// Numbas version: finer_feedback_settings
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"skill: 403, Arbitrary factors"], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "variables": {"groups": {"name": "groups", "group": "Ungrouped variables", "definition": "[\n  [3,\"4w^2 + w^3 = w^3(1+4w^n)\",1],\n  [2,\"2+w^2 = w^2(2w^n + 1)\",-2],\n  [-1,\"w+3/w = w^-1(3+w^n)\",2],\n  [3,\"4w-3w^3=w^3(4w^n-3)\",-2],\n  [-2,\"1/w + 1/w^2 = w^-2(w^n+1)\",1]\n]", "description": "", "templateType": "anything"}, "group": {"name": "group", "group": "Ungrouped variables", "definition": "random(groups)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "group[0]", "description": "", "templateType": "anything"}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "expression(group[1])", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "group[2]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["groups", "group", "a", "expr", "n"], "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>In the following, the factor $w^\\var{a}$ has been taken out of the left-hand side to give the right-hand side.</p>\n<p>\\[ \\var{expr} \\]</p>\n<p>Enter the value of the number $n$.</p>", "minValue": "n", "maxValue": "n", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, 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{"name": "answer", "group": "Ungrouped variables", "definition": "expression(pair[1])", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pairs", "pair", "expr", "answer"], "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>Collect terms in the following expression</p>\n<p>\\[ \\var{expr} \\]</p>\n<p>leaving your answer in the simplest possible form.</p>", "answer": "{answer}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "[\"n\": `+-$n`?] `@ ((n*x)`? + (n*y)`? + (n*u)`? + (n*v)`? + (n*p)`? + (n*q)`? + (n*g)`? + (n*h)`? + n)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "exam"}, {"name": "Collecting like terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["algebra", "Algebra", "collecting like terms", "like terms", "simplifying"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Simplify the following by collecting like terms.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10)+[0,0])[0..3]", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10)+[0,0])[0..2]", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10))[0..2]", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10))[0..2]", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10))[0..3]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "shuffle(list(-10..10)+[0,0,0,0])[0..3]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "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[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}$ =&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "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>Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x$ and $-2y$ have different variable parts and are therefore unlike terms (or not like terms).&nbsp;</p>\n<p>We can only collect like terms! Just like we can't say 2&nbsp;m + 3&nbsp;cm equals&nbsp;5&nbsp;m or 5&nbsp;cm, we can't say $2x+3y$ equals $5x$ or $5y$! We can, however, say $2a+3a=5a$.</p>\n<p>In our question we look at all the terms with a variable part of $x$ and add up all the corresponding&nbsp;coefficients, we do the same for the $y$ terms and the $z$ terms:</p>\n<p>\\[\\begin{align}<br/>&amp;\\simplify[!collectnumbers]{{a[1]}x+{b[1]}y+{c[1]}z+{b[2]}y+{a[2]}x+{c[2]}z+{a[0]}x+{c[0]}z+{b[0]}y}\\\\<br/>&amp;=\\simplify[basic]{({a[1]}+{a[2]}+{a[0]})x+({b[1]}+{b[2]}+{b[0]})y+({c[1]}+{c[2]}+{c[0]})z}\\end{align}\\]</p>\n<p>We present this as the sum of three unlike terms:</p>\n<p>\\[\\simplify{{sum(a)}x+{sum(b)}y+{sum(c)}z}\\]</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": "{sum(a)}*x+{sum(b)}*y+{sum(c)}*z", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "maxlength": {"length": "14", "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "sortAnswers": false}, {"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>$\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}$ =&nbsp;[[0]]</p>", "stepsPenalty": "1", "steps": [{"type": "information", "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>Like terms are terms where the variable part is the same. For example, $4x$ and $-x$ have the same variable part $x$. However, $3x^2$ and $-2x$ have different variable parts and are therefore unlike terms (or not like terms).&nbsp;</p>\n<p>We can only collect like terms! Just like we can't say 2&nbsp;m + 3&nbsp;cm equals&nbsp;5&nbsp;m or 5&nbsp;cm, we can't say $2x^2+3x$ equals $5x^2$ or $5x$! We can, however, say $2x^2+3x^2=5x^2$.</p>\n<p>In our question we look at all the terms with a variable part of $x^2$ and add up all the corresponding&nbsp;coefficients (the numbers in front of the variables), we do the same for the $x$ terms and the constant&nbsp;terms (the terms with no variable part):</p>\n<p>\\[\\begin{align}&amp;\\simplify[!collectnumbers]{{d[1]}x^2+{f[1]}x+{g[1]}+{d[0]}x^2+{f[0]}x+{g[0]}}\\\\&amp;=\\simplify[basic]{({d[1]}+{d[0]})x^2+({f[1]}+{f[0]})x+({g[1]}+{g[0]})}\\end{align}\\]</p>\n<p>We present this as the sum of three unlike terms:</p>\n<p>\\[\\simplify[!noleadingminus, basic]{{sum(d)}x^2+{sum(f)}x+{sum(g)}}\\]</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": 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"suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve the equation, and give the value of $x$:</p>\n<p>\\[ \\var{expr} \\]</p>", "minValue": "x", "maxValue": "x", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "exam"}]}, {"name": "112, Simple calculation", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Simple calculation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": 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{"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "variables": {"pairs": {"name": "pairs", "group": "Ungrouped variables", "definition": "[\n  [3+2i,4-3i],\n  [2-4i,5-5i],\n  [1-7i,2-4i],\n  [3+4i,2-5i],\n  [2-3i,3-2i]\n]", "description": "", "templateType": "anything"}, "pair": {"name": "pair", "group": "Ungrouped variables", "definition": "random(pairs)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "pair[0]", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "pair[1]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["pairs", "pair", "a", "b"], "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 $(\\var{a}) - (\\var{b})$.</p>", "answer": "{a-b}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "mustmatchpattern": {"pattern": "`+-$n`? + `+-$n*i`?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "exam"}]}, {"name": "404, L.c.d. of an algebraic 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[""], "variable_overrides": [[]], "questions": [{"name": "Partial Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"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/"}, {"name": "Krishna Kedia", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25454/"}], "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 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"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", "type": "question"}]}, {"name": "502, Solve equation using log", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Solving equations using logs", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"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/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "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}, "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", "type": "question"}]}, {"name": "402, Logs definition", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Logs - definition", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"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/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "tags": [], "metadata": {"description": "<p>Finding $x$ from a logarithmic equation of the form $\\log_ax = b$, where $a$ and $b$ are positive integers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the value of $x$:</p>\n<p>\\[ \\log_\\var{a}x = \\var{n} \\]</p>", "advice": "<p>To find the value of $x$, recall that $\\log_a(x)=b$ is equivalent to $x=a^b$.&nbsp;</p>\n<p>Therefore, \\[\\log_\\var{a}(x) = \\var{n} \\implies \\simplify[!collectNumbers]{x={a}^{n}}.\\]</p>\n<p>Hence, \\[x=\\var{a^n}\\,.\\]</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 logarithms.