^2y}{dx^2}$, we want to differentiate&nbsp;$\\tfrac{dy}{dx}$ again with respect to $x$:</p>\n<p>\\[ \\begin{split} &amp;\\frac{dy}{dx} = \\simplify{{3a}x^2+{2b}x+{c}}, \\\\ \\\\\\implies &amp;\\frac{d^2y}{dx^2} = \\simplify{{6a}x+{2b}}. \\end{split}\\]</p>\n<p>For $(\\simplify{{solx1dp}},\\, \\simplify{{soly1dp}})$, $\\frac{d^2y}{dx^2} = \\simplify{{check}}$, so it is a minimum.</p>\n<p>For $(\\simplify{{solx2dp}},\\, \\simplify{{soly2dp}})$, $\\frac{d^2y}{dx^2} = \\simplify{{check2}}$, so it is a maximum.</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/first-steps#Diff\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-7..7)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..5 except b)", "description": "", "templateType": "anything", "can_override": false}, "solx1": {"name": "solx1", "group": "Ungrouped variables", "definition": "(-2b+sqrt((2b)^2-12*a*c))/(6a)", "description": "", "templateType": "anything", "can_override": false}, "solx2": {"name": "solx2", "group": "Ungrouped variables", "definition": "(-2b-sqrt((2b)^2-12*a*c))/(6a)", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "precround(6a*solx1+2b,2)", "description": "", "templateType": "anything", "can_override": false}, "check2": {"name": "check2", "group": "Ungrouped variables", "definition": "precround(6a*solx2+2b,2)", "description": "", "templateType": "anything", "can_override": false}, "soly1": {"name": "soly1", "group": "Ungrouped variables", "definition": "a*(solx1)^3+b*(solx1)^2+c*solx1+d", "description": "", "templateType": "anything", "can_override": false}, "soly2": {"name": "soly2", "group": "Ungrouped variables", "definition": "a*(solx2)^3+b*(solx2)^2+c*solx2+d", "description": "", "templateType": "anything", "can_override": false}, "solx1dp": {"name": "solx1dp", "group": "Ungrouped variables", "definition": "precround(solx1,2)", "description": "", "templateType": "anything", "can_override": false}, "solx2dp": {"name": "solx2dp", "group": "Ungrouped variables", "definition": "precround(solx2,2)", "description": "", "templateType": "anything", "can_override": false}, "soly1dp": {"name": "soly1dp", "group": "Ungrouped variables", "definition": "precround(soly1,2)", "description": "", "templateType": "anything", "can_override": false}, "soly2dp": {"name": "soly2dp", "group": "Ungrouped variables", "definition": "precround(soly2,2)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "if(round(solx1)=solx1,'','(2 d.p.)')", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "if(round(solx2)=solx2,'','(2 d.p.)')", "description": "", "templateType": "anything", "can_override": false}, "y1": {"name": "y1", "group": "Ungrouped variables", "definition": "if(round(soly1)=soly1,'','(2 d.p.)')", "description": "", "templateType": "anything", "can_override": false}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "if(round(soly2)=soly2,'','(2 d.p.)')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "b^2>3*a*c", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c", "d", "solx1", "soly1", "solx2", "soly2", "check", "check2", "solx1dp", "solx2dp", "soly1dp", "soly2dp", "x1", "x2", "y1", "y2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>There is a minimum point at ([[0]], [[1]]) and a maximum point at ([[2]] , [[3]]).</p>\n<p><em>(Give the coordinates of the stationary points to 2 decimal places where necessary.)</em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{solx1dp}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{soly1dp}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{solx2dp}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{soly2dp}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD05 Differentiating Trig 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Find the derivative of a function of the form $y=a \\cos(bx+c)$ using a table of derivatives.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of $y=\\simplify[unitFactor]{{a}cos({b}x+{c})}.$</p>\n<p></p>\n<p></p>", "advice": "<p>From the Table of Derivatives we see that a function of the form \\[ f(x)=a \\cos(kx+c) \\] has a derivative \\[-ak \\sin (kx+c).\\]</p>\n<p>Therefore, the function&nbsp; \\[y=\\simplify[unitFactor]{{a}*cos({b}x+{c})}\\] has a derivative\\[ \\begin{split} \\frac{dy}{dx} &amp;=-(\\var{a}\\times \\var{b})\\sin(\\simplify[unitFactor]{{b}x+{c}})\\\\ &amp;= \\simplify[unitFactor]{{-a*b}sin({b}x+{c})}.\\end{split}\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{a*b}sin({b}x+{c})", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD06 Differentiating Trig 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Find the derivative of a function of the form $y=a \\sin(bx+c)$ using a table of derivatives.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Using the Table of Derivatives, calculate the derivative of $y=\\simplify[unitFactor]{{a}sin({b}x+{c})}.$</p>\n<p></p>\n<p></p>", "advice": "<p>From the Table of Derivatives we see that a function of the form \\[ f(x)=a \\sin(kx+c) \\] has a derivative \\[ak \\cos (kx+c).\\]</p>\n<p>Therefore, the function&nbsp; \\[y=\\simplify[unitFactor]{{a}*sin({b}x+{c})}\\] has a derivative\\[ \\begin{split} \\frac{dy}{dx} &amp;=(\\var{a}\\times \\var{b})\\cos(\\simplify[unitFactor]{{b}x+{c}})\\\\ &amp;= \\simplify[unitFactor]{{a*b}cos({b}x+{c})}.\\end{split}\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*b}cos({b}x+{c})", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD07 Differentiating with Trig 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}], "tags": [], "metadata": {"description": "<p>Find the derivative of a function of the form $y=a \\tan(bx+c)$ using a table of derivatives.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Using the Table of Derivatives, calculate the derivative of $y=\\simplify[unitFactor]{{a}tan({b}x+{c})}.$</p>\n<p></p>\n<p></p>", "advice": "<p>From the Table of Derivatives we see that a function of the form \\[ f(x)=a \\tan(kx+c) \\] has a derivative \\[ak \\sec^2(kx+c).\\]</p>\n<p>Therefore, the function&nbsp; \\[y=\\simplify[unitFactor]{{a}*tan({b}x+{c})}\\] has a derivative\\[ \\begin{split} \\frac{dy}{dx} &amp;=(\\var{a}\\times \\var{b})\\sec^2(\\simplify[unitFactor]{{b}x+{c}})\\\\ &amp;= \\simplify[unitFactor]{{a*b}}\\sec^2(\\simplify[unitFactor]{{b}x+{c}}).\\end{split}\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-15..15)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "{a*b}sec^2({b}x+{c})", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "sec", "value": ""}, {"name": "x", "value": ""}]}], "answer": "{a*b}sec({b}x+{c})^2", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD04 Differentiating with Exponentials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Calculating the derivative of an exponential function of the form $ae^{bx}$, using a table of derivatives.