// Numbas version: exam_results_page_options
{"name": "DCS23 Mathematics Entrance Test", "metadata": {"description": "<p>DCS 2023 Maths Entrance Test to be used September 2023</p>\n<p>1.5 hours with 3 randomised questions from each of these groups:</p>\n<ul>\n<li>Notation and Algebra</li>\n<li>Calculus - Differentiation</li>\n<li>Calculus - Integration</li>\n<li>Trigonometry</li>\n</ul>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "duration": 5400, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Notation and Algebra", "pickingStrategy": "all-ordered", "pickQuestions": "3", "questionNames": ["", "", "", "", ""], "variable_overrides": [[], [], [], [], []], "questions": [{"name": "N1", "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": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Mobius - ENG</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Four students spent {x1}, {x2}, {x3} and {x4} pounds on train tickets last month.</p>\n<p>Let $y$ denote last month's train ticket expenses for a student.</p>", "advice": "<p>The symbol $\\Sigma$ means 'sum of' or 'total'.</p>\n<p></p>\n<p></p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(1 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "random(10 .. 15#1)", "description": "", "templateType": "randrange", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(1 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "x4": {"name": "x4", "group": "Ungrouped variables", "definition": "random(15 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["x2", "x3", "x1", "x4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find&nbsp; $\\sum y = $ &nbsp;[[0]]</p>\n<p></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>Remember that&nbsp;</p>\n<p>$\\sum y = $&nbsp; means the total of the values&nbsp;</p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{x1}+{x2}+{x3}+{x4}", "maxValue": "{x1}+{x2}+{x3}+{x4}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find&nbsp; $(\\sum y)^2 = $ [[0]]</p>\n<p></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>$(\\sum y)^2 = $ means the square of the total of the values</p>\n<p></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "({x1}+{x2}+{x3}+{x4})^2", "maxValue": "({x1}+{x2}+{x3}+{x4})^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find</p>\n<p>$\\sum y^2 = $ [[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>$\\sum y^2 = $ means the total of the squares of the values</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{x1}^2+{x2}^2+{x3}^2+{x4}^2", "maxValue": "{x1}^2+{x2}^2+{x3}^2+{x4}^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "A1", "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": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Solve a simple linear equation algebraically. The unknown appears on both sides of the equation.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>We are asked to solve the equation</p>\n<p>\\[ \\var{d}x-\\var{f}=\\var{g}x+\\var{h} \\]</p>\n<p>In this equation, there are $x$ terms and constant terms on both sides of the equals sign.</p>\n<p>To solve this equation, we must rearrange it&nbsp;to get $x$ on its own.</p>\n<p>\\begin{align}<br/>\\var{d}x-\\var{f} &amp;= \\var{g}x+\\var{h} \\\\[0.5em]<br/>\\var{d}x-\\var{g}x &amp;= \\var{h}+\\var{f} &amp; \\text{Move } x \\text{ terms to the left, and constant terms to the right.}\\\\[0.5em]<br/>\\simplify{{d-g}*x} &amp;= {\\var{h+f}} &amp; \\text{Collect like terms together.}\\\\[0.5em]<br/>x &amp;=\\frac{\\var{h+f}}{\\var{d-g}} &amp; \\text{Divide both sides by } \\var{d-g} \\text{.} \\\\[0.5em]<br/>x &amp;= \\simplify{{h+f}/{d-g}}<br/>\\end{align}</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"dg_coprime": {"name": "dg_coprime", "group": "Ungrouped variables", "definition": "(d-g)/gcd_hfdg", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "gcd_hfdg": {"name": "gcd_hfdg", "group": "Ungrouped variables", "definition": "gcd((h+f),(d-g))", "description": "", "templateType": "anything", "can_override": false}, "hf_coprime": {"name": "hf_coprime", "group": "Ungrouped variables", "definition": "(h+f)/gcd_hfdg", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "(x*(d-g))-f", "description": "", "templateType": "anything", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "finalb": {"name": "finalb", "group": "Ungrouped variables", "definition": "hf_coprime/dg_coprime", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(g+2..8)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["d", "f", "g", "h", "x", "gcd_hfdg", "hf_coprime", "dg_coprime", "finalb"], "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>$\\var{d}x-\\var{f}=\\var{g}x+\\var{h}$</p>\n<p>What is the value of $x$?</p>\n<p>$x = $ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "finalb", "maxValue": "finalb", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "A2", "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Some quadratics are to be solved by factorising</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve the quadratic equation by factorising. Use fractions and integers instead od decimals for answers wherever possible.</p>\n<p><em>For parts b, If there is more than solution, enter them all separated by a comma</em>.</p>", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "map(num[j][3],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "solmin": {"name": "solmin", "group": "Ungrouped variables", "definition": "vector([min(-d[0]/c[0],-b[0]/a[0])]+[min(-d[1]/c[1],-b[1]/a[1])]+[min(-d[2]/c[2],-b[2]/a[2])]+[min(-d[3]/c[3],-b[3]/a[3])]+[min(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "map(num[j][1],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "map(num[j][2],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "solmax": {"name": "solmax", "group": "Ungrouped variables", "definition": "vector([max(-d[0]/c[0],-b[0]/a[0])]+[max(-d[1]/c[1],-b[1]/a[1])]+[max(-d[2]/c[2],-b[2]/a[2])]+[max(-d[3]/c[3],-b[3]/a[3])]+[max(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "map(num[j][0],j,0..4)", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "[[1,random(6..7),1,random(2..5)]] +\nshuffle([\n  [1,random(-4..-6),1,random(-3..3 except 0)],\n  [1,random(-4..-6),random(2..3),random(-1..1 except 0)],\n  [1,random(1,3,5),4,random(1,3,5)*random([1,-1])],\n  [1,random(-4..-6),3,random([-1,1,-2,2])]\n  ])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["num", "a", "b", "c", "d", "solmax", "solmin"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Factorise $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}$.&nbsp;[[0]]</p>\n<p></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>examples&nbsp;</p>\n<p>$x^2$ + 5$x$ + 6 = ($x$ + 2) ($x$ + 3)</p>\n<p>3$x^2$ + $x$ - 10 = (3$x$ - 5) ($x$ + 2)</p>\n<p></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a[1]}x+{b[1]})({c[1]}x+{d[1]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "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>Hence solve $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}=0$.