</a></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "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>$x=$ [[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^n}", "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", "type": "question"}]}, {"name": "442b, Integration of Powers, Trig and Exp", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Integration - powers, trig and exp", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "functions": {}, "tags": ["Calculus", "Steps", "calculus", "constant of integration", "exponential function", "indefinite integration", "integrals", "integrating powers", "integration", "integration of exponential function", "integration of powers", "integration of trigonometric functions", "standard integrals", "steps", "trigonometric functions"], "advice": "\n  <p>Note that \\[\\begin{eqnarray*} &amp;\\int&amp; \\;x^n\\;dx&amp;=&amp;\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &amp;\\int&amp; \\;\\sin(ax)\\;dx &amp;=&amp; -\\frac{1}{a}\\cos(ax)+C\\\\ &amp;\\int&amp; \\;e^{ax}\\;dx &amp;=&amp; \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]</p>\n  <p>Splitting the integral into three parts and using the above information we have: <br/>\\[\\begin{eqnarray*}\\simplify[std]{Int({b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3},x)}&amp;=&amp;\\simplify[std]{Int({b} * e ^ ({a}*x),x)+Int({b1} * Sin({a1}*x),x)+Int({a2} * x ^ {c3},x) }\\\\ &amp;=&amp;\\simplify[std]{({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C} \\end{eqnarray*}\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepspenalty": 0.0, "prompt": "\n      <p>$\\simplify[std]{f(x) = {b} * e ^ ({a}*x) + {b1} * Sin({a1}*x) + {a2} * x ^ {c3}}$</p>\n        <p>$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]</p>\n        <p>Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.</p>\n        <p>Click on Show steps to get more information. You will not lose any marks by doing so.</p>\n      ", "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions and not decimals.</p>", "showstrings": false, "strings": ["."], "partialcredit": 0.0}, "checkingaccuracy": 0.001, "vsetrange": [1.0, 2.0], "vsetrangepoints": 5.0, "checkingtype": "absdiff", "answersimplification": "std", "marks": 3.0, "answer": "({b}/{a}) * (e ^({a}*x)) + (({(-b1)}/{a1}) * Cos({a1}*x)) + ({a2}/{c3+1}) * (x ^ {(c3 + 1)})+C", "type": "jme"}], "steps": [{"prompt": "<p>Note that \\[\\begin{eqnarray*} &amp;\\int&amp; \\;x^n\\;dx&amp;=&amp;\\frac{x^{n+1}}{n+1}+C,\\;\\;n \\neq -1\\\\ &amp;\\int&amp; \\;\\sin(ax)\\;dx &amp;=&amp; -\\frac{1}{a}\\cos(ax)+C\\\\ &amp;\\int&amp; \\;e^{ax}\\;dx &amp;=&amp; \\frac{1}{a}e^{ax}+C\\\\ \\end{eqnarray*}\\]</p>", "type": "information", "marks": 0.0}], "marks": 0.0, "type": "gapfill"}], "statement": "\n  <p>Integrate the following function $f(x)$.</p>\n  <p>&nbsp;<br/>Input the constant of integration as $C$.</p>\n  ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "s1*random(2..5)", "name": "a"}, "b": {"definition": "s2*random(2..9)", "name": "b"}, "s3": {"definition": "random(1,-1)", "name": "s3"}, "s2": {"definition": "random(1,-1)", "name": "s2"}, "s1": {"definition": "random(1,-1)", "name": "s1"}, "s5": {"definition": "random(1,-1)", "name": "s5"}, "s4": {"definition": "random(1,-1)", "name": "s4"}, "a1": {"definition": "random(2..5)", "name": "a1"}, "a2": {"definition": "s4*random(3..9)", "name": "a2"}, "b1": {"definition": "s3*random(2..9)", "name": "b1"}, "c3": {"definition": "s5*random(2..8)", "name": "c3"}}, "metadata": {"notes": "\n    \t\t<p><strong>2/08/2012:</strong></p>\n    \t\t<p>Added tags.</p>\n    \t\t<p>Added description.</p>\n    \t\t<p>Corrected mistake in formula for integrating $\\sin(ax)$ in Steps and Advice.</p>\n    \t\t<p>Checked calculation. OK.</p>\n    \t\t<p>Added decimal point to forbidden strings along with message to user re input of numbers.</p>\n    \t\t<p>Message about Show steps included. Also another message about including the constant of integration.</p>\n    \t\t<p>Changed checking range from 0 to 1 to 1 to 2 as we can have negative powers of $x$.</p>\n    \t\t<p>Improved display of Steps by aligning integral signs.</p>\n    \t\t", "description": "<p>Find $\\displaystyle \\int ae ^ {bx}+ c\\sin(dx) + px ^ {q}\\;dx$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"name": "443, Definite Integrals", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Definite Integral - Powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"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": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "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}, "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", "type": "question"}]}, {"name": "444, Integration by substitution", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Integration - Substitution", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"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/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "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}, "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", "type": "question"}]}, {"name": "505, Integration by parts", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Integration - by Parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"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/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "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}, "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", "type": "question"}]}, {"name": "341b, Differentiation of polynomials", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Differentiation - Polynomials", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/353/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "functions": {}, "ungrouped_variables": ["ac", "bc", "cc", "dc", "d"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p style=\"text-align: center;\">If $y=ax^n$,</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=anx^{n-1}$ for all rational $n$.</p>\n<p style=\"text-align: left;\">We'll take the following term as an example:</p>\n<p style=\"text-align: left;\">$\\frac{3}{8}x^2$</p>\n<p style=\"text-align: left;\">All we have to do to terms where $x$ is to a power of anything is times the coefficient of $x$ by the original power, and then take one away from the original power.</p>\n<p style=\"text-align: left;\">If you are not familiar with this kind of work, these instructions may sound confusing, but it is much easier once you have seen it in practice.</p>\n<p style=\"text-align: left;\">We take</p>\n<p style=\"text-align: left;\">$\\frac{3}{8}x^2$</p>\n<p style=\"text-align: left;\">and times $\\frac{3}{8}$ by $2$, to get</p>\n<p style=\"text-align: left;\">$(\\frac{3}{8}\\times2)x^2=\\frac{6}{8}x^2=\\frac{3}{4}x^2$.</p>\n<p style=\"text-align: left;\">We then subtract one from the original power, $2$.</p>\n<p style=\"text-align: left;\">This gives us the final answer of</p>\n<p style=\"text-align: left;\"><strong>$\\frac{3}{4}x^1=\\frac{3}{4}x$</strong>.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Remember, don't be confused if there is no coefficient. The fact the term is there means the coefficient must be $1$, but we don't tend to write it out as, for example $1x$, we just say $x$.</p>", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "<p>$\\simplify{({ac[0]}/{d[0]})x^3+({bc[0]}/{d[1]})x^2+({cc[0]}/{d[2]})x+({dc[0]}/{d[3]})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3{ac[0]}/{d[0]}x^2+2{bc[0]}/{d[1]}x+{cc[0]}/{d[2]}", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>$\\simplify{({ac[1]}/{d[4]})x^3+({bc[1]}/{d[5]})x^2+({cc[1]}/{d[6]})x+({dc[1]}/{d[7]})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3{ac[1]}/{d[4]}x^2+2{bc[1]}/{d[5]}x+{cc[1]}/{d[6]}", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>$\\simplify{({ac[2]}/{d[8]})x^3+({bc[2]}/{d[9]})x^2+({cc[2]}/{d[10]})x+({dc[2]}/{d[11]})}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3{ac[2]}/{d[8]}x^2+2{bc[2]}/{d[9]}x+{cc[2]}/{d[10]}", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "<p><strong>Differentiate the following polynomials.</strong></p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"cc": {"definition": "repeat(random(-15..15),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc", "description": ""}, "ac": {"definition": "repeat(random(-3..3),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "ac", "description": ""}, "d": {"definition": "repeat(random(1..9),12)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "dc": {"definition": "repeat(random(-30..30),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "dc", "description": ""}, "bc": {"definition": "repeat(random(-10..10),3)", "templateType": "anything", "group": "Ungrouped variables", "name": "bc", "description": ""}}, "metadata": {"description": "<p>More work on differentiation with fractional coefficients.