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the derivative of $y=\\simplify[all]{{a}*e^({b}x)}.$</p>", "advice": "<p>From the Table of Derivatives we see that a function of the form \\[ f(x)=a e^{kx} \\] has a derivative \\[ak e^{kx}.\\]</p>\n<p>Therefore, the function&nbsp; \\[y=\\simplify[unitFactor]{{a}*e^({b}x)}\\] has a derivative\\[ \\begin{split} \\frac{dy}{dx} &amp;=(\\var{a}\\times \\var{b})e^{\\simplify[unitFactor]{{b}x}}\\\\ &amp;= \\simplify[unitFactor]{{a*b}e^({b}x)}.\\end{split}\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-20..20 except 0)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*b}e^({b}x)", "answerSimplification": "fractionNumbers, basic, unitFactor", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD09 Chain Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $\\sin(ax^m+bx^n)$ using the chain rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the derivative of $y=\\simplify[all]{sin({a}*x^{n}+{b}*x^{m})}$.</p>", "advice": "<p>If we have a function of the form $y=f(g(x))$, sometimes described as a function of a function, to calculate its derivative we need to use the chain rule:</p>\n<p>\\[ \\frac{dy}{dx} = \\frac{du}{dx} \\times \\frac{dy}{du}.\\]</p>\n<p></p>\n<p>This can be split up into steps:</p>\n<ul>\n<li>Let $u=g(x)$;</li>\n<li>Rewrite $y$ in terms of $u$, such that $y=f(u)$;</li>\n<li>Calculate $\\frac{du}{dx}$ and $\\frac{dy}{du}$;</li>\n<li>Write $\\frac{dy}{dx}$ as a product of&nbsp;$\\frac{du}{dx}$ and $\\frac{dy}{du}$;</li>\n<li>Make sure $\\frac{dy}{dx}$ is&nbsp;<em>only</em> in terms of $x$. Ensure any $u$ terms have been replaced using the initial substitution.</li>\n</ul>\n<p></p>\n<p>Following this process, we must first identify $g(x)$. Since the function is of the form $y=f(g(x))$, we are looking for the 'inner' function.</p>\n<p>So, for $y=\\simplify[all,fractionNumbers]{sin({a}*x^{n}+{b}*x^{m})}$, \\[g(x)=\\simplify[all, fractionNumbers, unitFactor]{{a}*x^{n}+{b}*x^{m}}.\\]</p>\n<p>If we now set $u=g(x)$, we can rewrite $y$ in terms of $u$ such that $y=f(u)$:</p>\n<p>\\[y=\\simplify[all, fractionNumbers,unitFactor]{sin(u)}.\\]</p>\n<p>Next, we calculate the two derivatives $\\frac{du}{dx}$ and $\\frac{dy}{du}$:</p>\n<p>\\[\\frac{du}{dx}=\\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}}, \\quad \\frac{dy}{du}=\\simplify[all, fractionNumbers, unitFactor]{cos(u)}.\\]</p>\n<p>Plugging these into the chain rule:</p>\n<p>\\[ \\begin{split} \\frac{dy}{dx} &amp;= \\frac{du}{dx} \\times \\frac{dy}{du}, \\\\&amp;=(\\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}}) \\times\\simplify[all, fractionNumbers, unitFactor]{cos(u)}. \\end{split} \\]</p>\n<p>Finally, we need to express $\\frac{dy}{dx}$&nbsp;<em>only&nbsp;</em>in terms of $x$, so we must replace the $u$ term using the initial substitution $u=\\simplify[all, fractionNumbers, unitFactor]{{a}*x^{n}+{b}*x^{m}}$:</p>\n<p>\\[ \\frac{dy}{dx} =(\\simplify[all,fractionNumbers]{{a*n}x^{n-1}+{b*m}x^{m-1}})\\simplify[all, fractionNumbers, unitFactor]{cos({a}*x^{n}+{b}*x^{m})}.\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\" target=\"_blank\" rel=\"noopener\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>\n<p></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "n>m", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n", "m"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\frac{dy}{dx}=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a*n}*x^{n-1}+{b*m}*x^{m-1})*cos({a}x^{n}+{b}x^{m})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD10 Product Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Calculating the derivative a function of the form $ax^n \\sin(bx)$ using the product rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of \\[ \\simplify{y={a}x^{n} sin({b}x)}. \\]</p>", "advice": "<p>If we have a function of the form $y=u(x)v(x)$, to calculate its derivative we need to use the product rule:</p>\n<p>\\[ \\dfrac{dy}{dx} = u(x) \\times \\dfrac{dv}{dx} + v(x) \\times\\dfrac{du}{dx}.\\]</p>\n<p>This can be split up into steps:</p>\n<ol>\n<li>Identify the functions $u(x)$ and $v(x)$;</li>\n<li>Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;</li>\n<li>Substitute these into the formula for the product rule to obtain an expression for $\\tfrac{dy}{dx}$;</li>\n<li>Simplify $\\tfrac{dy}{dx}$ where possible.</li>\n</ol>\n<p>Following this process, we must first identify $u(x)$ and $v(x)$.</p>\n<p>As \\[ \\simplify{y={a}x^{n} sin({b}x)}, \\]</p>\n<p>let \\[ u(x) = \\simplify{{a}x^{n}} \\quad \\text{and} \\quad v(x)=\\simplify{sin({b}x)}.\\]</p>\n<p>Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:</p>\n<p>\\[ \\dfrac{du}{dx} = \\simplify{{a*n}x^{n-1}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{b}cos({b}x)}.\\]</p>\n<p>Substituting these results into the product rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,= \\dfrac{du}{dx}\\times v(x) + u(x) \\times\\dfrac{dv}{dx} \\\\ &amp;\\,=\\simplify{{a*n}x^{n-1}} \\times\\simplify{sin({b}x)} +\\simplify{{a}x^{n}} \\times \\simplify{{b}cos({b}x)}.&nbsp; \\end{split}\\]</p>\n<p>Simplifying,</p>\n<p>\\[\\dfrac{dy}{dx} = \\simplify{{n*a}x^{n-1}sin({b}x) + {a*b}x^{n}cos({b}x)}. \\]</p>\n<p></p>\n<p><span style=\"font-weight: 400;\"><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\">Use this link to find some resources which will help you revise this topic</a></span></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\dfrac{dy}{dx}=$[[0]]&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{n*a}x^{n-1}sin({b}x) + {a*b}x^{n}cos({b}x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CD11 Quotient Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Calculating the derivative of a function of the form $\\frac{ax^n}{bx+c}$ using the quotient rule.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the derivative of \\[ \\simplify{y={a}x^{n}/({b}x+{c})}. \\]</p>", "advice": "<p>If we have a function of the form $y=\\tfrac{u(x)}{v(x)}$, to calculate its derivative we need to use the quotient rule:</p>\n<p>\\[ \\dfrac{dy}{dx} = \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2}\\,.\\]</p>\n<p>This can be split up into steps:</p>\n<ol>\n<li>Identify the functions $u(x)$ and $v(x)$;</li>\n<li>Calculate their derivatives $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$;</li>\n<li>Substitute these into the formula for the quotient rule to obtain an expression for $\\tfrac{dy}{dx}$;</li>\n<li>Simplify $\\tfrac{dy}{dx}$ where possible.