</p>\n<p>[[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>example&nbsp;</p>\n<p>$x^2$ + 5$x$ + 6 = 0</p>\n<p>($x$ + 2) ($x$ + 3) = 0</p>\n<p>($x$ +2) = 0 ,&nbsp; so $x$ = -2</p>\n<p>($x$ + 3) = 0 , so $x$ = -3</p>\n<p>enter this as&nbsp; &nbsp;-2,-3&nbsp; &nbsp; or&nbsp; &nbsp;-3,-2&nbsp;</p>"}], "gaps": [{"type": "list-of-numbers", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "[solmin[1],solmax[1]]", "allowFractions": true, "correctAnswerFractions": true, "isSet": false, "show_input_hint": true, "separator": ","}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Solving quadratics - factorisation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}, {"name": "John Wick", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3326/"}], "tags": [], "metadata": {"description": "<p>Some quadratics are to be solved by factorising</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve the quadratic equations by factorising. Enter answers into the answer matrix in <strong>ASCENDING ORDER.</strong></p>", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"solmax": {"name": "solmax", "group": "Ungrouped variables", "definition": "vector([max(-d[0]/c[0],-b[0]/a[0])]+[max(-d[1]/c[1],-b[1]/a[1])]+[max(-d[2]/c[2],-b[2]/a[2])]+[max(-d[3]/c[3],-b[3]/a[3])]+[max(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "vector([random(1..3)]+[random(2..3)]+[random(1..3)]+[random(-2..-4)]+[random(3..6)])", "description": "", "templateType": "anything", "can_override": false}, "solmin": {"name": "solmin", "group": "Ungrouped variables", "definition": "vector([min(-d[0]/c[0],-b[0]/a[0])]+[min(-d[1]/c[1],-b[1]/a[1])]+[min(-d[2]/c[2],-b[2]/a[2])]+[min(-d[3]/c[3],-b[3]/a[3])]+[min(-d[4]/c[4],-b[4]/a[4])])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "vector([random(1..5)] + [random(1..5)] + [random(-1..-5)] + [random(-1..-5)] + [random(2..5)] )", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "vector([1]+[random(2..4)]+[1]+[1]+[random(3..4)])", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "vector([1]+[1]+[1]+[1]+[1])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "solmax", "solmin"], "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>Factorise $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+ {b[0]*d[0]}}$.&nbsp;[[0]]</p>\n<p>Hence solve $\\simplify{{a[0]*c[0]}x^2+{a[0]*d[0]+b[0]*c[0]}x+ {b[0]*d[0]}}=0$.</p>\n<p>[[1]]</p>\n<p>Use fractions or integers for your answers.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a[0]}x+{b[0]})({c[0]}x+{d[0]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([{solmin[0]}],[{solmax[0]}])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "1", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "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>Factorise $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}$.&nbsp;[[0]]</p>\n<p>Hence solve $\\simplify{{a[1]*c[1]}x^2+{a[1]*d[1]+b[1]*c[1]}x+ {b[1]*d[1]}}=0$.</p>\n<p>[[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({a[1]}x+{b[1]})({c[1]}x+{d[1]})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": [")("], "showStrings": false, "partialCredit": 0, "message": ""}, "notallowed": {"strings": ["x^2", "x*x", "x x", "x(", "x*("], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([{solmin[1]}],[{solmax[1]}])", "correctAnswerFractions": false, "numRows": "2", "numColumns": "1", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Simple simultaneous equations", "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": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Paul Hancock", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1738/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}, {"name": "heike hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2960/"}], "tags": [], "metadata": {"description": "<p>This question tests the student's ability to solve simple linear equations by elimination. Part a) involves only having to manipulate one equation in order to solve, and part b) involves having to manipulate both equations in order to solve.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"g": {"name": "g", "group": "Part b", "definition": "d*x2+f*y2", "description": "", "templateType": "anything", "can_override": false}, "y2": {"name": "y2", "group": "Part b", "definition": "random(-5..-1)*sgn(x2)", "description": "<p>Value of $y$ in part b</p>", "templateType": "anything", "can_override": false}, "numerator": {"name": "numerator", "group": "part a", "definition": "m*(j/h)-n", "description": "", "templateType": "anything", "can_override": false}, "j": {"name": "j", "group": "part a", "definition": "random(2..3)", "description": "<p>$x$ coefficient of the second equation in part a. An integer multiple of the $x$ coefficient of the second equation.</p>", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Part b", "definition": "random(-5..5)", "description": "<p>Value of $x$ in part b</p>", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "part a", "definition": "random(1..6)", "description": "<p>Constant part of the LHS of the first equation in part a</p>", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "part a", "definition": "h*x1 + k*y1", "description": "<p>RHS of the first equation in part a</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Part b", "definition": "random(a+1..9 except map(j*a,j,0..10/a))", "description": "<p>$x$ coefficient in the second equation of part b. Never an integer multiple of the $x$ coefficient in the first equation.</p>", "templateType": "anything", "can_override": false}, "divisor": {"name": "divisor", "group": "part a", "definition": "k*(j/h)+l", "description": "", "templateType": "anything", "can_override": false}, "y1": {"name": "y1", "group": "part a", "definition": "random(-5..-1)*(x1)", "description": "<p>Value of $x$ in part a</p>", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "part a", "definition": "j*x1 + l*y1", "description": "<p>RHS of the second equation in part a</p>", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Part b", "definition": "random(max(a,b)+1..12 except map(j*b,j,0..10/b))", "description": "<p>$y$ coefficient of the second equation in part b. Never an integer multiple of the $y$ coefficient in the first equation.</p>", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Part b", "definition": "a*x2+b*y2", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Part b", "definition": "random(a+1..7)", "description": "<p>Coefficient of $y$ in the first equation of part b.