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"name": "341c, Differentiation of exponential, log, trig functions", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Differentiation - exponentials, log, trig functions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lois Rollings", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/326/"}, {"name": "Sid Gurjar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3870/"}, {"name": "Venkata Lakshmipathi Raju Chinthalapati", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4574/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "tags": ["calculus", "Calculus", "Differentiation", "differentiation"], "metadata": {"description": "<p>A question to test basic&nbsp;differentiation of functions, including powers of x, trig, log and exponential functions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Differentiate the&nbsp;following functions with respect to $x$, simplifying your answers where possible.</p>", "advice": "<p>This&nbsp;<a href=\"http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-table1-2009-1.pdf\">leaflet</a> from Mathcentre will remind you of the rules for differentiating these functions.</p>\n<p></p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)//Use as a power", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-5..5 except -1..1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$x^\\var{a}-\\sin(\\var{b}x)$</p>", "answer": "{a}x^{a-1}-{b}cos({b}x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\var{a}e^{-\\var{b}x}+\\frac{\\var{c}}{x}$</p>", "answer": "-{a}{b}e^(-{b}x)-{c}/x^2", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\var{d}\\ln(\\var{c}x)-\\var{a*b}\\cos(x/\\var{b})$</p>", "answer": "{d}/x+{a}sin(x/{b})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\var{a}\\sqrt{x}-\\var{c}e^{\\var{d}x}$</p>", "answer": "{a}/(2sqrt(x))-{c}{d}e^({d}x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\var{d}-\\frac{\\var{b}}{x^2}$</p>", "answer": "2{b}/x^3", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\var{10*d}\\sin(\\var{c/10}x)+\\var{b}x$</p>", "answer": "{c}*{d}cos({c}/10*x)+{b}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "type": "question"}]}, {"name": "503, Chain rule - powers", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Chain rule - powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "functions": {}, "ungrouped_variables": ["c"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p><strong>This question is the chain rule again.</strong></p>\n<p>This time, the function that is being differentiated is the term inside the brackets.</p>\n<p>This is another 'chain rule by inspection' kind of question to save time and paper.</p>\n<p>Firstly, differentiate everything inside the brackets.</p>\n<p>Then multiply the existing coefficient of the bracket by this result.</p>\n<p>Finally, multiply by the original magnitude of the power and decrease the power by one.</p>\n<p>If these steps confuse you, look back at <em>'Differentiation - Basic Polynomial Expressions'</em>&nbsp;and make sure you understand fully how to work those types of questions out.</p>", "rulesets": {}, "parts": [{"prompt": "<p>$y=(\\var{c[0]}x-1)^3$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "3*{c[0]}({c[0]}x-1)^2", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=(\\var{c[1]}-x)^5$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "-5({c[1]}-x)^4", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=(x^2+\\var{c[2]})^4$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "8x*(x^2+{c[2]})^3", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=(x^3-\\var{c[3]})^2$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "6x^2*(x^3-{c[3]})", "marks": "2", "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$y=(1-\\var{c[4]}x^2)^3$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrangepoints": 5, "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "-6{c[4]}x*(1-{c[4]}x^2)^2", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p><strong>Differentiate the following using the chain rule.</strong></p>\n<p>Do not write out $dy/dx$; only input the differentiated right hand side of each equation.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"definition": "repeat(random(2..9),8)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}}, "metadata": {"notes": "", "description": "<p>Using the chain rule with polynomials</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"name": "504, Quotient rule", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Differentiation - Quotient Rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "functions": {}, "ungrouped_variables": ["c", "p"], "tags": [], "advice": "<p><strong>These questions use the&nbsp;quotient&nbsp;rule.</strong></p>\n<p>The quotient&nbsp;rule is defined as</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}+u\\frac{dv}{dx}}{v^2}$</p>\n<p style=\"text-align: center;\">when $y=\\frac{u}{v}$</p>\n<p><strong>Worked example using Part a:</strong></p>\n<p style=\"text-align: center;\"><fraction><numerator>$x$</numerator><denominator>$\\simplify{x+{c[0]}}$</denominator></fraction></p>\n<p style=\"text-align: left;\">This expression is the result of&nbsp;$x$ divided by&nbsp;($\\simplify{x+{c[0]}}$).</p>\n<p style=\"text-align: left;\">We can therefore say:</p>\n<p style=\"text-align: left;\">$u=x$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$v=\\simplify{x+{c[0]}}$,</p>\n<p style=\"text-align: left;\">Hence meaning that $y=\\frac{u}{v}$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">We already have what $u$ and $v$ equal, so all we have to do is find what $\\frac{du}{dx}$ and $\\frac{dv}{dx}$ are, and then substitute everything into the rule.</p>\n<p style=\"text-align: left;\">Differentiating with respect to $x$, we get:</p>\n<p style=\"text-align: left;\">$\\frac{du}{dx}=1$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$\\frac{dv}{dx}=1$.</p>\n<p style=\"text-align: left;\">As there are no powers or coefficients of $x$ that are $&gt;1$, this is a very simple version of the quotient&nbsp;rule, but knowing how to work out this equation formally will make more difficult looking problems just as simple.</p>\n<p style=\"text-align: left;\">Substituting in all the results we've found, we get:</p>\n<p style=\"text-align: left;\">$\\frac{dy}{dx}=\\frac{1(\\simplify{x+{c[0]}})+1(x)}{\\simplify{(x+{c[0]})^2}}$</p>\n<p style=\"text-align: left;\">We then simplify, collecting all the terms, to get our final answer of:</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=\\simplify{((2x+{c[0]}))/(x+{c[0]})^2}$</p>", "rulesets": {"std": ["all"]}, "parts": [{"variableReplacements": [], "prompt": "<p><fraction><numerator>$\\simplify{{c[1]}x+{c[2]}}$</numerator><denominator>$\\simplify{x+{c[3]}}$</denominator></fraction></p>", "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({c[1]}*{c[3]}-{c[2]})/(x+{c[3]})^2", "marks": "2", "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}, {"variableReplacements": [], "prompt": "<p><fraction><numerator>$\\simplify{{c[6]}-sqrt(x)}$</numerator><denominator>$\\simplify{({c[7]}+x)^2}$</denominator></fraction></p>", "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(-1({c[7]}+x)-4(sqrt(x)*({c[6]}-sqrt(x))))/(2sqrt(x)({c[7]}+x)^3)", "marks": "3", "checkvariablenames": true, "checkingtype": "absdiff", "vsetrange": [0, 1]}], "statement": "<p><strong>Differentiate the following expressions with respect to $x$ using the quotient rule.</strong></p>\n<p><strong></strong>Simplify your answers as much as possible.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}\n", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "variables": {"p": {"definition": "repeat(random(2..7),6)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "c": {"definition": "repeat(random(-9..9 except 0),12)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}}, "metadata": {"notes": "", "description": "<p>An introduction to using the quotient rule</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}, {"name": "240, Poisson Distribution", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Poisson Distribution (Dentist)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Patricia Cogan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3359/"}], "tags": [], "metadata": {"description": "<p>The number of patients arriving at a dentist&rsquo;s surgery each afternoon follows<br />a Poisson distribution, with a mean of four patients per hour.<br />Calculate the probability that in a particular one-hour period</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p><span>T</span><span>he number of patients arriving at a dentist&rsquo;s surgery each afternoon follows a Poisson distribution, with a mean of $\\var{l}$ patients per hour.