</li>\n</ol>\n<p></p>\n<p>Following this process, we must first identify $u(x)$ and $v(x)$.</p>\n<p>As \\[ \\simplify{y={a}x^{n}/({b}x+{c})}, \\]</p>\n<p>let \\[ u(x) = \\simplify{{a}x^{n}} \\quad \\text{and} \\quad v(x)=\\simplify{{b}x+{c}}.\\]</p>\n<p>Next, we need to find the derivatives, $\\tfrac{du}{dx}$ and $\\tfrac{dv}{dx}$:</p>\n<p>\\[ \\dfrac{du}{dx} = \\simplify{{a*n}x^{n-1}}\\quad \\text{and} \\quad\\dfrac{dv}{dx}=\\simplify{{b}}.\\]</p>\n<p>Substituting these results into the quotient rule formula we can obtain an expression for $\\tfrac{dy}{dx}$:</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,= \\dfrac{v(x) \\times \\frac{du}{dx} - u(x) \\times\\frac{dv}{dx}}{[v(x)]^2} \\\\ \\\\&amp;\\,=\\dfrac{(\\simplify{{b}x+{c}}) \\times\\simplify{{a*n}x^{n-1}} - \\simplify{{a}x^{n}} \\times \\simplify{{b}}}{\\simplify{({b}x+{c})^2}}.&nbsp; \\end{split}\\]</p>\n<p>Simplifying,</p>\n<p>\\[ \\begin{split} \\dfrac{dy}{dx} &amp;\\,=\\dfrac{(\\simplify{{b}x+{c}})\\simplify{{a*n}x^{n-1}} - \\simplify{{b*a}x^{n}}}{\\simplify{({b}x+{c})^2}} \\\\ \\\\&amp;\\,=\\dfrac{\\simplify[all,!cancelTerms]{{b*a*n}x^{n}+{c*a*n}x^{n-1} - {b*a}x^{n}}}{\\simplify{({b}x+{c})^2}}\\\\ \\\\ &amp;\\,=\\dfrac{\\simplify{{b*a*n}x^{n}+{c*a*n}x^{n-1} - {b*a}x^{n}}}{\\simplify{({b}x+{c})^2}} \\\\ \\\\ &amp;\\,=\\dfrac{\\simplify{{simp}x^{n-1}({(b*a*n-b*a)/simp}x+{c*a*n/simp})}}{\\simplify{({b}x+{c})^2}} \\end{split} \\]</p>\n<p></p>\n<p><span style=\"font-weight: 400;\"><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\">Use this link to find some resources which will help you revise this topic.</a></span></p>\n<p><span style=\"font-weight: 400;\"></span></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,b])", "description": "", "templateType": "anything", "can_override": false}, "simp": {"name": "simp", "group": "Ungrouped variables", "definition": "gcd(b*a*n-b*a,c*a*n)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "simp>1", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "n", "simp"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\dfrac{dy}{dx}=$[[0]]&nbsp;</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({simp}x^{n-1}({(b*a*n-a*b)/simp}x+{c*a*n/simp}))/({b}x+{c})^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CI04 Integration - trig identities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Using the various versions of $\\cos{2x}$ identity to integrate $\\sin^2{x}$ and $\\cos^2{x}$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Integrate $f(x)=\\var{Func}$.</p>", "advice": "<div data-jme-visible=\"SorC='sin(x)^2'\">\n<p>We can't integrate $\\var{Coeff}\\sin^2(x)$ directly so first we have to use the double angle formula $\\cos(2x)=1-2\\sin^2(x)$. We re-arrange using the double angle formula to give us,</p>\n<p>\\[\\var{Coeff}\\sin^2(x)=\\frac{\\var{Coeff}}2-\\frac{\\var{Coeff}}2\\cos(2x).\\]</p>\n<p>From the Table of Integrals we see that a function of the form \\[ f(x)= \\cos(nx) \\] has the integral \\[ \\int \\cos(nx) dx&nbsp; = \\frac{1}{n}\\sin(nx)+c\\]</p>\n<p></p>\n<p>So, for the function</p>\n<p>\\[f(x)=\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}cos(2x)},\\]</p>\n<p>the integral is</p>\n<p>\\[ \\begin{split} \\int\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}cos(2x)} dx \\,= \\simplify[unitFactor,fractionNumbers]{{-Coeff/2}int(cos(2x),x)} &amp;\\,=\\simplify[unitFactor,fractionNumbers]{{-Coeff/2}(1/2 sin({2}x))} +c, \\\\ &amp;\\,=\\simplify[unitFactor,fractionNumbers]{{-Coeff/4} sin(2x)+c}. \\end{split}&nbsp; \\]</p>\n<p>The integral of $\\frac{\\var{Coeff}}2$ is</p>\n<p>\\[\\int\\frac{\\var{Coeff}}2dx=\\frac{\\var{Coeff}}2x+c,\\]</p>\n<p>so combining these our final answer is</p>\n<p>\\[\\int\\frac{\\var{Coeff}}2-\\frac{\\var{Coeff}}2\\cos(2x)dx=\\simplify[unitFactor,fractionNumbers]{{Coeff/2}x-{Coeff/4} sin(2x)+c}\\]</p>\n</div>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">We can't integrate $\\var{Coeff}\\cos^2(x)$ directly so first we have to use the double angle formula $\\cos(2x)=2\\cos^2(x)-1$. We re-arrange using the double angle formula to give us,</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">\\[\\var{Coeff}\\cos^2(x)=\\frac{\\var{Coeff}}2+\\frac{\\var{Coeff}}2\\cos(2x).\\]</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">From the Table of Integrals we see that a function of the form \\[ f(x)= \\cos(nx) \\] has the integral \\[ \\int \\cos(nx) dx&nbsp; = \\frac{1}{n}\\sin(nx)+c\\]</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">So, for the function</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">\\[f(x)=\\simplify[unitFactor,fractionNumbers]{{Coeff/2}cos(2x)},\\]</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">the integral is</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">\\[ \\begin{split} \\int\\simplify[unitFactor,fractionNumbers]{{Coeff/2}cos(2x)} dx \\,= \\simplify[unitFactor,fractionNumbers]{{Coeff/2}int(cos(2x),x)} &amp;\\,=\\simplify[unitFactor,fractionNumbers]{{Coeff/2}(1/2 sin({2}x))} +c, \\\\ &amp;\\,=\\simplify[unitFactor,fractionNumbers]{{Coeff/4} sin(2x)+c}. \\end{split}&nbsp; \\]</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">The integral of $\\frac{\\var{Coeff}}2$ is</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">\\[\\int\\frac{\\var{Coeff}}2dx=\\frac{\\var{Coeff}}2x+c,\\]</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">so combining these our final answer is</span></p>\n<p><span data-jme-visible=\"SorC='cos(x)^2'\">\\[\\int\\frac{\\var{Coeff}}2+\\frac{\\var{Coeff}}2\\cos(2x)dx=\\simplify[unitFactor,fractionNumbers]{{Coeff/2}x+{Coeff/4} sin(2x)+c}\\]</span><br/><br/><br/><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Diff\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"SorC": {"name": "SorC", "group": "Ungrouped variables", "definition": "random('sin(x)^2','cos(x)^2')", "description": "", "templateType": "anything", "can_override": false}, "Coeff": {"name": "Coeff", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "Func": {"name": "Func", "group": "Ungrouped variables", "definition": "simplify((expression(Coeff+\"*\"+SorC)),[\"unitFactor\",\"trig\"])", "description": "", "templateType": "anything", "can_override": false}, "OneIfCosMinusOneIfSine": {"name": "OneIfCosMinusOneIfSine", "group": "Ungrouped variables", "definition": "1 if(SorC='sin(x)^2',-1,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["SorC", "Coeff", "Func", "OneIfCosMinusOneIfSine"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": true, "customName": "+k instead of +c", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "1/2{Coeff}*(x+{OneIfCosMinusOneIfSine}/2*sin(2x))+k", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "k", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Forgotten constant", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>It looks like you forgot to include the integration constant. You should always remember the \"+C\" when doing an indefinite integral.