</p>\n<p></p>", "templateType": "anything", "can_override": false}, "l": {"name": "l", "group": "part a", "definition": "random(1..6)", "description": "<p>Constant part of the LHS of the second equation in part a</p>", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "part a", "definition": "random(2..5)", "description": "<p>$x$ coefficient of the first equation in part a</p>", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Part b", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "part a", "definition": "3", "description": "<p>Value of $x$ in part a</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "lcm(a,d)<lcm(b,f)", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part b", "variables": ["a", "b", "c", "d", "f", "g", "x2", "y2"]}, {"name": "part a", "variables": ["h", "j", "k", "l", "m", "n", "y1", "x1", "numerator", "divisor"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solve this set of simultaneous equations and give your answers for $x$ and $y$ below.</p>\n<p>\\begin{align}<br/>\\simplify{{h}x+{k}y} &amp;= \\var{m} \\text{,} \\\\<br/>\\simplify{{j}x+{l}y} &amp;= \\var{n} \\text{.}<br/>\\end{align}</p>\n<p>$x =$&nbsp;[[0]]</p>\n<p>$y =$ [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "x1", "maxValue": "x1", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "y1", "maxValue": "y1", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Calculus - Differentiation", "pickingStrategy": "all-ordered", "pickQuestions": "3", "questionNames": ["", "", "", "", ""], "variable_overrides": [[], [], [], [], []], "questions": [{"name": "C1", "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": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Differentiate $\\displaystyle ax^b+cx^{1/d} with respect to $x$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Differentiate the following with respect to $x$.</p>\n<p><em>Input all numbers as <span style=\"text-decoration: underline;\">fractions or integers</span>, not as decimals.</em></p>", "advice": "", "rulesets": {"std": ["all", "fractionNumbers"]}, "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}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "ee": {"name": "ee", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "f", "ee"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n      <p>$f(x)=\\var{a}x^{\\var{b}}+\\var{c}x^{1/\\var{d}}$</p>\n        <p>$\\displaystyle \\frac{df}{dx}=\\;$[[0]]</p>\n      ", "stepsPenalty": 0, "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>The derivative of&nbsp; &nbsp; $x^n$&nbsp; &nbsp; &nbsp;is&nbsp; &nbsp; &nbsp; $nx^{n-1}$</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a*b}*x^{b-1}+{c}/{d}*x^({1-d}/{d})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [1, 2], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers, not as decimals.</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Table_of_Derivatives_UV2rNbD.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_UV2rNbD.pdf"], ["question-resources/C2_example.JPG", "/srv/numbas/media/question-resources/C2_example.JPG"], ["question-resources/C2_example_938dLmT.JPG", "/srv/numbas/media/question-resources/C2_example_938dLmT.JPG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "Instructional questions (non-randomized) to emphasize the method.", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "<p>We are asked to differentiate:</p>\n<p>\\[&nbsp;y=\\var{bCF}x^{\\var{bP1}} \\cos{(\\var{bCF2}x)}\\]</p>\n<p></p>\n<p>Recognising that the function to differentiate is the product of two functions, we identify the two functions that are involved.</p>\n<p></p>\n<p>$u$ is the first function, $v$ is second:</p>\n<p></p>\n<p style=\"text-align: center;\">$\\large u=\\var{bCF}x^{\\var{bP1}} $&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\large v=\\cos{(\\var{bCF2}x)} $</p>\n<p>&nbsp;</p>\n<p>Now, we need to use the approriate techniques to differentiate each of these, for $u$ we can use the <strong>Power Rule</strong> and for $v$ your<strong> <a href=\"resources/question-resources/Table_of_Derivatives_UV2rNbD.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">Table of Derivatives</a></strong>:</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">This gives us:</p>\n<p style=\"text-align: center;\">$\\large \\frac{du}{dx}=\\simplify{ {bP1}*{bCF}*x^({bP1}-1) }$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; and&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $&nbsp;\\large \\frac{dv}{dx}=-&nbsp;\\var{bCF2} \\sin{(\\var{bCF2}x)}$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">We now use the formula:</p>\n<p style=\"text-align: center;\">$&nbsp;\\large &nbsp;\\frac{dy}{dx}=u \\frac{dv}{dx} + v \\frac{du}{dx} $</p>\n<p style=\"text-align: left;\">&nbsp;Make the appropriate substitutions into the formula:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\">$ \\large&nbsp; \\frac{dy}{dx}= \\var{bCF}x^{\\var{bP1}} \\times -&nbsp;\\var{bCF2}\\sin{(\\var{bCF2}x)} + \\cos{(\\var{bCF2}x)} &nbsp;\\times \\simplify{ {bP1}*{bCF}*x^({bP1}-1) } $</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: left;\">Finally, we need to use our basic algebra to simplify this as much as possible. Multiply out any brackets where it would simplify and collect like terms:</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: center;\">$&nbsp;&nbsp;\\large&nbsp;\\frac{dy}{dx}= \\simplify{ - {bCF2}* {bCF} x^{{bP1}}} \\sin{(\\var{bCF2}x)} + \\simplify{ {bP1}*{bCF}*x^({bP1}-1) } \\cos{(\\var{bCF2}x)} $</p>\n<p style=\"text-align: center;\">&nbsp;</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"bCF": {"name": "bCF", "group": "Part (b)", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "bP1": {"name": "bP1", "group": "Part (b)", "definition": "random(2..6 except bCF)", "description": "", "templateType": "anything", "can_override": false}, "bCF2": {"name": "bCF2", "group": "Part (b)", "definition": "random(2..6 except bCF)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Part (b)", "variables": ["bCF", "bP1", "bCF2"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Use the product rule to differentiate&nbsp; $&nbsp; y=\\var{bCF}x^{\\var{bP1}} \\cos{(x)}$</p>\n<p></p>\n<p></p>\n<p style=\"text-align: center;\">$&nbsp; \\large \\frac{dy}{dx}=&nbsp; &nbsp;$[[0]]</p>\n<p style=\"text-align: left;\"><em>When inputting trig functions brackets should be used e.g. </em>cos(<em>x</em>)</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>Hint:</p>\n<p>Example with product rule</p>\n<p><img src=\"resources/question-resources/C2_example_938dLmT.JPG\" width=\"612\" height=\"477\"/></p>\n<p></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({bCF}x^{{bP1}})*(-sin(x))+(cos(x))*({bP1}*{bCF}x^{{bP1}-1})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C3", "extensions": [], "custom_part_types": [], "resources": [["question-resources/C3_example.JPG", "/srv/numbas/media/question-resources/C3_example.JPG"], ["question-resources/mathcentre_CC_pyzfvWo.JPG", "/srv/numbas/media/question-resources/mathcentre_CC_pyzfvWo.JPG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Differentiate $\\displaystyle \\frac{ax+b}{cx+d}$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Differentiate the following function $y$ using the quotient rule.