<br/><br/></span></p>\n<p><span></span></p>\n<p>&nbsp;The Poisson distribution formula: $P(r)=\\frac{\\lambda^re^{-\\lambda}}{r!}$ or&nbsp;$P(r)=e^{-\\lambda}\\left[\\frac{\\lambda^r}{r!}\\right]$</p>\n<p></p>\n<p></p>\n<p><strong>Please give your answer to at least 3 decimal places.</strong></p>", "advice": "<p><strong>Part (a)</strong></p>\n<p>Remember that for a Poisson random variable:<br/>\\begin{align}<br/>\\operatorname{P}(X=x)&amp;=\\dfrac{\\lambda^x\\times e^{-\\lambda}}{x!}\\\\<br/>\\end{align}</p>\n<p>1.\\[ \\begin{eqnarray*}\\operatorname{P}(X = \\var{x}) &amp;=&amp; \\frac{\\var{l} ^ {\\var{x}}e ^ { -\\var{l}}} {\\var{x}!}\\\\&amp; =&amp; \\var{answer1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<p>&nbsp;</p>\n<p><strong>Part (b)</strong></p>\n<p><span>The probability that in a particular one hour period, the number of patients entering the waiting room</span><span>&nbsp;will be between&nbsp;$\\var{x}$ and $\\var{y}$ inclusive is given by:</span></p>\n<p>$P(\\var{x} \\leq X\\leq\\var{y}) = P(X=\\var{x}) + P(X=\\var{x+1}) +P(X=\\var{y})$</p>\n<p>where&nbsp;</p>\n<p>$P(X=\\var{x}) =\\frac{\\var{l}^{\\var{x}}e^{-\\var{l}}}{\\var{x}!}=\\var{prx}$</p>\n<p><span>$P(X=\\var{x+1}) =\\frac{\\var{l}^{\\var{x+1}}e^{-\\var{l}}}{\\var{x+1}!}=\\var{prx1}$</span></p>\n<p><span>$P(X=\\var{y}) =\\frac{\\var{l}^{\\var{y}}e^{-\\var{l}}}{\\var{y}!}=\\var{pry}$</span></p>\n<p>Hence&nbsp;</p>\n<p><span>$P(\\var{x} \\leq X \\leq \\var{y})=$ $\\var{prx}+\\var{prx1}+\\var{pry}=\\var{answer2}$</span></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(10..50#5)", "description": "<p>time interval</p>", "templateType": "anything", "can_override": false}, "answer1": {"name": "answer1", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!", "description": "", "templateType": "anything", "can_override": false}, "prx": {"name": "prx", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!", "description": "", "templateType": "anything", "can_override": false}, "answer2": {"name": "answer2", "group": "Ungrouped variables", "definition": "((e^-l)*(l^x))/x!+((e^-l)*(l^(x+1)))/(x+1)!+((e^-l)*(l^y))/y!", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "x+2", "description": "<p>upper value of X</p>", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..5#1)", "description": "<p>time hour</p>", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(3..12)", "description": "<p>number of customers entering the shop</p>", "templateType": "anything", "can_override": false}, "prx1": {"name": "prx1", "group": "Ungrouped variables", "definition": "((e^-l)*(l^(x+1)))/(x+1)!", "description": "", "templateType": "anything", "can_override": false}, "pry": {"name": "pry", "group": "Ungrouped variables", "definition": "((e^-l)*(l^y))/y!", "description": "", "templateType": "anything", "can_override": false}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "random(5..10#1)", "description": "<p>average, lambda</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["l", "x", "n", "y", "t", "answer1", "answer2", "prx", "prx1", "pry"], "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>Write down the value of $\\lambda$ for one hour.</p>\n<p></p>\n<p>$\\lambda = $ [[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": "l", "maxValue": "l", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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><span>Calculate the probability that in a particular one-hour period exactly $\\var{x}$ patients will arrive. Please give your answer to at least 3 decimal places.<br/><br/><br/></span></p>\n<p>$P(r = \\var{x}) =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "minValue": "answer1-0.005", "maxValue": "answer1+0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "answer1-0.001", "maxValue": "answer1+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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><span>Find the probability that in a particular one hour period, the number of patients entering the waiting room</span><span> will be between $\\var{x}$ and $\\var{y}$ inclusive. Please give your answer to at least 3 decimal places.</span></p>\n<p><span>&nbsp;</span></p>\n<p>$P(\\var{x}\\leq r \\leq\\var{y}) =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "minValue": "answer2 -0.005", "maxValue": "answer2 +0.005", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "answer2 -0.005", "maxValue": "answer2 +0.005", "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", "type": "question"}]}, {"name": "241, Binomial Distribution", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Binomial Distribution (Cycling)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}], "tags": ["binomial", "Binomial", "REBEL", "rebel", "Rebel", "rebelmaths"], "metadata": {"description": "<p>It is estimated that 30% of all CIT students cycle to college. If a random sample of eight CIT students is chosen, calculate the probability that...</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Please give your answer to at least 3 decimal places.</strong></p>\n<p>It is estimated that $\\var{p_perc}$%&nbsp;of all CIT students cycle to college. A&nbsp;random sample of $\\var{n}$ CIT students is chosen.</p>\n<p></p>", "advice": "<p><strong>Part (a)</strong></p>\n<p>If a&nbsp;random variable $X$&nbsp;follows a&nbsp;binomial distribution with parameters $n$&nbsp;and $p$.&nbsp;The probability of $r$ successes out of $n$ trials is given by:</p>\n<p>$P(X=r)=P(r,n)=C^n_{r}p^{r}q^{n-r}$</p>\n<p>where $p$ is the probability of success for each trial and $q$ is the probability of failure for each trial.</p>\n<p>The probability that a student cycles to college&nbsp;is $\\var{p}$, therefore $p=\\var{p}$ and $q=1-\\var{p}=\\var{q}$.</p>\n<p>We are interested in claculating the probability that none of the sample of $\\var{n}$ students cycle to college so $r=0$ and $n=\\var{n}$</p>\n<p><span>$P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}$&nbsp;</span></p>\n<p><span>$P(\\var{r0}, \\var{n})= \\var{pr0}$</span></p>\n<p><span></span></p>\n<p><strong>Part (b)</strong></p>\n<p><span>We are interested in claculating the probability that&nbsp;at least&nbsp;$\\var{r}$ of the&nbsp;<span>$\\var{n}$</span><span> students cycle to college. Let $X$ represent the number of students that cycle to college. We need to calculate:</span></span></p>\n<p><span>$P(X \\geq \\var{r}) = P(X= \\var{r}) + P(X= \\var{r+1})+...+ P(X=\\var{n})$</span></p>\n<p><span></span></p>\n<p>Since $P(X=\\var{r0})+P(X=\\var{r0+1})+...+P(X=\\var{n})=\\var{r0+1}$&nbsp;</p>\n<p><span>We may write&nbsp;</span></p>\n<p><span>$P(X \\geq \\var{r}) = 1-P(X= \\var{r0}) - P(X=\\var{r0+1})-...- P(X=\\var{r-1})$</span></p>\n<p><span></span></p>\n<p>where</p>\n<p><span>$P(X= \\var{r0})=P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}=\\var{pr0}$&nbsp;</span></p>\n<p><span>$P(X=1) =P(1, \\var{n})= C^\\var{n}_{1}$ $\\var{p}^{1}$ $\\var{q}^{\\var{n}-1}$ $=\\var{pr1}$</span></p>\n<p><span>$P(X=2) = P(2, \\var{n})=$ $C^\\var{n}_{2}$&nbsp;</span>$\\var{p}^{2}$ $\\var{q}^{\\var{n}-2}$ $=\\var{pr2}$</p>\n<p></p>\n<p>Then&nbsp;</p>\n<p><span>$P(X \\geq \\var{r}) = 1-\\var{qn}-\\var{pr1}-\\var{pr2}=\\var{answer2}$</span></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"pr1": {"name": "pr1", "group": "Ungrouped variables", "definition": "n*p*q^(n-1)", "description": "<p>probability that r = 1</p>", "templateType": "anything", "can_override": false}, "r0": {"name": "r0", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "anything", "can_override": false}, "p_perc": {"name": "p_perc", "group": "Ungrouped variables", "definition": "p*100", "description": "<p>percentage of students that cycle to college</p>", "templateType": "anything", "can_override": false}, "pr2": {"name": "pr2", "group": "Ungrouped variables", "definition": "((n*(n-1))/2)*(p^2)*q^(n-2)", "description": "<p>probability that r = 2</p>", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "1-p", "description": "<p>probability tha an individual does not cycle to college</p>", "templateType": "anything", "can_override": false}, "answer2": {"name": "answer2", "group": "Ungrouped variables", "definition": "1-answer1", "description": "", "templateType": "anything", "can_override": false}, "pr3": {"name": "pr3", "group": "Ungrouped variables", "definition": "((n*(n-1)*(n-2))/6)*(p^3)*(q^(n-3))", "description": "<p>probability that r = 3</p>", "templateType": "anything", "can_override": false}, "r": {"name": "r", "group": "Ungrouped variables", "definition": "3", "description": "<p>more than r of the students cycle to college</p>", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "n-2", "description": "", "templateType": "anything", "can_override": false}, "answer1": {"name": "answer1", "group": "Ungrouped variables", "definition": "if(r=2,pr0+pr1, pr0+pr1+pr2)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(0.1..0.4#0.05)", "description": "<p>the probability that an individual student cycles to college</p>", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..12)", "description": "<p>sample size</p>", "templateType": "anything", "can_override": false}, "qn": {"name": "qn", "group": "Ungrouped variables", "definition": "q^n", "description": "", "templateType": "anything", "can_override": false}, "pr0": {"name": "pr0", "group": "Ungrouped variables", "definition": "q^n", "description": "<p>probability that&nbsp;r = 0</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["p", "p_perc", "n", "q", "r", "pr0", "pr1", "pr2", "pr3", "answer1", "answer2", "qn", "r0", "n2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the probability that none of the <span>$\\var{n}$</span>&nbsp;students in the sample cycle to college.