</p>", "useAlternativeFeedback": false, "answer": "1/2{Coeff}*(x+{OneIfCosMinusOneIfSine}/2*sin(2x))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "1/2{Coeff}*(x+{OneIfCosMinusOneIfSine}/2*sin(2x))+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CI05 Integration - Substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $\\frac{nx^{n-1}}{x^n+a}$ using integration by substitution.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate \\[ \\simplify[all]{int(({n}x^{n-1})/(x^{n}+{a}),x)}\\]</p>\n<p>by using the substitution \\[ \\simplify[all]{u=x^{n}+{a}}.\\]</p>", "advice": "<p>Since this integral is of the form \\[ \\int g'(x)f(g(x))\\,dx,\\] we can use the method of substitution to calculate the solution.&nbsp;</p>\n<p>Firstly, we must make a change of variables from $x$ to $u$, where $u$ is equal to the 'inner' function $g(x)$.</p>\n<p>So, for \\[\\simplify[fractionNumbers]{int(({n}x^{n-1})/((x^{n}+{a})),x)}\\]</p>\n<p>let $\\color{red}{u=\\simplify[fractionNumbers]{x^{n}+{a}}}.$</p>\n<p>Now, we need to calculate the differential, $du$, where \\[ du = \\left(\\frac{du}{dx}\\right)dx. \\]</p>\n<p>Differentiating $u$ with respect to $x$:</p>\n<p>\\[ \\frac{du}{dx}= \\simplify[fractionNumbers]{{n}x^{n-1}}.\\]</p>\n<p>Therefore, \\[ \\color{blue}{du = \\simplify[fractionNumbers]{{n}x^{n-1}}\\, dx}.\\]</p>\n<p>We can now rewrite the original integral in terms of $u$:</p>\n<p>\\[ \\int \\frac{\\color{blue}{\\simplify{{n}x^{n-1}}}}{\\color{red}{\\simplify{x^{n}+{a}}}}\\color{blue}{\\text{d}x} =&nbsp;\\int \\frac{1}{\\color{red}{u}}\\color{blue}{\\text{d}u}.\\]</p>\n<p><strong>(Note: It is important to see that both the function we are integrating, and the variable we are integrating with respect to, has changed.)</strong></p>\n<p>\\[ \\simplify[fractionNumbers]{int(1/u,u) = ln(abs(u)) + c}.\\]</p>\n<p>Finally, we must rewrite our solution back in terms of the original variable $x$:</p>\n<p>\\[ \\simplify[fractionNumbers]{ln(abs(u)) + c = ln(abs(x^{n}+{a}))&nbsp;+ c}.\\]</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Int\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": "100"}, "ungrouped_variables": ["a", "n"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Correct answer", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": true, "customName": "Alternative using brackets", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>Technically we should use the absolute value symbols for the logs. This can be done in NUMBAS by using \"abs(*function*)\".</p>", "useAlternativeFeedback": false, "answer": "ln(x^{n}+{a})+c", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Alternative using \"+k\"", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "ln(abs(x^{n}+{a})) + k", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "k", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Alternative using brackets and \"+k\"", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>Technically we should use the absolute value symbols for the logs. This can be done in NUMBAS by using \"abs(*function*)\".</p>", "useAlternativeFeedback": false, "answer": "ln(x^{n}+{a})+k", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "k", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Forgotten constant", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>It looks like you forgot to include the integration constant. You should always remember the \"+C\" when doing an indefinite integral.</p>", "useAlternativeFeedback": false, "answer": "ln(abs(x^{n}+{a}))", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "ln(abs(x^{n}+{a}))+c", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "CI06 Integration - Parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}, {"name": "Andrew Neate", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21832/"}, {"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "tags": [], "metadata": {"description": "<p>Calculating the integral of a function of the form $ax^2 \\cos(bx)$ using integration by parts.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the integral \\[ \\simplify{int({a}x^2 cos({b}x),x)}\\]</p>", "advice": "<p>If we have a function of $x$ which is the product of two functions of $x$, to integrate such a function it is often necessary to use Integration by Parts. The formula for Integration by Parts is:</p>\n<p>\\[ \\int u(x) \\frac{dv}{dx} dx = u(x)v(x) - \\int v(x) \\frac{du}{dx} dx.\\]</p>\n<p>Using this method can be broken down into steps:</p>\n<ol>\n<li>Identify $u(x)$ and $\\tfrac{dv}{dx}$ (The function you pick for each is important, in general you want $u(x)$ to become simpler when differentiating it, and you must be able to integrate $\\tfrac{dv}{dx}$ to find $v(x)$);</li>\n<li>Calculate $\\tfrac{du}{dx}$ and $v(x)$;</li>\n<li>Put the functions $u(x)$, $v(x)$, and their derivatives into the Integration by Parts formula;</li>\n<li>Calculate the integral $\\int v(x) \\tfrac{du}{dx} dx$ (This may require you to use Integration by Parts again, this is OK!);</li>\n<li>Simplify your answer where possible and don't forget to add the constant of integration.</li>\n</ol>\n<p></p>\n<p>For the integral</p>\n<p>\\[ \\simplify{int({a}x^2 cos({b}x),x)},\\]</p>\n<p>we must first identify $u(x)$ and $\\tfrac{dv}{dx}$. In this case, let \\[ u(x)=\\simplify{{a}x^2},\\quad \\frac{dv}{dx}= \\simplify{cos({b}x)}. \\]</p>\n<p>Next, we need to calculate $\\tfrac{du}{dx}$ and $v(x)$:</p>\n<p>\\[ \\begin{split} u(x) = \\var{a}x^2 \\quad &amp;\\implies \\frac{du}{dx} = \\simplify{{2a}x}; \\\\ \\frac{dv}{dx} = \\cos(\\var{b}x) &amp;\\implies v(x) = \\simplify[fractionNumbers]{1/{b} sin({b}x)}. \\end{split} \\]</p>\n<p>Plugging these 4 terms into the integration by parts formula:</p>\n<p>\\[&nbsp; \\begin{split} \\simplify{int({a}x^2 cos({b}x),x)} &amp;\\,= \\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) - int({2a/b}x sin({b}x),x)},&nbsp; \\\\ \\\\ &amp;\\,= \\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) -{2a/b}int(x sin({b}x),x)}.