</p>", "advice": "\n\t  \n\t  \n\t  <p>The quotient rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u/v,x,1) = (v * Diff(u,x,1) - u * Diff(v,x,1))/v^2}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>For this example:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[std]{u = ({a}x+{b})}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {a}}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\simplify[std]{v = ({c} * x+{d})} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {c}}\\]</p>\n\t  \n\t  \n\t  \n\t  <p>Hence on substituting into the quotient rule above we get:</p>\n\t  \n\t  \n\t  \n\t  <p>\\[\\begin{eqnarray*} \\frac{df}{dx}&#38;=&#38;\\simplify[std]{({a}({c}x+{d})-{c}({a}x+{b}))/({c}x+{d})^2}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{({a*c}x+{a*d}-{c*a}x-{c*b})/({c}x+{d})^2}\\\\\n\t  \n\t  &#38;=&#38;\\simplify[std]{{det}/({c}x+{d})^2}\n\t  \n\t  \\end{eqnarray*}\\]</p>\n\t  \n\t  \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"result": "(sqrt(b)*a)/b", "pattern": "a/sqrt(b)"}]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "if(a*d=b*c1,c1+1,c1)", "description": "", "templateType": "anything", "can_override": false}, "det": {"name": "det", "group": "Ungrouped variables", "definition": "a*d-b*c", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "d", "s2", "s1", "det", "c1"], "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[std]{y = ({a} * x+{b})/({c}*x+{d})}\\]</p>\n<p>$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]</p>", "stepsPenalty": "2", "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>Example of quotient rule:</p>\n<p><img src=\"resources/question-resources/C3_example.JPG\" width=\"607\" height=\"393\"/></p>\n<p><img src=\"resources/question-resources/mathcentre_CC_pyzfvWo.JPG\" width=\"568\" height=\"30\"/></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{det}/({c}x+{d})^2", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [10, 11], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C5", "extensions": [], "custom_part_types": [], "resources": [["question-resources/C5_example_start.JPG", "/srv/numbas/media/question-resources/C5_example_start.JPG"], ["question-resources/C5i_example.JPG", "/srv/numbas/media/question-resources/C5i_example.JPG"], ["question-resources/C5ii_example.JPG", "/srv/numbas/media/question-resources/C5ii_example.JPG"], ["question-resources/mathcentre_CC_uGtA73X.JPG", "/srv/numbas/media/question-resources/mathcentre_CC_uGtA73X.JPG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}, {"name": "Johnny Yi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2810/"}], "tags": [], "metadata": {"description": "<p>Differentiate $f(x) = x^m(a x+b)^n$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Differentiate $y$ using the product and function of a function rules.</p>", "advice": "\n  \n  \n  <p>The product rule says that if $u$ and $v$ are functions of $x$ then<br/>\\[\\simplify[std]{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}\\]</p>\n  \n  \n  \n  <p>For this example:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{u = x ^ {m}}\\Rightarrow \\simplify[std]{Diff(u,x,1) = {m}x ^ {m -1}}\\]</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{v = ({a} * x+{b})^{n}} \\Rightarrow \\simplify[std]{Diff(v,x,1) = {n*a} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  \n  <p>Hence on substituting into the product rule above we get:</p>\n  \n  \n  \n  <p>\\[\\simplify[std]{Diff(f,x,1) = {m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}}\\]</p>\n  \n  \n  ", "rulesets": {"surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}], "std": ["all", "!collectNumbers", "fractionNumbers"]}, "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}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "s1", "b", "m", "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>$\\displaystyle \\simplify[std]{y = x ^ {m} * ({a} * x+{b})^{n}}$</p>\n<p>$\\displaystyle \\frac{dy}{dx}=\\;$[[0]]</p>\n<p></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>Example of Function of Function Rule (also known as Chain Rule):</p>\n<p><img src=\"resources/question-resources/C5_example_start.JPG\" width=\"607\" height=\"194\"/></p>\n<p><img src=\"resources/question-resources/C5i_example.JPG\" width=\"603\" height=\"49\"/></p>\n<p><img src=\"resources/question-resources/C5ii_example.JPG\" width=\"609\" height=\"187\"/></p>\n<p><img src=\"resources/question-resources/mathcentre_CC_uGtA73X.JPG\" width=\"587\" height=\"31\"/></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{m}x ^ {m-1} * ({a} * x+{b})^{n}+{n*a}x^{m} * ({a} * x+{b})^{n-1}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C6", "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": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Find the stationary points of a cubic which has 2 turning points.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the coordinates of the stationary points of the function and determine their nature:&nbsp;</p>\n<p>$y=\\simplify[all,!collectNumbers]{{a}x^3+{b}x^2+{c}x+{d}}$</p>\n<p>&nbsp;</p>", "advice": "<p>On differentiating we get $\\displaystyle \\frac{df}{dx}=\\simplify[std]{{3*a}x^2+{2*b}x+{c}}$.</p>\n<p>To find the stationary points we have to solve $\\displaystyle \\frac{df}{dx}=0$&nbsp;for $x$.</p>\n<p>So we have to solve $\\simplify[std]{{3*a}x^2+{2*b}x+{c}=0}$.</p>\n<p>Note that the quadratic factorises and the equation becomes $\\simplify[std]{({3a}x-{r1})(x-{r2})=0}$.</p>\n<p>Hence we have two stationary points: $x=\\simplify[std]{{r1}/{3a}}$ and $x=\\var{r2}$.</p>\n<p>To find out the types of these stationary points we look at the sign of $\\displaystyle \\frac{d^2y}{dx^2} = \\simplify{{6a}*x+{2*b}}$ at &nbsp;the stationary points.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2y}{dx^2} \\lt 0 $ at a stationary point then it is a MAXIMUM.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2y}{dx^2} \\gt 0 $ at a stationary point then it is a MINIMUM.</p>\n<p>If &nbsp;$\\displaystyle \\frac{d^2y}{dx^2} = 0 $ at a stationary point then we have to do more work!</p>\n<p>At $x=\\var{r2}$ we have $\\displaystyle \\frac{d^2y}{dx^2} = \\simplify{{6*a*r2+2*b}}${lg1}$0$ hence is a {type1}.</p>\n<p>At $\\displaystyle x=\\simplify[std]{{r1}/{3a}}$ we have $\\displaystyle \\frac{d^2y}{dx^2} = \\simplify{{2*r1+2*b}}${lg2}$0$ hence is a {type2}.