</p>", "minValue": "(q^n)-0.001", "maxValue": "(q^n)+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the probability that at least&nbsp;$\\var{r}$ of the&nbsp;<span>$\\var{n}$</span> students cycle to college.</p>", "minValue": "answer2 -0.001", "maxValue": "answer2 +0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "342, Probability Complement rule", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "probability complement 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": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Calculate probability using P(A) = 1-P(not A)</p>", "licence": "None specified"}, "statement": "<p>Two $\\var{n}$-sided dice, each with sides labelled $1,2...\\var{n}$ are rolled and their scores are added.<br/><br/><br/><em></em></p>", "advice": "<h3>part a)</h3>\n<p>We are looking for pairs of numbers than total to make $\\var{min[choice]}$ or less, which is quite a small total, so it doesn't take too long to find that the pairs of numbers are:</p>\n<p>\\[ \\var{pairstruncate} \\]</p>\n<p>so there {isorare} $\\var{pairschoice[choice]+1}$ {event} we are interested in, and therefore the probability is&nbsp;<br/><br/>\\[ P(\\text{total is} \\leq \\var{min[choice]}) = \\frac{\\var{pairschoice[choice]+1}}{\\var{total}} \\]&nbsp;</p>\n<h3>part b)</h3>\n<p>There are a very large number of ways to totals of $\\var{min[choice]+1}$ or greater, so in this case we use the complement rule;<br/><br/>\\[P(A') = 1-P(A)\\]</p>\n<p>which in this case means</p>\n<p>\\[P(\\text{total is}\\ge \\var{min[choice]+1}) = 1-P(\\text{total is} \\leq \\var{min[choice]}) = 1-\\frac{\\var{pairschoice[choice]+1}}{\\var{total}} = \\var[fractionnumbers]{1-options[choice]/{total}}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"min": {"name": "min", "group": "Ungrouped variables", "definition": "list(2..4)", "description": "", "templateType": "anything", "can_override": false}, "options": {"name": "options", "group": "Ungrouped variables", "definition": "[1,3,6]", "description": "", "templateType": "anything", "can_override": false}, "choice": {"name": "choice", "group": "Ungrouped variables", "definition": "random(0..2)", "description": "", "templateType": "anything", "can_override": false}, "total": {"name": "total", "group": "Ungrouped variables", "definition": "n*n", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(5..10)", "description": "", "templateType": "anything", "can_override": false}, "pairs": {"name": "pairs", "group": "Ungrouped variables", "definition": "[[1,1],[1,2],[2,1],[2,2],[1,3],[3,1]]", "description": "", "templateType": "anything", "can_override": false}, "pairschoice": {"name": "pairschoice", "group": "Ungrouped variables", "definition": "[0,2,5]", "description": "", "templateType": "anything", "can_override": false}, "pairstruncate": {"name": "pairstruncate", "group": "Ungrouped variables", "definition": "pairs[0..pairschoice[choice]+1]", "description": "", "templateType": "anything", "can_override": false}, "event": {"name": "event", "group": "Ungrouped variables", "definition": "if(pairschoice[choice]+1=1,'event','events')", "description": "", "templateType": "anything", "can_override": false}, "isorare": {"name": "isorare", "group": "Ungrouped variables", "definition": "if(pairschoice[choice]+1=1,'is','are')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["min", "options", "choice", "total", "n", "pairs", "pairschoice", "pairstruncate", "event", "isorare"], "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>What is the probability that the score is less than or equal to $\\var{min[choice]}$?</p>\n<p></p>\n<p>[[0]]&nbsp;<em>(Give your answer as a fraction)</em></p>\n<p></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": "{options[choice]}/{total}", "maxValue": "{options[choice]}/{total}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": "100", "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"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>What is the probability that the score is $\\var{min[choice]+1}$ or greater?<br/><br/><em>Hint: It will probably save you time if you use the complement rule.</em></p>\n<p><em></em></p>\n<p>[[0]]&nbsp; &nbsp;<em>(Give your answer as a fraction)</em></p>\n<p><em></em></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": "1-{options[choice]}/{total}", "maxValue": "1-{options[choice]}/{total}", "correctAnswerFraction": true, "allowFractions": true, "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", "type": "question"}]}, {"name": "606, Forces", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Vectors: Resolving Forces 2", "extensions": [], "custom_part_types": [], "resources": ["question-resources/force_component_image_4.png"], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Find the $x$ and $y$ components of the resultant force on an object, when multiple forces are&nbsp;applied at&nbsp;different angles.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><img width=\"430\" height=\"338\" style=\"display: block; margin-left: auto; margin-right: auto;\" src=\"resources/question-resources/force_component_image_4.png\"/></p>\n<p>$F = \\var{force1} \\, \\mathrm{N}$ at an angle of $\\theta = \\var{theta1}^{\\circ}$ clockwise from the left horizonatal.<br/>$P = \\var{force2} \\, \\mathrm{N}$ vertically upwards.<br/>$Q = \\var{force3} \\, \\mathrm{N}$ at an able of $\\theta^{\\ast} = \\var{theta2}^{\\circ}$ clockwise from the right horizontal.&nbsp;</p>\n<p><em>Give your answers to the following questions in Newtons to 3 decimal places.</em></p>", "advice": "<h4>a) - c)</h4>\n<p>Resolve each force in the&nbsp;HORIZONTAL:</p>\n<p>$P$ doesn't contribute.</p>\n<p>The force $F$ is at $\\var{theta1}^{\\circ}$ to the horizontal,&nbsp;therefore has a contribution of $F \\times \\cos \\var{theta1}^{\\circ} = \\var{force1} \\times \\cos \\var{theta1}^{\\circ} = \\var{precround(force1*cos(radians(theta1)),3)}$. &nbsp;This&nbsp;force is acting to the left.</p>\n<p>The force $Q$ is at $\\var{theta2}^{\\circ}$ to the horizontal,&nbsp;therefore has a contribution of $Q \\times cos&nbsp;\\var{theta2}^{\\circ} = \\var{force3} \\times \\cos\\var{theta2}^{\\circ} = \\var{precround(force3*cos(radians(theta2)),3)}$. This is acting to the right.</p>\n<p>Therefore the sum of components in the $x$-direction is</p>\n<p>forces acting to the right - forces acting to the left</p>\n<p>\\[\\var{precround(force3*cos(radians(theta2)),3)} - \\var{precround(-force1*cos(radians(180-theta1)),3)}\\]<br/>\\[= \\var{precround(force1*cos(radians(180-theta1)) + force3*cos(radians(theta2)),3)}\\]</p>\n<p>(if this value is positive then it is acting to the right, if it is negative it is acting to the left)</p>\n<h4>d) - g)</h4>\n<p>Resolve each force from the VERTICAL:</p>\n<p>For the force $P$ this is acting completely in the positive direction, at no angle. Therefore it's contribution is $\\var{force2}$. </p>\n<p>The force $F$ is at $\\var{theta1}^{\\circ}$ to the horizontal,&nbsp;therefore has a contribution of $F \\times \\sin \\var{theta1}^{\\circ} = \\var{force1} \\times \\sin \\var{theta1}^{\\circ} = \\var{precround(force1*sin(radians(theta1)),3)}$. &nbsp;This&nbsp;force is acting upwards.</p>\n<p>The force $Q$ is at $\\var{theta2}^{\\circ}$ to the horizontal,&nbsp;therefore has a contribution of $Q \\times cos&nbsp;\\var{theta2}^{\\circ} = \\var{force3} \\times \\cos\\var{theta2}^{\\circ} = \\var{precround(force3*cos(radians(theta2)),3)}$. This is acting to the right.