\\end{split} \\]</p>\n<p>Since the integral on the right-hand side is still the product of two functions of $x$, we need to use integration by parts again.&nbsp;</p>\n<p>So, for&nbsp;</p>\n<p>\\[ \\simplify{int(x sin({b}x),x)}, \\]</p>\n<p>&nbsp;Let $u=x$ and $\\tfrac{dv}{dx} = \\sin(\\var{b}x)$. Therefore, $\\tfrac{du}{dx}=1$ and $v(x)=\\simplify{-1/{b} cos({b}x)}$.</p>\n<p>Hence,</p>\n<p>\\[ \\begin{split} \\simplify{int(x sin({b}x),x)} &amp;\\,= \\simplify{-1/{b}x cos({b}x)- int(-1/{b} cos({b}x),x)} \\\\ \\\\ &amp;\\,= \\simplify{-1/{b}x cos({b}x)+1/{b^2}sin({b}x)}. \\end{split}\\]</p>\n<p>Plugging this back into the original calculation:</p>\n<p>\\[&nbsp; \\begin{split} \\simplify{int({a}x^2 cos({b}x),x)} &amp;\\,= \\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) -{2a/b}int(x cos({b}x),x)} \\\\ \\\\ &amp;\\,=&nbsp;\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) -{2a/b}[-1/{b}x cos({b}x)+1/{b^2}sin({b}x)]} \\\\ \\\\ &amp;\\,=\\simplify[fractionNumbers]{{a/b}x^2 sin({b}x) +{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)} + c.\\end{split} \\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/maths-resources/next-steps#Int\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "gcd(a,b)=1 and b>a", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Correct Answer", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": true, "customName": "Alt constant +k", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "{a/b}x^2 sin({b}x)+{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)+k", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "k", "value": ""}, {"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "Forgotten constant", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>It looks like you forgot to include the integration constant. You should always remember the \"+C\" when doing an indefinite integral.</p>", "useAlternativeFeedback": false, "answer": "{a/b}x^2 sin({b}x)+{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)", "answerSimplification": "all,!collectLikeFractions,fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{a/b}x^2 sin({b}x)+{2a/b^2}x cos({b}x)-{2a/b^3}sin({b}x)+c", "answerSimplification": "fractionNumbers, basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "SM07 Identify measures of spread/location", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Gareth Woods", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/978/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Michael Pan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23528/"}], "tags": [], "metadata": {"description": "<p>Identifying measures of spread or location (average)</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Match each of the following with what they measure.</p>", "advice": "<p>The mean is a measure of location or central tendancy. It is calcuated by summing all of the data values and dividing by the number of values.</p>\n<p>The median is a measure of location or central tendancy. It is the middle value of an ordered data set.</p>\n<p>The inter-quartile range is a measure of spread. The interquartile range is the difference between upper and lower quartiles.The lower quartile, or first quartile (Q1), is the value under which 25% of data points are found when they are arranged in increasing order. The upper quartile, or third quartile (Q3), is the value under which 75% of data points are found when arranged in increasing order. The inter-quartile range therefore gives us an idea of the middle 50% of the ordered data set.</p>\n<p>The standard deviation is a measure of spread. It measures the dispersion of a data set relative to its mean.&nbsp;</p>\n<p>The variance is a measure spread because it is the square of the standard deviation.</p>\n<p>A p-value the probability that a particular statistical measure, such as the mean or standard deviation, of an assumed probability distribution will be greater than or equal to (or less than or equal to in some instances) observed results. A p-value is used to determine statistical significance, not measures of spread or location.</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/stats-resources/first-steps#mean,-median-and-mode\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {"std": ["all", "fractionNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "function dragpoint_board() {\n  var scope = question.scope;\n\n  JXG.Options.text.display = 'internal';\n  \n  var yo0 = scope.variables.yo0.value;\n  var yo1 = scope.variables.yo1.value;\n  var yo2 = scope.variables.yo2.value;\n  var yo3 = scope.variables.yo3.value;\n  var yo4 = scope.variables.yo4.value;\n  var yo5 = scope.variables.yo5.value;\n  var yo6 = scope.variables.yo6.value;\n  var yo7 = scope.variables.yo7.value; \n  var yo8 = scope.variables.yo8.value;\n  var yo9 = scope.variables.yo9.value; \n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('550px','550px',{boundingBox:[-0.8,82,16,-8], axis:false, grid:true});\n \n  $(question.display.html).find('#dragpoint').append(div);\n  \n  var board = div.board;\n  \nboard.suspendUpdate();           \n\n  \n var dataArr = [yo0,yo5,0,yo1,yo6,0,yo2,yo7,0,yo3,yo8,0,yo4,yo9]; \n  \n var xaxis = board.create('axis', [[0, 0], [12, 0]], {withLabel: true, name: \"Bank\", label: {offset: [250,-30]}});\n  \n  xaxis.removeAllTicks();  \n   \n  board.create('axis', [[0, 0], [0, 10]], {hideTicks:true, withLabel: false, name: \"\", label: {offset: [-110,300]}});\n  \n  var pop0 = board.create('point', [1.5,0],{name:'Morgan',fixed:true,size:0,color:'black',face:'diamond', label:{offset:[-20,-8]}});\n  var pop1 = board.create('point',[4.5,0],{name:'Strome',fixed:true,size:0,color:'black',face:'diamond', label:{offset:[-20,-8]}});\n  var pop2 = board.create('point',[7.5,0],{name:'Bentley',fixed:true,size:0,color:'black', face:'diamond', label:{offset:[-15,-8]}});\n  var pop3 = board.create('point',[10.5,0],{name:'Sand',fixed:true,size:0,color:'black', face:'diamond', label:{offset:[-15,-8]}});\n  var pop4 = board.create('point',[13.5,0],{name:'Karchen',fixed:true,size:0,color:'black', face:'diamond', label:{offset:[-15,-8]}});\n\n  var leg1 = board.create('point',[12,75],{name:'last year',fixed:true,size:6,color:'#DA2228', face:'square', label:{offset:[9,0]}});\n  var leg2 = board.create('point',[12,72],{name:'this year',fixed:true,size:6,color:'#6F1B75', face:'square', label:{offset:[9,0]}});\n  \n  \n//  var chart = board.createElement('chart', dataArr, \n  //                              {chartStyle:'bar', fillOpacity:1, width:1,\n    //     