</p>\n<p>&nbsp;</p>", "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-2..2 except 0)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "r1*r2", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "round(-(3*a*r2+r1)/2)", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "random(-4..4#2 except 0)-3*a*r2", "description": "", "templateType": "anything", "can_override": false}, "r2": {"name": "r2", "group": "Ungrouped variables", "definition": "random(-6..6 except 0)", "description": "", "templateType": "anything", "can_override": false}, "mn": {"name": "mn", "group": "Ungrouped variables", "definition": "if(3a*r2+b<0,1,0)", "description": "", "templateType": "anything", "can_override": false}, "type2": {"name": "type2", "group": "Ungrouped variables", "definition": "if(mx=1, 'maximum','minimum')", "description": "", "templateType": "anything", "can_override": false}, "lg2": {"name": "lg2", "group": "Ungrouped variables", "definition": "if(mx=0,'$\\\\gt$','$\\\\lt$')", "description": "", "templateType": "anything", "can_override": false}, "lg1": {"name": "lg1", "group": "Ungrouped variables", "definition": "if(mx=0,'$\\\\lt$','$\\\\gt$')", "description": "", "templateType": "anything", "can_override": false}, "type1": {"name": "type1", "group": "Ungrouped variables", "definition": "if(mx=0, 'maximum','minimum')", "description": "", "templateType": "anything", "can_override": false}, "mx": {"name": "mx", "group": "Ungrouped variables", "definition": "if(3a*r2+b<0,0,1)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-10..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "r1", "r2", "mn", "d", "lg2", "lg1", "type1", "mx", "type2"], "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>$x$-coordinate of the stationary point giving a minimum: [[0]]</p>\n<p>$x$-coordinate of the stationary point giving a maximum: [[1]]</p>\n<p><em>Input your answers as <span style=\"text-decoration: underline;\">fractions or integers</span> and not as decimals.</em></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>To locate the stationary points:</p>\n<ol>\n<li>Find the first derivative $dy/dx$ and set it equal to 0.</li>\n<li>Solving this equation will give you the $x$ co-ordinates.&nbsp;</li>\n</ol>\n<p><br/>To determine the nature of the stationary points:</p>\n<ol>\n<li>Find the second derivative $d^2y/dx^2$&nbsp;</li>\n<li>Substitute the values of x you found above one at a time.</li>\n<li>If $d^2y/dx^2$ is positive, then the stationary point is minimum.</li>\n<li>If $d^2y/dx^2$ is negative, then the stationary point is maximum.</li>\n</ol>\n<p><em>Note that you will not have to deal with stationary points that are points of inflexion in this question.&nbsp;</em></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{mx*r2}+{(1-mx)}*{(r1)}/{(3*a)}", "answerSimplification": "std", "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": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{mn*r2}+{(1-mn)}*{(r1)}/{(3*a)}", "answerSimplification": "std", "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": "Calculus - Integration", "pickingStrategy": "all-ordered", "pickQuestions": "3", "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "C8", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Indefinite integration of basic functions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Integrate the follwing expression with respect to ${x}$.</p>\n<p><em>Type the letter e for the exponential constant.</em></p>\n<p><em>Don't forget the constant of integration + c</em></p>", "advice": "<ol>\n<li>\\[\\int\\simplify[all]{{a}+{b}*cos(x)}\\,dx=\\simplify{{a}*x+{b}*sin(x)+c}\\]</li>\n<li>\\[\\int\\simplify[all]{{c}x+{b}*exp({a}*x)}\\,dx=\\simplify{{c/2}*x^2+{b/a}*exp({a}*x)+c}\\]</li>\n<li>\\[\\int\\simplify[all]{{c+1}*sin({b}*x)-{a}/x}\\,dx=\\simplify{{-(c+1)/b}*cos({b}*x)-{a}*ln(x)+c}\\]</li>\n<li>\\[\\int\\simplify[all]{{c}/(x^2)+{b+1}/{a+1}*x^{b}}\\,dx=\\simplify{{-c}/x+{1/(a+1)}*x^{b+1}+c}\\]</li>\n<li>\\[\\int\\simplify[all]{{b+2}*x^{b-1}-{d}*sinh(x)+{c}*exp({a}*x)}\\,dx=\\simplify{{1+2/b}*x^{b}-{d}*cosh(x)+{c/a}*exp({a}*x)+c}\\]</li>\n</ol>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0])", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\">\\[\\int\\simplify[all]{{c}x+{b}*exp(x)}\\,dx\\] <br/>= [[0]]</p>", "stepsPenalty": 0, "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>Remember that the intgeral of&nbsp; &nbsp; $x^n$&nbsp; &nbsp; is&nbsp; &nbsp;${x^{(n+1)}}/{n+1}$</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You have forgotten the constant of integration, which is needed when doing indefinite integration.</p>", "useAlternativeFeedback": false, "answer": "{c}*x^2/2+{b/a}*exp({a}*x)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "answer": "{c}*x^2/2+{b}*exp(x)+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C9", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Definite integation of basic functions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p><em>Make sure your calculator is in radians for this question.</em></p>\n<p></p>", "advice": "<p>(a) and (b):</p>\n<p>\\[\\int_\\var{d-2}^\\var{d}\\simplify[all]{{a}*x-sin({b}*x)}\\,dx=\\left[\\simplify{{a}*x^2/2+cos({b}*x)/{b}}\\right]_\\var{d-2}^\\var{d}=\\var{a1}\\]</p>\n<p>(c) and (d):</p>\n<p>\\[\\int_\\var{d}^\\var{d+3}\\simplify[all]{{c}/x^{a+1}+{b}*sqrt(x)}\\,dx=\\left[\\simplify{{-c/a}/x^{a}+{2*b/3}*x^(3/2)}\\right]_\\var{d}^\\var{d+3}=\\var{a2}\\]</p>\n<p>(e) and (f):</p>\n<p>\\[\\int_{-2}^\\var{a-2}\\simplify[all]{{c+1}*exp(x/{b})-{a}}\\,dx=\\left[\\simplify{{b*(c+1)}*exp(x/{b})-{a}*x}\\right]_{-2}^\\var{a-2}=\\var{a3}\\]</p>\n<p>(g) and (h):</p>\n<p>\\[\\int_\\var{b}^\\var{b+a}\\simplify[all]{{c}/({b}*x)+{b}/{a+1}*cos({c}*x)}\\,dx=\\left[\\simplify{{c}/{b}*ln(x)+{b}/{c*(a+1)}*sin({c}*x)}\\right]_\\var{b}^\\var{b+a}=\\var{a4}\\]</p>\n<p>(i) and (j):</p>\n<p>\\[\\int_\\var{a}^\\var{a+1}\\simplify[all]{x^{b-1}/{b+2}-{d}+{c}*exp({-a}*x)}\\,dx=\\left[\\simplify{x^{b}/{b*(b+2)}-{d}*x-{c/a}*exp({-a}*x)}\\right]_\\var{a}^\\var{a+1}=\\var{a5}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a2": {"name": "a2", "group": "Ungrouped variables", "definition": "-c/(a*(d+3)^a)+(2*b/3)*(d+3)^(3/2)+c/(a*d^a)-(2*b/3)*d^(3/2)", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "b*(c+1)*exp((a-2)/b)-a*(a-2)-b*(c+1)*exp(-2/b)-2*a", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-5..5 except [-1,0])", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "a*d^2/2+cos(b*d)/b-a*(d-2)^2/2-cos(b*(d-2))/b", "description": "", "templateType": "anything", "can_override": false}, "a5": {"name": "a5", "group": "Ungrouped variables", "definition": "(a+1)^b/(b*(b+2))-d*(a+1)-c/a*exp(-a*(a+1))-a^{b}/(b*(b+2))+d*a+c/a*exp(-a^2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..15)", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "c/b*ln(b+a)+b/(c*(a+1))*sin(c*(b+a))-c/b*ln(b)-b/(c*(a+1))*sin(c*b)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "a1", "a2", "a3", "a4", "a5"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Integrate this expression with respect to x</p>\n<p><em>Don't forget the constant of integration + c</em></p>\n<p>\\[\\int\\simplify[all]{{a}*x-sin({b}*x)}\\,dx\\]</p>", "stepsPenalty": 0, "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>Remember that</p>\n<p><br/>the integral of&nbsp; &nbsp; $x^n$&nbsp; &nbsp;is&nbsp; &nbsp; $x^{n+1} / {n+1}$ ,&nbsp; and</p>\n<p>the integral of&nbsp; &nbsp; sin$(ax)$&nbsp; &nbsp;is&nbsp; &nbsp; $-$cos$(ax) / a$</p>"}], "alternatives": [{"type": "jme", "useCustomName": true, "customName": "No constant", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>You have forgotten the constant of integration&nbsp;(condoned on this occasion).