</p>\n<p>Therefore the sum of components in the $y$-direction is</p>\n<p>forces acting up - forces acting down</p>\n<p>\\[\\var{precround(force3*sin(radians(theta2)),3)} - \\var{precround(-force1*sin(radians(180-theta1)),3)}\\]<br/>\\[= \\var{precround(force1*sin(radians(180-theta1)) + force3*sin(radians(theta2)),3)}\\]</p>\n<p>(if this value is positive then it is acting to the right, if it is negative it is acting to the left)</p>\n<p></p>\n<p>&nbsp;</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"f1y": {"name": "f1y", "group": "components", "definition": "force1*sin(radians(theta1))", "description": "", "templateType": "anything", "can_override": false}, "f1x": {"name": "f1x", "group": "components", "definition": "-force1*cos(radians(theta1))", "description": "", "templateType": "anything", "can_override": false}, "f3y": {"name": "f3y", "group": "components", "definition": "-force3*sin(radians(theta2))", "description": "", "templateType": "anything", "can_override": false}, "f3x": {"name": "f3x", "group": "components", "definition": "force3*cos(radians(theta2))", "description": "", "templateType": "anything", "can_override": false}, "force2": {"name": "force2", "group": "Ungrouped variables", "definition": "random(2 .. 7#0.25)", "description": "", "templateType": "randrange", "can_override": false}, "theta2": {"name": "theta2", "group": "Ungrouped variables", "definition": "random(5 .. 85#1)", "description": "", "templateType": "randrange", "can_override": false}, "theta1": {"name": "theta1", "group": "Ungrouped variables", "definition": "random(2 .. 88#1)", "description": "", "templateType": "randrange", "can_override": false}, "force3": {"name": "force3", "group": "Ungrouped variables", "definition": "random(2 .. 10#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "force1": {"name": "force1", "group": "Ungrouped variables", "definition": "random(3 .. 15#1)", "description": "", "templateType": "randrange", "can_override": false}, "yangle1": {"name": "yangle1", "group": "Ungrouped variables", "definition": "90-theta1", "description": "", "templateType": "anything", "can_override": false}, "yangle2": {"name": "yangle2", "group": "Ungrouped variables", "definition": "90+theta2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["force1", "force2", "force3", "theta1", "theta2", "yangle1", "yangle2"], "variable_groups": [{"name": "components", "variables": ["f1x", "f1y", "f3x", "f3y"]}], "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>Find the component of $F$ in the $x$-direction</p>", "minValue": "f1x", "maxValue": "f1x", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "<p>You have not given your answer to the correct precision.</p>", "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, "prompt": "<p>Find the component of $Q$ in the $x$-direction.</p>", "minValue": "f3x", "maxValue": "f3x", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "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": [{"variable": "f1x", "part": "p0", "must_go_first": false}, {"variable": "f3x", "part": "p1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find&nbsp;the resultant force in the $x$-direction.</p>", "minValue": "f1x+f3x", "maxValue": "f1x+f3x", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "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, "prompt": "<p>Find the component of $P$ in the $y$-direction.</p>", "minValue": "force2", "maxValue": "force2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "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, "prompt": "<p>Find the component of $F$ in the $y$-direction.</p>", "minValue": "f1y", "maxValue": "f1y", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "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, "prompt": "<p>Find the component of $Q$ in the $y$-direction.</p>", "minValue": "f3y", "maxValue": "f3y", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "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": [{"variable": "f1y", "part": "p4", "must_go_first": false}, {"variable": "f3y", "part": "p5", "must_go_first": false}, {"variable": "force2", "part": "p3", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find&nbsp;the resultant force in the $y$-direction.</p>", "minValue": "f1y+force2+f3y", "maxValue": "f1y+force2+f3y", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "605, Forces (easy)", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Vectors: Resolving forces 1a", "extensions": [], "custom_part_types": [], "resources": ["question-resources/force_component_image.png", "question-resources/force_component_image_PgpiR1U.png", "question-resources/Picture2_7CTf2tv.png", "question-resources/Picture2_4njOTSR.png", "question-resources/Picture1_kv6rtT4.png"], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Amy Chadwick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/505/"}, {"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Find the $x$ and $y$ components of a force which is&nbsp;applied at an angle to a particle. Resolve using $F \\cos \\theta$. The force acts in the positive $x$ and positive $y$ direction.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p style=\"text-align: center;\"><img src=\"resources/question-resources/force_component_image_PgpiR1U.png\" width=\"552\" height=\"350\"/></p>\n<p style=\"text-align: left;\">In the above diagram the force $F=\\var{force} \\ \\mathrm{N}$ and the angle $\\theta = \\var{angle}^{\\circ}$.</p>", "advice": "<h4><img src=\"resources/question-resources/Picture2_7CTf2tv.png\" width=\"212\" height=\"96\"/><img src=\"resources/question-resources/Picture1_kv6rtT4.png\" width=\"257\" height=\"112\"/></h4>\n<h4></h4>\n<h4>a)</h4>\n<p>The component of force in the $x$-direction can be found using $\\cos\\theta \\times F$. Remember to set&nbsp;your calculator to use&nbsp;degrees and not radians.</p>\n<p>\\begin{align} \\text{component in the }x \\text{-direction } &amp; = F \\cos \\theta \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &amp; = \\var{force} \\times \\cos \\var{angle} \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = \\var{precround(force*cos(radians(angle)),3)}\\end{align}</p>\n<h4>b)</h4>\n<p>This time we are using sin</p>\n<p>\\begin{align} \\text{component in the y-direction } &amp; = F \\sin \\theta \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &amp; = \\var{force} \\times \\sin \\var{angle} \\\\<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&amp; = \\var{precround(force*sin(radians(angle)),3)}\\end{align}</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"yangle": {"name": "yangle", "group": "Ungrouped variables", "definition": "90-angle", "description": "", "templateType": "anything", "can_override": false}, "force": {"name": "force", "group": "Ungrouped variables", "definition": "random(3 .. 15#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "angle": {"name": "angle", "group": "Ungrouped variables", "definition": "random(1 .. 89#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["force", "angle", "yangle"], "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>Find the component of the force in the $x$-direction, in Newtons to 3 decimal places.</p>", "minValue": "force*cos(radians(angle))", "maxValue": "force*cos(radians(angle))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "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, "prompt": "<p>Find the component of the force in the $y$-direction,&nbsp;in Newtons to 3 decimal places.</p>", "minValue": "force*cos(radians(yangle))", "maxValue": "force*cos(radians(yangle))", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "604, Vector magnitude", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Vectors: Magnitude 2b", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Calculate the magnitude of a 3-dimensional vector $\\mathbf v$, where $\\mathbf v$ is written in the form $v_1 \\mathbf i+v_2 \\mathbf j + v_3 \\mathbf k$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the magnitude of the vector $\\mathbf v =\\simplify[all,!noLeadingMinus]{{vi}*mathbf:i+{vj}*mathbf:j+{vk}*mathbf:k}$ .</p>", "advice": "<p>For a vector of the form $\\mathbf a = \\simplify{a_1*mathbf:i+a_2*mathbf:j+a_3*mathbf:k}$, the magnitude is found by calculating the square root of the sum of the components squared:</p>\n<p>\\[ |\\mathbf a| = \\sqrt{a_1^2+a_2^2+a_3^2}.\\]</p>\n<p>Therefore, for the vector $\\mathbf v =\\simplify[!noLeadingMinus]{{vi}*mathbf:i+{vj}*mathbf:j+{vk}*mathbf:k}$,</p>\n<p><span data-jme-visible=\"{magv_dp}=0\">\\[ \\begin{split} |\\mathbf v| &amp;\\,= \\simplify[!collectNumbers]{sqrt({vi}^2+{vj}^2+{vk}^2)} \\\\ &amp;\\,= \\sqrt{\\var{vi^2+vj^2+vk^2}} \\\\ &amp;\\, = \\var{magv} .\\end{split} \\]</span></p>\n<p><span data-jme-visible=\"{magv_dp}=1\">\\[ \\begin{split} |\\mathbf v| &amp;\\,= \\simplify[!collectNumbers]{sqrt({vi}^2+{vj}^2+{vk}^2)} \\\\ &amp;\\,= \\sqrt{\\var{vi^2+vj^2+vk^2}} \\\\ &amp;\\, = \\var{magv2} \\text{ (2 d.p.)