colorArray:['#8E1B77','#8E1B77','Red','Red','blue','red','blue','red','red','blue', 'red','blue','red','red'], shadow:false});\n \n//var chart = board.createElement('chart', dataArr, \n  //                              {chartStyle:'bar', width:1,fillOpacity:1, fillColor:'red', shadow:false});  \n  \n  \n    var a = board.create('chart', [[1,2,3],[yo0,yo5,0]], {chartStyle:'bar',colors:['#DA2228','#6F1B75','#6F1B75'],width:1,fillOpacity:1});\n    var b = board.create('chart', [[4,5,6],[yo1,yo6,0]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75','#6F1B75'],fillOpacity:1});\n    var c = board.create('chart', [[7,8,9],[yo2,yo7,0]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75','#6F1B75'],fillOpacity:1});\n    var d = board.create('chart', [[10,11,12],[yo3,yo8,0]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75','#6F1B75'],fillOpacity:1});\n    var e = board.create('chart', [[13,14],[yo4,yo9]], {chartStyle:'bar',width:1,colors:['#DA2228','#6F1B75'],fillOpacity:1});\n  \n    board.unsuspendUpdate();\n  \n  var txt1 = board.create('text',[-0.3,30, 'Investment \u00a3(m)'], {fontColor:'black', fontSize:14, rotate:90});\n  \n // var txt = board.create('text',[0.5,75, 'Investment (m)'], {fontSize:14, rotate:90});\n  \n // var txt1 = board.create('text',[8,76, 'red bars represent 2010'], {fontColor:'red', fontSize:14, rotate:90});\n  \n // var txt2 = board.create('text',[8,73, 'blue bars represents 2011'], {fontSize:14, rotate:90});\n\n // var myColors = new Array('red', 'blue', 'white','red', 'blue', 'white','red', 'blue', 'white','red', 'blue', 'white','red', 'blue');\n  \n  \n  \n  //board.unsuspendUpdate();\n\n    // Rotate text around the lower left corner (-2,-1) by 30 degrees.\n   // var tRot = board.create('transform', [90.0*Math.PI/180.0, -1,40], {type:'rotate'}); \n  //  tRot.bindTo(txt);\n  //  board.update();\n\n  \n//var chart2 = board.createElement('chart', dataArr, {chartStyle:'line,point'});\n//chart2[0].setProperty('strokeColor:black','strokeWidth:2','shadow:true');\n//for(var i=0; i<11;i++) {\n //   chart2[1][i].setProperty({strokeColor:'black',fillColor:'white',face:'[]', size:4, strokeWidth:2});\n//}\n//board.unsuspendUpdate();  \n                       \n  //board.unsuspendUpdate();\n\n}\n\nquestion.signals.on('HTMLAttached',function() {\n  dragpoint_board();\n});", "css": "table#values th {\n  background: none;\n  text-align: center;\n}"}, "parts": [{"type": "m_n_x", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "minAnswers": 0, "maxAnswers": 0, "shuffleChoices": true, "shuffleAnswers": true, "displayType": "radiogroup", "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["Variance", "Mean", "Median", "Inter-quartile range", "P-value", "Standard deviation"], "matrix": [["1", 0, 0], [0, "1", 0], [0, "1", 0], ["1", 0, 0], [0, 0, "1"], ["1", 0, 0]], "layout": {"type": "all", "expression": ""}, "answers": ["Measure of Spread", "Measure of location (average)", "Neither measure of location nor measure of spread"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "SN01 Correlation", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Lauren Desoysa", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21504/"}, {"name": "Upuli Wickramaarachchi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23527/"}], "tags": [], "metadata": {"description": "<p>Tests understanding of scatter plots and related concepts.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><span style=\"font-weight: 400;\">The scatter plot below shows the relationship between an employee&rsquo;s height in centimetres and how long it takes them to walk to work in minutes.</span></p>\n<table>\n<tbody>\n<tr>\n<td>time (mins)</td>\n<td>{drawgraph()}</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">height (cm)</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p></p>\n<p></p>\n<p></p>", "advice": "<p>The graph shows that there is a positive correlation between a person's height and how long it takes them to walk to work.</p>\n<p>A postive correlation is a relationship between two variables where both variables move in the same diection.</p>\n<p>This tells us that as a person's height increases, the time it takes to walk to work increases.</p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/stats-resources/next-steps#Introduction%20to%20correlation\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"slope": {"name": "slope", "group": "Regression variables", "definition": "(6*sumxy-sumx*sumy)/(6*sumxx-(sumx)^2)", "description": "<p>s</p>", "templateType": "anything", "can_override": false}, "timemax": {"name": "timemax", "group": "Calculation variables", "definition": "max([p1y,p2y,p3y,p4y,p5y,p6y])", "description": "", "templateType": "anything", "can_override": false}, "minx": {"name": "minx", "group": "Graph Limits", "definition": "140", "description": "", "templateType": "anything", "can_override": false}, "miny": {"name": "miny", "group": "Graph Limits", "definition": "-10", "description": "", "templateType": "anything", "can_override": false}, "p3x": {"name": "p3x", "group": "Points", "definition": "random(166..175)", "description": "", "templateType": "anything", "can_override": false}, "p3y": {"name": "p3y", "group": "Points", "definition": "random(26..35)", "description": "", "templateType": "anything", "can_override": false}, "p5x": {"name": "p5x", "group": "Points", "definition": "random(146..155 except p1x)", "description": "", "templateType": "anything", "can_override": false}, "p5y": {"name": "p5y", "group": "Points", "definition": "random(6..15)", "description": "", "templateType": "anything", "can_override": false}, "p1x": {"name": "p1x", "group": "Points", "definition": "random(146..155)", "description": "", "templateType": "anything", "can_override": false}, "p1y": {"name": "p1y", "group": "Points", "definition": "random(6..15)", "description": "", "templateType": "anything", "can_override": false}, "timediff": {"name": "timediff", "group": "Calculation variables", "definition": "timemax-timemin", "description": "", "templateType": "anything", "can_override": false}, "maxx": {"name": "maxx", "group": "Graph Limits", "definition": "188", "description": "", "templateType": "anything", "can_override": false}, "maxy": {"name": "maxy", "group": "Graph Limits", "definition": "63", "description": "", "templateType": "anything", "can_override": false}, "roundedslope": {"name": "roundedslope", "group": "Regression variables", "definition": "precround(slope,2)", "description": "", "templateType": "anything", "can_override": false}, "yintercept": {"name": "yintercept", "group": "Regression