</p>", "useAlternativeFeedback": false, "answer": "{a}*x^2/2+cos({b}*x)/{b}", "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}*x^2/2+cos({b}*x)/{b}+c", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Hence evaluate</p>\n<p>\\[\\int_\\var{d-2}^\\var{d}\\simplify[all]{{a}*x-sin({b}*x)}\\,dx\\]</p>\n<p></p>\n<p><em>Remember to use radians and give your answer correct to 3 significant figures</em>.</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>Example:</p>\n<p>The definite intgeral is calculated by substituting $x$ with the values of the limits.<br/>e.g.&nbsp;</p>\n<p>$ \\int_1^4\\ {{x}^4}\\,dx = 4^5 / 5 - 1^4/5=204.6=205$ (3 significant figures)</p>"}], "minValue": "{a1}", "maxValue": "{a1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C10", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Vaughan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1135/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Find $\\displaystyle I=\\int \\frac{2 a x + b} {a x ^ 2 + b x + c}\\;dx$ by substitution or otherwise.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Use substitution to find the following integral.</p>\n<p>Note that $\\displaystyle \\int \\frac{1}{x}\\;dx=\\ln(|x|)+c$.</p>\n<p><em>Don't forget the constant of integration</em> $+c$.</p>\n<p><em>Input all numbers as integers or fractions - do not use decimals.</em></p>\n<p></p>\n<p></p>\n<p></p>", "advice": "\n  <p>This exercise is best solved by using substitution.</p>\n  <p>Note that the numerator $\\simplify[std]{{2 * a} * x + {b}}$ of \\[\\simplify[std]{({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c})}\\] is the derivative of the denominator $\\simplify[std]{{a} * x ^ 2 + {b} * x + {c}}$</p>\n  <p>So if you use as your substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$ you then have $\\simplify[std]{ du = ({2 * a} * x + {b}) * dx}$</p>\n  <p>Hence we can replace $\\simplify[std]{ ({2 * a} * x + {b}) * dx}$ by $du$</p>\n  <p>Hence the integral becomes:</p>\n  <p>\\[\\begin{eqnarray*} I&amp;=&amp;\\int\\;\\frac{du}{u}\\\\ &amp;=&amp;\\ln(|u|)+C\\\\ &amp;=&amp; \\simplify[std]{ln(abs({a} * (x ^ 2) + ({b} * x) + {c}))+C} \\end{eqnarray*}\\]<br/><br/></p>\n  <p><strong>A Useful Result</strong><br/>This example can be generalised.</p>\n  <p>Suppose \\[I = \\int\\; \\frac{f'(x)}{f(x)}\\;dx\\]</p>\n  <p>The using the substitution $u=f(x)$ we find that $du=f'(x)\\;dx$ and so using the same method as above:<br/>\\[I = \\int \\frac{du}{u} = \\ln(|u|)+ C = \\ln(|f(x)|)+C\\]</p>\n  ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "ceil(b^2/4a+random(1..5))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "b", "s1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\[\\simplify[std]{Int(({2 * a} * x + {b}) / ({a} * x ^ 2 + {b} * x + {c}),x)}\\]</p>\n<p style=\"text-align: center;\">$=\\;$[[0]]</p>\n<p></p>\n<p><em>You should input</em> $|x|$ <em>as</em> abs(x)</p>\n<p><em>You should use brackets after&nbsp;</em> ln&nbsp; &nbsp;e.g ln x = ln(x)</p>\n<p><em>Use the format</em>&nbsp; &nbsp;ln ( abs(x) ) for $ln (|x|)$</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>Try the substitution $u=\\simplify[std]{{a} * (x ^ 2) + ({b} * x) + {c}}$</p>\n<p></p>\n<p></p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "ln(abs((({a} * (x ^ 2)) + ({b} * x) + {c})))+c", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["abs"], "showStrings": false, "partialCredit": 0, "message": "<p>Note that \\[\\int \\frac{1}{x}\\;dx=\\ln(|x|)+C\\] and you must input the absolute value of the argument of the natural logarithm. You input the absolute value using abs, for example abs(x)=$\\simplify{abs(x)}$</p>"}, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not input numbers as decimals, only as integers without the decimal point, or fractions</p>"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "C13", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "TEAME UCC", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/351/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Integration by Parts</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<p>Use Integration by Parts</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..4 except a)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Evaluate $\\int_0^\\pi x \\cos(x) \\mathrm{dx}$ using integration by parts.</p>\n<p></p>", "stepsPenalty": "2", "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": "<ol>\n<li>Start by letting $u = x$ and $dv = \\cos(x)$</li>\n<li>Next find $du$ and $v$</li>\n<li>Substitute expressions from parts 1 and 2 above into the formula for integration by parts:</li>\n</ol>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\int (u*dv)&nbsp; dx= u*v-\\int(v*du)&nbsp; dx$</p>\n<p></p>"}], "minValue": "-2", "maxValue": "-2", "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"}]}, {"name": "Trigonometry ", "pickingStrategy": "all-ordered", "pickQuestions": "3", "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "T2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/T2.JPG", "/srv/numbas/media/question-resources/T2.JPG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}], "tags": [], "metadata": {"description": "<p>Simple trig equations with radians</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve the following trigonometric equation in <em>radians</em>, for $\\theta$ in the range $0\\leq\\theta\\leq2\\pi$.</p>\n<p></p>", "advice": "<p>Please refer to the advice section of&nbsp;<em>'Trigonometric Equations 1 - Simple (Degrees)'</em> for help.</p>\n<p>To convert from degrees to radians, multiply by $\\frac{\\pi}{180}$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"f2": {"name": "f2", "group": "Q2", "definition": "c/b", "description": "", "templateType": "anything", "can_override": false}, "s42": {"name": "s42", "group": "Q3", "definition": "precround(2*pi+s4,2)", "description": "", "templateType": "anything", "can_override": false}, "f4": {"name": "f4", "group": "Q3", "definition": "-c/(c+3)", "description": "", "templateType": "anything", "can_override": false}, "s31": {"name": "s31", "group": "Q4", "definition": "precround(s3,2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Q2", "definition": "random(2..b-1)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Q1", "definition": "random(3..