} \\end{split} \\]</span></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"vi": {"name": "vi", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything", "can_override": false}, "vj": {"name": "vj", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "magv": {"name": "magv", "group": "Ungrouped variables", "definition": "(vi^2+vj^2+vk^2)^0.5", "description": "", "templateType": "anything", "can_override": false}, "magv2": {"name": "magv2", "group": "Ungrouped variables", "definition": "precround(magv,2)", "description": "", "templateType": "anything", "can_override": false}, "vk": {"name": "vk", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}, "magv_dp": {"name": "magv_dp", "group": "Ungrouped variables", "definition": "if(countdp(string(magv))<3,0,1)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "if(magv_dp=0,magv,magv2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["vi", "vj", "vk", "magv", "magv2", "magv_dp", "ans"], "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>$|\\mathbf v |= $[[0]]</p>\n<p>(<em>Give your 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": "{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, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "603, Vectors linear combination", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Vectors: Linear Combinations 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Given 3 vectors $\\mathbf v$, $\\mathbf a$ and $\\mathbf b$, find the constants $c_1$ and $c_2$ such that $\\mathbf v = c_1 \\mathbf a + c_2 \\mathbf b$ .</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Find the values of the constants $c_1$ and $c_2$ such that the vector $\\mathbf v = \\var{v}$ is a linear combination of the vectors $\\mathbf a=\\var{a}$ and $\\mathbf b=\\var{b}$: \\[ \\mathbf v = c_1 \\mathbf a + c_2 \\mathbf b. \\]</p>", "advice": "<p>We are trying to find the constants $c_1$ and $c_2$ such that we can express $\\mathbf v$ as a linear combination of the vectors $\\mathbf a$ and $\\mathbf b$:</p>\n<p>\\[ \\var{v} = c_1 \\var{a} + c_2 \\var{b} ,\\]</p>\n<p></p>\n<p>We can calculate $c_1$ and $c_2$ by solving the simultaneous equations</p>\n<p>\\[ \\begin{split} \\var{v[0]} &amp;\\,= \\simplify[all,!noLeadingMinus]{{a[0]}c_1 + {b[0]} c_2}&nbsp; \\\\ \\var{v[1]} &amp;\\,= \\simplify[all,!noLeadingMinus]{{a[1]}c_1 + {b[1]} c_2} . \\end{split} \\]</p>\n<p></p>\n<p>We find $c_1 = \\var{c1}$ and $c_2 = \\var{c2}$, and therefore,</p>\n<p>\\[\\var{v} = \\simplify[!noLeadingMinus,unitFactor]{{c1}{a}+{c2}{b}}. \\]</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": "vector(repeat(random(-5..5 except 0),2))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(random(-5..-1),random(-5..5))", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "c1*a+c2*b", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(-4..5 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(-5..6 except [0,c1])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "v", "c1", "c2"], "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>$c_1= $[[0]]</p>\n<p>$c_2= $[[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": "{c1}", "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": "{c2}", "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": "602, Vector addition and scalars", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Vectors: Scalars and Addition 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p>Calculating several linear combinations of three 2-dimensional vectors.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given three vectors,</p>\n<p>\\[ \\mathbf a = \\var{a},\\quad \\mathbf b = \\var{b}, \\quad \\mathbf c = \\var{c}, \\]</p>\n<p>calculate the following:</p>", "advice": "<p>When multiplying a vector by a scalar, and when adding or subtracting vectors, we must use the following rules:</p>\n<p>For the vectors $\\mathbf v = \\begin{pmatrix} v_1 \\\\ v_2 \\end{pmatrix}$ and&nbsp;$\\mathbf w = \\begin{pmatrix} w_1 \\\\ w_2 \\end{pmatrix}$,&nbsp;</p>\n<ul>\n<li>$\\alpha \\,\\mathbf v =&nbsp;\\begin{pmatrix} \\alpha \\,v_1 \\\\ \\alpha \\, v_2 \\end{pmatrix},\\, \\text{ where $\\alpha$ is a&nbsp;constant,}$</li>\n</ul>\n<p></p>\n<ul>\n<li>\n<p>$\\mathbf v \\pm \\mathbf w =&nbsp;\\begin{pmatrix} v_1 \\pm w_1 \\\\ v_2 \\pm w_2 \\end{pmatrix}.$</p>\n</li>\n</ul>\n<p>Combining these rules:</p>\n<p>\\[&nbsp;\\alpha \\,\\mathbf v \\pm \\beta \\, \\mathbf w=&nbsp;\\begin{pmatrix} \\alpha \\,v_1 \\\\ \\alpha \\, v_2 \\end{pmatrix} \\pm \\begin{pmatrix} \\beta \\, w_1 \\\\ \\beta \\, w_2 \\end{pmatrix} =&nbsp;\\begin{pmatrix} \\alpha \\,v_1 \\pm \\beta \\, w_1 \\\\ \\alpha \\, v_2 \\pm \\beta \\, w_2 \\end{pmatrix}. \\]</p>\n<p></p>\n<p>Part a)</p>\n<p><span data-jme-visible=\"{n1}&gt;0\">\\[ \\begin{split} \\simplify{{m1}*mathbf:a +{n1}*mathbf:b} \\, &amp;\\,= \\simplify[!collectNumbers]{{m1}*{a}+{n1}*{b}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{m1*a}+{n1*b}} \\\\ &amp;\\,= \\var{sola} \\end{split} \\]</span></p>\n<p><span data-jme-visible=\"{n1}&lt;0\">\\[ \\begin{split} \\simplify{{m1}*mathbf:a +{n1}*mathbf:b} \\, &amp;\\,= \\simplify[!collectNumbers]{{m1}*{a}+{n1}*{b}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{m1*a}-{abs(n1)*b}} \\\\ &amp;\\,= \\var{sola} \\end{split} \\]</span></p>\n<p></p>\n<p>Part b)</p>\n<p><span data-jme-visible=\"{p1}&gt;0\">\\[ \\begin{split} \\simplify{{m2}*mathbf:a +{p1}*mathbf:c} \\, &amp;\\,= \\simplify[!collectNumbers]{{m2}*{a}+{p1}*{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{m2*a}+{p1*c}} \\\\ &amp;\\,= \\var{solb} \\end{split} \\]</span></p>\n<p><span data-jme-visible=\"{p1}&lt;0\">\\[ \\begin{split} \\simplify{{m2}*mathbf:a +{p1}*mathbf:c} \\, &amp;\\,= \\simplify[!collectNumbers]{{m2}*{a}+{p1}*{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{m2*a}-{abs(p1)*c}} \\\\ &amp;\\,= \\var{solb} \\end{split} \\]</span></p>\n<p></p>\n<p>Part c)</p>\n<p><span data-jme-visible=\"{p2}&gt;0\">\\[ \\begin{split} \\simplify{{n2}*mathbf:b +{p2}*mathbf:c} \\, &amp;\\,= \\simplify[!collectNumbers]{{n2}*{b}+{p2}*{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{n2*b}+{p2*c}} \\\\ &amp;\\,= \\var{solc} \\end{split} \\]</span></p>\n<p><span data-jme-visible=\"{p2}&lt;0\">\\[ \\begin{split} \\simplify{{n2}*mathbf:b +{p2}*mathbf:c} \\, &amp;\\,= \\simplify[!collectNumbers]{{n2}*{b}+{p2}*{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{n2*b}-{abs(p2)*c}} \\\\ &amp;\\,= \\var{solc} \\end{split} \\]</span></p>\n<p></p>\n<p>Part d)</p>\n<p><span data-jme-visible=\"{p3}&gt;0\">\\[ \\begin{split} \\simplify{{k}*(mathbf:a+mathbf:b) +{p3}*mathbf:c} \\, &amp;\\,= \\simplify[!collectNumbers]{{k}*({a}+{b})+{p3}*{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{k}*{a+b}+{p3}*{c}} \\\\ &amp;\\,= \\simplify[!collectNumbers]{{k*(a+b)}+{p3*c}} \\\\ &amp;\\,= \\var{sold} \\end{split} \\]</span></p>\n<p><span data-jme-visible=\"{p3}&lt;0\">\\[ \\begin{split} \\simplify{{k}*(mathbf:a+mathbf:b) +{p3}*mathbf:c} \\, &amp;\\,= \\simplify[!collectNumbers]{{k}*({a}+{b})+{p3}*{c}} \\\\ &amp;\\,=\\simplify[!collectNumbers]{{k}*{a+b}+{p3}*{c}} \\\\&amp;\\,= \\simplify[!collectNumbers]{{k*(a+b)}-{abs(p3)*c}} \\\\ &amp;\\,= \\var{sold} \\end{split} \\]</span></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": "vector(repeat(random(-5..10),2))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(random(-5..10 except 0),random(-5..10))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "vector(random(-5..10),random(-5..10 except 0))", "description": "", "templateType": "anything", "can_override": false}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "n1": {"name": "n1", "group": "Ungrouped variables", "definition": "random(1,-1)*random(2..6 except m1)", "description": "", "templateType": "anything", "can_override": false}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(1,-1)*random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "random(1,-1)*random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "solb": {"name": "solb", "group": "Ungrouped variables", "definition": "m2*a+p1*c", "description": "", "templateType": "anything", "can_override": false}, "solc": {"name": "solc", "group": "Ungrouped variables", "definition": "n2*b+p2*c", "description": "", "templateType": "anything", "can_override": false}, "sold": {"name": "sold", "group": "Ungrouped variables", "definition": "k*(a+b)+p3*c", "description": "", "templateType": "anything", "can_override": false}, "p3": {"name": "p3", "group": "Ungrouped variables", "definition": "random(1,-1)*random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "sola": {"name": "sola", "group": "Ungrouped variables", "definition": "vector(m1*a+n1*b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a[1]>0 ", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "m1", "m2", "n1", "n2", "p1", "p2", "p3", "k", "sola", "solb", "solc", "sold"], "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{{m1}*mathbf:a+{n1}*mathbf:b}=$ [[0]]</p>", "gaps": [{"type": "matrix", "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, "correctAnswer": "vector(sola[0],sola[1])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"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>$\\simplify{{m2}*mathbf:a+{p1}*mathbf:c}=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "vector(solb[0],solb[1])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"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>$\\simplify{{n2}*mathbf:b+{p2}*mathbf:c}=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "vector(solc[0],solc[1])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"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>$\\simplify{{k}*(mathbf:a+mathbf:b)+{p3}*mathbf:c}=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "Gap 3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "vector(sold[0],sold[1])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "601, Vector between points", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Vectors: Finding Vectors Between Points 1", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "tags": [], "metadata": {"description": "<p class=\"p1\">Given the coordinates of three 2-dimensional points $A$, $B$ and $C$, find the vectors $\\vec{AB}$, $\\vec{AC}$ and $\\vec{CB}$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>The points A, B and C have coordinates $(\\var{a[0]},\\, \\var{a[1]})$, $(\\var{b[0]},\\, \\var{b[1]})$, and $(\\var{c[0]},\\, \\var{c[1]})$ respectively.&nbsp;</p>\n<p>Find:</p>", "advice": "<p>To find the vector between two points, it can help to mark the points and their corresponding position vectors on a graph. In this case,&nbsp;let $\\bf a$, $\\bf b$ and $\\bf c$ be the position vectors of $A$, $B$, and $C$, such that&nbsp;</p>\n<p>\\[ \\mathbf a = \\var{a}, \\quad&nbsp;\\mathbf b = \\var{b}, \\quad&nbsp;\\mathbf c = \\var{c}.\\]</p>\n<p>{geogebra_applet('https://www.geogebra.org/m/jrrcctdj',defs)}</p>\n<p>To find the vector $\\vec{AB}$, consider how you can get between $A$ and $B$ only 'travelling' along the position vectors. Firstly, we need to go from $A$ to the origin $O$, which is the vector $\\mathbf{-a}$, and then from the $O$ to $B$, which is the vector $\\mathbf b$. Therefore, $\\vec{AB} = \\mathbf{-a}+\\mathbf{b}$, or more simply, \\[ \\vec{AB} = \\mathbf {b} - \\mathbf{a}.\\] The same method applies for finding the vectors $\\vec{AC}$ and $\\vec{CB}$.</p>\n<p>Therefore,</p>\n<p>\\[ \\vec{AB} = \\mathbf{b}-\\mathbf{a} = \\var{sola}, \\]</p>\n<p>\\[ \\vec{AC} = \\mathbf{c}-\\mathbf{a} = \\var{solb}, \\]</p>\n<p>\\[ \\vec{CB} = \\mathbf{b}-\\mathbf{c} = \\var{solc}, \\]</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "vector(random(-5..-3),random(2..3))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector(random(2..5),random(3..4 except a[1]))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "vector(random(1..2),random(-2..-4))", "description": "", "templateType": "anything", "can_override": false}, "sola": {"name": "sola", "group": "Ungrouped variables", "definition": "b-a", "description": "", "templateType": "anything", "can_override": false}, "solb": {"name": "solb", "group": "Ungrouped variables", "definition": "c-a", "description": "", "templateType": "anything", "can_override": false}, "solc": {"name": "solc", "group": "Ungrouped variables", "definition": "b-c", "description": "", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n  ['A',a],\n  ['B',b],\n  ['C',c]\n]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "sola", "solb", "solc", "defs"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": 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"minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"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>$\\vec{AC}$</p>\n<p>[[0]]</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, "correctAnswer": "vector(solb[0],solb[1])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}, {"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>$\\vec{CB}$</p>\n<p>[[0]]</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, "correctAnswer": "vector(solc[0],solc[1])", "correctAnswerFractions": false, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "505, Chain rule - trigonometry", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "Chain rule - trigonometry", "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": "", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Alessandro Palazio", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/25453/"}], "tags": [], "metadata": {"description": "<p>More work on differentiation with&nbsp;trigonometric functions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Differentiate the following&nbsp;trigonometric functions using the chain rule.</strong></p>\n<p>Do not write out $dy/dx$; only input the differentiated right hand side of each equation.</p>", "advice": "<p><strong></strong>If you don't know how to differentiate trigonometric functions, please see&nbsp;<em>'Differentiation 4 - Trigonometric Functions'</em>.</p>\n<p><strong></strong></p>\n<p><strong>These questions use the chain rule.</strong></p>\n<p>The earlier questions are easy to do by inspection, e.g using Part a:</p>\n<p style=\"text-align: center;\">$y=sin(\\var{c[0]}x)$.</p>\n<p>We differentiate the term(s) inside the function, here the term is $\\var{c[0]}x$.</p>\n<p>Then we derive $sin$ of any function, giving us $cos$.</p>\n<p>Putting our results together, we get</p>\n<p>$\\var{c[0]}cos(\\var{c[0]}x)$.</p>\n<p></p>\n<p></p>\n<p></p>\n<p><strong>We will now go through an entire worked example of the formal method of the chain rule using Part e.</strong></p>\n<p>The expression we will be differentiating here is</p>\n<p style=\"text-align: center;\">$y=tan^\\var{p[0]}(x)$.</p>\n<p style=\"text-align: left;\">As a reminder, the chain rule is defined as</p>\n<p style=\"text-align: center;\">$\\frac{dy}{dx}=\\frac{dy}{du}\\times\\frac{du}{dx}$.</p>\n<p style=\"text-align: left;\">Now we let $u=tanx$, so then $y=u^\\var{p[0]}$</p>\n<p style=\"text-align: left;\">This becomes an easy differentiation using $\\frac{dy}{du}\\times\\frac{du}{dx}$:</p>\n<p style=\"text-align: left;\">Differentiate $y$ with respect to $u$, giving $\\simplify{{p[0]}u^{{p[0]}-1}}$.</p>\n<p style=\"text-align: left;\">Then differentiate $u$ with respect to $x$, giving $sec^2x$.</p>\n<p style=\"text-align: left;\">Multiply these results together, and substitue $tan$ back in for $u$.</p>\n<p style=\"text-align: left;\">Your final result is therefore</p>\n<p style=\"text-align: center;\">$\\simplify{{p[0]}(tan^{{p[0]}-1}(x))*sec^2(x)}$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "repeat(random(3..6),2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "shuffle(2..8)[0..5]", "description": "<p>coefficients</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["c", "p"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", 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0, "exploreObjective": null, "prompt": "<p>$y=-\\sin(\\var{c[2]}x^2)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-2{c[2]}x*cos({c[2]}x^2)", "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, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"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>$y=-5\\cos(\\var{c[3]}x)+\\sin(\\var{c[4]}x)$</p>\n<p>$\\frac{dy}{dx}=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "5{c[3]}sin({c[3]}x)+{c[4]}cos({c[4]}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", 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They can help you assess the best way to improve your skills and fill in the gaps. Save the page with your results to have a reference on what are your weaker and stronger topics, and bring it to the appointment.</p>\n<p>Use the \"print this result summary\" button and save the file, it will contain all your attempted questions with your answers; it will be useful to go through the questions with the tutor.</p>\n<p>Click <a href=\"https://helpdesk.herts.ac.uk:8443/helpdesk/WebObjects/Helpdesk.woa/wa/TicketActions/new?rt=146\">here</a> to book your appointment.</p>", "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": [], "reviewshowexpectedanswer": true, "showanswerstate": false, "reviewshowfeedback": true, "showactualmark": true, "showtotalmark": true, "reviewshowscore": true, "reviewshowadvice": true}, "diagnostic": {"knowledge_graph": {"topics": [{"name": "101, Multiplication of negative numbers", "description": "", "depends_on": ["102, Multiply negative and positive"], "learning_objectives": ["Numbers"]}, {"name": "102, Multiply 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