variables", "definition": "(sumy-slope*sumx)/6", "description": "", "templateType": "anything", "can_override": false}, "timemin": {"name": "timemin", "group": "Calculation variables", "definition": "min([p1y,p2y,p3y,p4y,p5y,p6y])", "description": "", "templateType": "anything", "can_override": false}, "tallest": {"name": "tallest", "group": "Calculation variables", "definition": "max([p1x,p2x,p3x,p4x,p5x,p6x])", "description": "", "templateType": "anything", "can_override": false}, "regy1": {"name": "regy1", "group": "Regression variables", "definition": "slope*minx+yintercept", "description": "", "templateType": "anything", "can_override": false}, "regy2": {"name": "regy2", "group": "Regression variables", "definition": "slope*maxx+yintercept", "description": "", "templateType": "anything", "can_override": false}, "sumy": {"name": "sumy", "group": "Regression variables", "definition": "p1y+p2y+p3y+p4y+p5y+p6y", "description": "", "templateType": "anything", "can_override": false}, "sumx": {"name": "sumx", "group": "Regression variables", "definition": "p1x+p2x+p3x+p4x+p5x+p6x", "description": "", "templateType": "anything", "can_override": false}, "p6y": {"name": "p6y", "group": "Points", "definition": "random(46..55)", "description": "<p>p6y</p>", "templateType": "anything", "can_override": false}, "p6x": {"name": "p6x", "group": "Points", "definition": "random(176..185 except p4x)", "description": "", "templateType": "anything", "can_override": false}, "p4y": {"name": "p4y", "group": "Points", "definition": "random(36..45)", "description": "", "templateType": "anything", "can_override": false}, "p4x": {"name": "p4x", "group": "Points", "definition": "random(176..185)", "description": "", "templateType": "anything", "can_override": false}, "p2y": {"name": "p2y", "group": "Points", "definition": "random(16..25)", "description": "", "templateType": "anything", "can_override": false}, "p2x": {"name": "p2x", "group": "Points", "definition": "random(156..165)", "description": "", "templateType": "anything", "can_override": false}, "sumxx": {"name": "sumxx", "group": "Regression variables", "definition": "p1x^2+p2x^2+p3x^2+p4x^2+p5x^2+p6x^2", "description": "", "templateType": "anything", "can_override": false}, "sumxy": {"name": "sumxy", "group": "Regression variables", "definition": "p1x*p1y+p2x*p2y+p3x*p3y+p4x*p4y+p5x*p5y+p6x*p6y", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Graph Limits", "variables": ["minx", "maxx", "miny", "maxy"]}, {"name": "Points", "variables": ["p1x", "p1y", "p2x", "p2y", "p3x", "p3y", "p4x", "p4y", "p5x", "p5y", "p6x", "p6y"]}, {"name": "Calculation variables", "variables": ["tallest", "timemax", "timemin", "timediff"]}, {"name": "Regression variables", "variables": ["sumx", "sumy", "sumxy", "sumxx", "slope", "yintercept", "regy1", "regy2", "roundedslope"]}], "functions": {"drawgraph": {"parameters": [], "type": "html", "language": "javascript", "definition": "      var miny = Numbas.jme.unwrapValue(scope.variables.miny);\n      var maxy = Numbas.jme.unwrapValue(scope.variables.maxy);\n      var minx = Numbas.jme.unwrapValue(scope.variables.minx);\n      var maxx = Numbas.jme.unwrapValue(scope.variables.maxx);\n      var regy1 = Numbas.jme.unwrapValue(scope.variables.regy1);\n      var regy2 = Numbas.jme.unwrapValue(scope.variables.regy2);\n\n      var p1x = Numbas.jme.unwrapValue(scope.variables.p1x);\n      var p1y = Numbas.jme.unwrapValue(scope.variables.p1y);\n      var p2x = Numbas.jme.unwrapValue(scope.variables.p2x);\n      var p2y= Numbas.jme.unwrapValue(scope.variables.p2y);\n      var p3x = Numbas.jme.unwrapValue(scope.variables.p3x);\n      var p3y= Numbas.jme.unwrapValue(scope.variables.p3y);\n      var p4x = Numbas.jme.unwrapValue(scope.variables.p4x);\n      var p4y= Numbas.jme.unwrapValue(scope.variables.p4y);\n      var p5x = Numbas.jme.unwrapValue(scope.variables.p5x);\n      var p5y= Numbas.jme.unwrapValue(scope.variables.p5y);\n      var p6x = Numbas.jme.unwrapValue(scope.variables.p6x);\n      var p6y= Numbas.jme.unwrapValue(scope.variables.p6y);\n      \n      var div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n      {boundingBox:[minx,maxy,maxx,miny],\n       axis:false,\n       showNavigation:false,\n       grid:true});\n        var brd = div.board;          \n        var xaxis=brd.createElement('axis', [[minx,0],[maxx,0]]);\n        var yaxis=brd.createElement('axis', [[minx+5,miny],[minx+5,maxy]]);\n        var li1=brd.create('line',[[minx,regy1],[maxx,regy2]],{fixed:true,withLabel:false});\n        var pt1=brd.create('point',[p1x,p1y],{visible:true,withLabel:false}); \n        var pt2=brd.create('point',[p2x,p2y],{visible:true,withLabel:false}); \n        var pt3=brd.create('point',[p3x,p3y],{visible:true,withLabel:false}); \n        var pt4=brd.create('point',[p4x,p4y],{visible:true,withLabel:false}); \n        var pt5=brd.create('point',[p5x,p5y],{visible:true,withLabel:false}); \n        var pt6=brd.create('point',[p6x,p6y],{visible:true,withLabel:false}); \nreturn div;\n      "}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Mark the statement that best <strong>describes</strong> what this scatter plot shows.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["<p>In general, there is a positive correlation between a person's height and how long it takes them to walk to work.</p>", "<p>In general, there is a negative correlation between a person's height and how long it takes them to walk to work.</p>", "<p>In general, there is a no correlation between a person's height and how long it takes them to walk to work.</p>"], "matrix": ["1", 0, 0], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "SP01 Probability - \"sample space\"", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}], "tags": [], "metadata": {"description": "<p>Calculate probability of selecting coloured counters from a bag.</p>", "licence": "None specified"}, "statement": "<p>A bag contains:<br/><br/> $\\var{srn}$ small, red tokens,&nbsp;<br/>$\\var{sbn}$ small,&nbsp;blue&nbsp;tokens,&nbsp;<br/>$\\var{brn}$ large, red&nbsp;tokens, and&nbsp;<br/>$\\var{bbn}$ large, blue tokens.</p>", "advice": "<h3>part a)</h3>\n<p>A probability is a fraction. &nbsp;You can give your answer as a fraction, decimal or percentage as these are all equivalent.<br/><br/>The formula for probability is:</p>\n<p>\\[ P(A) = \\frac{\\text{number of possibilities for A}}{\\text{number of total possible outcomes}} \\]</p>\n<p>For this question the total possible outcomes are $\\var{srn}+\\var{sbn}+\\var{brn}+\\var{bbn} = \\var{total}$.