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Q2", "definition": "random(5..12)", "description": "", "templateType": "anything", "can_override": false}, "s4": {"name": "s4", "group": "Q3", "definition": "arcsin(f4)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Q1", "definition": "arcsin(f1)", "description": "", "templateType": "anything", "can_override": false}, "s22": {"name": "s22", "group": "Q2", "definition": "precround(2*pi-s2,2)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Q1", "definition": "(a-1)/a", "description": "", "templateType": "anything", "can_override": false}, "s21": {"name": "s21", "group": "Q2", "definition": "precround(s2,2)", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Q4", "definition": "arccos(f3)", "description": "", "templateType": "anything", "can_override": false}, "s32": {"name": "s32", "group": "Q4", "definition": "precround(2*pi-s3,2)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Q2", "definition": "arccos(f2)", "description": "", "templateType": "anything", "can_override": false}, "f3": {"name": "f3", "group": "Q4", "definition": "(1-b)/(b+1)", "description": "", "templateType": "anything", "can_override": false}, "s11": {"name": "s11", "group": "Q1", "definition": "precround(s1,2)", "description": "", "templateType": "anything", "can_override": false}, "s41": {"name": "s41", "group": "Q3", "definition": "precround(pi-s4,2)", "description": "", "templateType": "anything", "can_override": false}, "s12": {"name": "s12", "group": "Q1", "definition": "precround(pi-s1,2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Q1", "variables": ["a", "f1", "s1", "s11", "s12"]}, {"name": "Q2", "variables": ["b", "c", "f2", "s2", "s21", "s22"]}, {"name": "Q4", "variables": ["f3", "s31", "s32", "s3"]}, {"name": "Q3", "variables": ["f4", "s4", "s41", "s42"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\[\\var{a}\\sin(\\theta)=\\var{a-1}\\]</p>\n<p>$\\sin(\\theta)=$ [[0]]&nbsp; <em>Write this answer as a fraction.</em>&nbsp;</p>\n<p>$\\theta=$ [[1]]&nbsp;or&nbsp;[[2]]&nbsp; <em>Give these answers correct to 2 decimal places and in ascending order (i.e. lowest first).</em></p>", "stepsPenalty": "2", "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>Example</p>\n<p><img src=\"resources/question-resources/T2.JPG\" width=\"781\" height=\"403\"/></p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{f1}-0.01", "maxValue": "{f1}+0.01", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{s11}", "maxValue": "{s11}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": "100", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{s12}", "maxValue": "{s12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": "100", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Trigonometric Equations  - Core (Degrees)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Trigonometric equations with degrees</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Solve the following equation in <em>degrees</em>, for $x$ in the range $0^\\circ\\leq x\\leq720^\\circ$.</strong></p>\n<p>\\[\\var{a+1}\\cos(x-\\var{b})=\\var{a-2}\\]</p>\n<p>Give your final answers correct to the nearest degree in ascending order.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(3..9)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "p", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(10..p-5)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "(a-2)/(a+1)", "description": "", "templateType": "anything", "can_override": false}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "a4+b", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "arccos(f)*180/pi", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "360+p", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "360-p", "description": "", "templateType": "anything", "can_override": false}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "720-p", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "a2+b", "description": "", "templateType": "anything", "can_override": false}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "a3+b", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "a1+b", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "a1", "a2", "a3", "a4", "b", "b1", "b2", "b3", "b4", "f", "p"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\cos(x-\\var{b})=$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{f}-0.01", "maxValue": "{f}+0.01", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": 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International"}, "statement": "<p><strong>Solve the following equations in <em>radians</em>, for $\\theta$ in the range $0\\leq\\theta\\leq\\pi$:</strong></p>\n<p>\\[\\var{b}\\cos(3\\theta+\\var{d})=\\var{c}\\]</p>\n<p>Give your answers for parts (b) (c) and (d) correct to 2 decimal places in ascending order.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a1": {"name": "a1", "group": "Ungrouped variables", "definition": "p", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..b-1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(5..12)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(0.1..p-0.1 #0.1)", "description": 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"", "maxRuns": 100}, "ungrouped_variables": ["a1", "a2", "a3", "a4", "b", "b1", "b2", "b3", "b4", "c", "c1", "c2", "c3", "c4", "d", "f", "p"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\cos(3\\theta+\\var{d})=$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", 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false, "advicethreshold": 0, "intro": "<ul>\n<li>You have 1.5 hours to complete the questions.</li>\n<li>Each question has its own page - you <span style=\"text-decoration: underline;\">must</span> submit a page before moving on to the next page or your answers will not be saved. You may wish to submit part questions inidividually which is also acceptable.</li>\n<li>You can go back to any question and submit a different answer at any point during the test.</li>\n<li>You may use a scientific calculator throughout the test.</li>\n<li>You will need paper and pem/pencil for your rough work.</li>\n<li>Its important to follow the instructions given carefully when inputting answers!</li>\n</ul>\n<p></p>", "end_message": "", "reviewshowscore": false, "reviewshowfeedback": false, "reviewshowexpectedanswer": false, "reviewshowadvice": false, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "extensions": [], "custom_part_types": [{"source": {"pk": 2, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/2/edit"}, "name": "List of numbers", "short_name": "list-of-numbers", "description": "<p>The answer is a comma-separated list of numbers.</p>\n<p>The list is marked correct if each number occurs the same number of times as in the expected answer, and no extra numbers are present.