<br/><br/>Therefore</p>\n<p>\\[ P(\\text{A large red token}) = \\frac{\\var{brn}}{\\var{total}} = \\var[fractionnumbers]{brn/total}\\]</p>\n<h3>part b)</h3>\n<p>For this question we need to know the total number of small tokens, i.e.&nbsp;$\\var{srn}+\\var{sbn} = \\var{srn+sbn}$.<br/><br/>Therefore</p>\n<p>\\[ P(\\text{A small token}) = \\frac{\\var{srn+sbn}}{\\var{total}} = \\var[fractionnumbers]{(srn+sbn)/total}\\]</p>\n<p></p>\n<p><a href=\"https://www.sheffield.ac.uk/mash/stats-resources/probability#equally-likely-outcomes\"><span style=\"font-weight: 400;\">Use this link to find some resources which will help you revise this topic.</span></a></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"srn": {"name": "srn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "brn": {"name": "brn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "sbn": {"name": "sbn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "bbn": {"name": "bbn", "group": "Ungrouped variables", "definition": "random(1..20)", "description": "", "templateType": "anything", "can_override": false}, "total": {"name": "total", "group": "Ungrouped variables", "definition": "brn+bbn+srn+sbn", "description": "", "templateType": "anything", "can_override": false}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "precround(brn/total,2)", "description": "", "templateType": "anything", "can_override": false}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "precround((srn+sbn)/total,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["srn", "brn", "sbn", "bbn", "total", "ans1", "ans2"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>You take a token at random.<br/><br/>What is the probability that it is a large, red&nbsp;token?<br/><br/><em>Give your answer as a fraction, or a decimal correct to 2dp.</em></p>", "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "minValue": "ans1", "maxValue": "ans1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "{brn}/{total}", "maxValue": "{brn}/{total}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>You take a token at random.<br/><br/>What is the probability that it is a small token?<br/><br/><em>Give your answer as a fraction, or a decimal correct to 2dp.</em></p>", "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "minValue": "ans2", "maxValue": "ans2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "{{srn}+{sbn}}/{total}", "maxValue": "{{srn}+{sbn}}/{total}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "SP02 Intuitive Probability", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Mash Sheffield", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4679/"}, {"name": "Upuli Wickramaarachchi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23527/"}, {"name": "Michael Pan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/23528/"}], "tags": ["taxonomy"], "metadata": {"description": "<p>Predicting the probability of an unbiased coin landing on heads based on the results of previous throws.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>When we flip an unbiased&nbsp;coin there are two possible events that we could measure: the coin lands on heads or&nbsp;the coin lands on tails.</p>\n<p>Each toss of the coin is independent; if we flip a coin once and it lands&nbsp;on heads then the next time we flip the coin it is still equally likely to land on either heads or tails.</p>\n<p>It doesn't matter what the coin landed on previously as this&nbsp;outcome does not affect the&nbsp;outcome of the next flip of the coin.</p>\n<p>Even when we flip&nbsp;an unbiased coin&nbsp;$\\var{no_flips}$ times and it lands on&nbsp;heads each time; the next time we flip the coin, it is still equally likely to land on either heads or tails.</p>\n<p>So the probability that the coin lands on heads the next time that the coin is flipped is still $\\displaystyle\\frac{1}{2}$.</p>\n<h4 style=\"text-align: center;\"><a href=\"https://www.sheffield.ac.uk/mash/stats-resources/probability#equally-likely-outcomes\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></h4>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"no_flips": {"name": "no_flips", "group": "Ungrouped variables", "definition": "random(6..9)", "description": "<p>Number of flips of the coin</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["no_flips"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>An unbiased coin is flipped $\\var{no_flips}$ times. Given that the coin landed on heads each time, what is the probability of the coin landing on heads the next time it is flipped?</p>", "minValue": "1/2", "maxValue": "1/2", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": true, "startpassword": "", "allowAttemptDownload": true, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "<p>This is a tool for you! It is here to help you diagnose whether there are any maths or statistics pre-requisites for your course that you may want to brush up on. If at any point you are struggling with any question you should find a link at the end of the \"reveal answer\" section that will take you to some recommended online resources on that subject area. You can also always contact the Maths and Stats Help team (MaSH) to arrange a one to one appointment or check out our workshop timetable to see if you can access the support you need that way. Find all this information via our website <a href=\"https://students.sheffield.ac.uk/mash\" target=\"_blank\" rel=\"noopener\" title=\"Link to MASH home page\">here</a>!</p>", "end_message": "<p>Thanks for completing the Skills Audit. You can attempt this as many times as you need. Remember the score is not what matters - this is in no way assessed work - this is simply a tool for working out whether you may need to brush up on anything to ensure that you can access all the material on your course and get off to the best possible start.</p>\n<p>Don't forget to look up what support is available to you through our web pages <a href=\"https://students.sheffield.ac.uk/mash\" target=\"_blank\" rel=\"noopener\" title=\"Link to MASH homepage\">here</a>!</p>", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": [], "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "inreview"}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "type": "exam", "contributors": [{"name": "Will Morgan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/21933/"}], "extensions": ["geogebra", "jsxgraph"], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": []}</script>
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