</p>\n<p>You can optionally treat the answer as a set, so the number of occurrences doesn't matter, only whether each number is included or not.</p>", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "join(\n  if(settings[\"correctAnswerFractions\"],\n    map(let([a,b],rational_approximation(x), string(a/b)),x,settings[\"correctAnswer\"])\n  ,\n    settings[\"correctAnswer\"]\n  ),\n  settings[\"separator\"] + \" \"\n)", "hint": {"static": false, "value": "if(settings[\"show_input_hint\"],\n  \"Enter a list of numbers separated by <code>{settings['separator']}</code>.\",\n  \"\"\n)"}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "bits:\nlet(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)\n\nexpected_numbers:\nlet(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)\n\nvalid_numbers:\nif(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )\n\nis_sorted:\nassert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )\n\nincluded:\nmap(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )\n\nall_included:\nall(included)\n\nno_extras:\nif(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )\n\ninterpreted_answer:\nif(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)\n\nmark:\nif(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)\n\nnotationStyles:\n[\"en\"]\n\nisSet:\nsettings[\"isSet\"]\n\nextra_numbers:\nfilter(not (x in expected_numbers),x,interpreted_answer)", "marking_notes": [{"name": "bits", "description": "<p>The separate items in the student's answer</p>", "definition": "let(b,filter(x<>\"\",x,split(studentAnswer,settings[\"separator\"])),\n    if(isSet,list(set(b)),b)\n)"}, {"name": "expected_numbers", "description": "", "definition": "let(l,settings[\"correctAnswer\"] as \"list\",\n    if(isSet,list(set(l)),l)\n)"}, {"name": "valid_numbers", "description": "<p>Is every number in the student's list valid?</p>", "definition": "if(all(map(not isnan(x),x,interpreted_answer)),\n    true,\n    let(index,filter(isnan(interpreted_answer[x]),x,0..len(interpreted_answer)-1)[0], wrong, bits[index],\n      warn(wrong+\" is not a valid number\");\n      fail(wrong+\" is not a valid number.\")\n    )\n  )"}, {"name": "is_sorted", "description": "<p>Are the student's answers in ascending order?</p>", "definition": "assert(sort(interpreted_answer)=interpreted_answer,\n    multiply_credit(0.5,\"Not in order\")\n  )"}, {"name": "included", "description": "<p>Is each number in the expected answer present in the student's list the correct number of times?</p>", "definition": "map(\n      let(\n        num_student,len(filter(x=y,y,interpreted_answer)),\n        num_expected,len(filter(x=y,y,expected_numbers)),\n        switch(\n          num_student=num_expected,\n            true,\n          num_student<num_expected,\n            incorrect(\n              if(isSet,\"Your answer does not contain \"+x+\" but should.\",\n                \"Your answer does not contain \"+x+\" enough times.\"\n              )\n            );false\n          ,\n            incorrect(\n              if(isSet,\"Your answer contains \"+x+\" but should not.\",\n                \"Your answer contains \"+x+\" too many times.\"\n              )\n            );false\n        )\n      )\n      ,\n      x,\n      sort(list(set(expected_numbers)))\n  )"}, {"name": "all_included", "description": "<p>Has every number been included the right number of times?</p>", "definition": "all(included)"}, {"name": "no_extras", "description": "<p>True if the student's list doesn't contain any numbers that aren't in the expected answer.</p>", "definition": "if(all(map(x in expected_numbers, x, interpreted_answer)),\n    true\n  ,\n    incorrect(\"Your answer contains \"+extra_numbers[0]+\" but should not.\");\n    false\n  )"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "if(lower(studentAnswer) in [\"empty\",\"\u2205\"],[],\n  map(\n    if(settings[\"allowFractions\"],parsenumber_or_fraction(x,notationStyles), parsenumber(x,notationStyles))\n    ,x\n    ,bits\n  )\n)"}, {"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=\"\",fail(\"You have not entered an answer\"),false);\napply(valid_numbers);\napply(included);\napply(no_extras);\ncorrectif(all_included and no_extras)"}, {"name": "notationStyles", "description": "", "definition": "[\"en\"]"}, {"name": "isSet", "description": "<p>Should the answer be considered as a set, so the number of times an element occurs doesn't matter?</p>", "definition": "settings[\"isSet\"]"}, {"name": "extra_numbers", "description": "<p>Numbers included in the student's answer that are not in the expected list.</p>", "definition": "filter(not (x in expected_numbers),x,interpreted_answer)"}], "settings": [{"name": "correctAnswer", "label": "Correct answer", "help_url": "", "hint": "The list of numbers that the student should enter. The order does not matter.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "allowFractions", "label": "Allow the student to enter fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "correctAnswerFractions", "label": "Display the correct answers as fractions?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": false}, {"name": "isSet", "label": "Is the answer a set?", "help_url": "", "hint": "If ticked, the number of times an element occurs doesn't matter, only whether it's included at all.", "input_type": "checkbox", "default_value": false}, {"name": "show_input_hint", "label": "Show the input hint?", "help_url": "", "hint": "", "input_type": "checkbox", "default_value": true}, {"name": "separator", "label": "Separator", "help_url": "", "hint": "The substring that should separate items in the student's list", "input_type": "string", "default_value": ",", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [["question-resources/Table_of_Derivatives_UV2rNbD.pdf", "/srv/numbas/media/question-resources/Table_of_Derivatives_UV2rNbD.pdf"], ["question-resources/C2_example.JPG", "/srv/numbas/media/question-resources/C2_example.JPG"], ["question-resources/C2_example_938dLmT.JPG", "/srv/numbas/media/question-resources/C2_example_938dLmT.JPG"], ["question-resources/C3_example.JPG", "/srv/numbas/media/question-resources/C3_example.JPG"], ["question-resources/mathcentre_CC_pyzfvWo.JPG", "/srv/numbas/media/question-resources/mathcentre_CC_pyzfvWo.JPG"], ["question-resources/C5_example_start.JPG", "/srv/numbas/media/question-resources/C5_example_start.JPG"], ["question-resources/C5i_example.JPG", "/srv/numbas/media/question-resources/C5i_example.JPG"], ["question-resources/C5ii_example.JPG", "/srv/numbas/media/question-resources/C5ii_example.JPG"], ["question-resources/mathcentre_CC_uGtA73X.JPG", "/srv/numbas/media/question-resources/mathcentre_CC_uGtA73X.JPG"], ["question-resources/T2.JPG", "/srv/numbas/media/question-resources/T2.JPG"]]}