// Numbas version: finer_feedback_settings
{"name": "Standard time -- 2hrs 2024-25 WM175 Assessment 1", "metadata": {"description": "<p>Topics: <br />Algebra, Calculus, Trigonometeric Equations, Complex numbers &amp; Partial Derivatives<br /><br />Students must complete the exam within 120 mins (standard time).<br /><br />Questions have variables to produce randomised questions.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 7200, "percentPass": "40", "showQuestionGroupNames": true, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Algebra and calculus", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"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", "type": "question"}, {"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", "type": "question"}, {"name": "Derivative (Function of a Function)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/chain_rule_example_mathcentre.JPG", "/srv/numbas/media/question-resources/chain_rule_example_mathcentre.JPG"]], "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": "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 $\\displaystyle (ax^m+b)^{n}$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the derivative of $y$ using the function of a function rule (also known as <em>the chain rule</em>).</p>\n<p>Note, you should check that the preview of your answer shown is correct before submitting.&nbsp;&nbsp;</p>", "advice": "", "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}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(5..9)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..5)", "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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>\\[\\simplify[std]{y = ({a} * x^{m}+{b})^{n}}\\]</p>\n<p></p>\n<p>$\\displaystyle{u}=$&nbsp; [[0]]</p>\n<p>$\\displaystyle \\frac{\\mathrm{d}u}{\\mathrm{d}x}=$&nbsp; [[1]]</p>\n<p>$\\displaystyle \\frac{\\mathrm{d}y}{\\mathrm{d}u}=$&nbsp; [[2]]</p>\n<p>Hence,</p>\n<p>$\\displaystyle \\frac{\\mathrm{d}y}{\\mathrm{d}x}=$&nbsp; [[3]]</p>\n<p></p>\n<p>For help, you may wish to click on <em>Show steps</em> - but be aware this will cost you 2 marks.&nbsp;</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>This example shows how to differentiate a similar function of a function - use it to help you answer the one above.</p>\n<p><img src=\"resources/question-resources/chain_rule_example_mathcentre.JPG\"/></p>\n<p></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} * x^{m} + {b}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"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*m}x ^ {m-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": ""}]}, {"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": "{n}u ^ {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": "u", "value": ""}]}, {"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*m*n}x ^ {m-1}) (({a} * x^{m}+{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", "type": "question"}, {"name": "Integration", "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": "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</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", "type": "question"}]}, {"name": "Trigonometric equations", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Trigonometry: degrees and radians", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Sanka Balasuriya", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/738/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p>Convert degrees to radians and radians to degrees.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the value of</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"radians2": {"name": "radians2", "group": "Ungrouped variables", "definition": "radians1/pi", "description": "", "templateType": "anything", "can_override": false}, "degrees": {"name": "degrees", "group": "Ungrouped variables", "definition": "random(1..360)", "description": "", "templateType": "anything", "can_override": false}, "radians": {"name": "radians", "group": "Ungrouped variables", "definition": "random(2,3,4,5,9,12,15,18,20,30,36,45,60,90)", "description": "", "templateType": "anything", "can_override": false}, "radians1": {"name": "radians1", "group": "Ungrouped variables", "definition": "(pi/radians)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["radians", "degrees", "radians1", "radians2"], "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>An angle of $\\var{degrees}^{\\circ}$ corresponds in radians to [[0]] radians.</p>\n<p>(Note: you should leave your answer as a fraction in terms of $\\pi,$ which can be entered by typing \"pi\" without the quotation marks)</p>\n<p>An angle of $\\frac{\\pi}{\\var{radians}}$ radians corresponds to&nbsp;[[1]]&nbsp;degrees.</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": "(pi*{degrees})/180", "answerSimplification": "all ", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}, {"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": "180*{radians2}", "answerSimplification": "all ", "showPreview": true, "checkingType": "dp", "checkingAccuracy": "2", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Trigonometric Equations  - Simple (Radians)", "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>Simple trig equations with radians</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Solve the following equation in <em>radians</em>, for $\\theta$ in the range $0\\leq\\theta\\leq2\\pi$.</strong></p>\n<p>Give your answers correct to 2 decimal places in ascending order.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Q1", "definition": "random(3..9)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Q1", "definition": "(a-1)/a", "description": "", "templateType": "anything", "can_override": false}, "f2": {"name": "f2", "group": "Q2", "definition": "c/b", "description": "", "templateType": "anything", "can_override": false}, "f3": {"name": "f3", "group": "Q4", "definition": "(1-b)/(b+1)", "description": "", "templateType": "anything", "can_override": false}, "f4": {"name": "f4", "group": "Q3", "definition": "-c/(c+3)", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Q4", "definition": "arccos(f3)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Q2", "definition": "arccos(f2)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Q1", "definition": "arcsin(f1)", "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}, "s4": {"name": "s4", "group": "Q3", "definition": "arcsin(f4)", "description": "", "templateType": "anything", "can_override": false}, "s22": {"name": "s22", "group": "Q2", "definition": "precround(2*pi-s2,2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Q2", "definition": "random(5..12)", "description": "", "templateType": "anything", "can_override": false}, "s41": {"name": "s41", "group": "Q3", "definition": "precround(pi-s4,2)", "description": "", "templateType": "anything", "can_override": false}, "s42": {"name": "s42", "group": "Q3", "definition": "precround(2*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}, "s21": {"name": "s21", "group": "Q2", "definition": "precround(s2,2)", "description": "", "templateType": "anything", "can_override": false}, "s32": {"name": "s32", "group": "Q4", "definition": "precround(2*pi-s3,2)", "description": "", "templateType": "anything", "can_override": false}, "s11": {"name": "s11", "group": "Q1", "definition": "precround(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{c+3}\\sin(\\theta)=-\\var{c}\\]</p>\n<p>$\\sin(\\theta)=$ [[0]]</p>\n<p>$\\theta=$ [[1]]&nbsp;or&nbsp;[[2]]</p>\n<p>Discard any solutions out of range.</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": "{f4}-0.01", "maxValue": "{f4}+0.01", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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": "{s41}", "maxValue": "{s41}", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{s42}", "maxValue": "{s42}", "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", "type": "question"}, {"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", 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"scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{re(conj(z3^d1))}/{abs(z3^(2*d1))}", "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, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Make sure that you input the real and imaginary parts as fractions and not as decimals</p>"}, "valuegenerators": []}, {"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": "{im(conj(z3^d1))}/{round(abs(z3)^(2*d1))}", "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}, {"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 integers or letters only for part (c) answers below.</p>\n<p></p>\n<p>when&nbsp; \\[z=i^0\\] $z=\\;\\;$ [[0]]</p>\n<p>when&nbsp; \\[z=i^1\\] $z=\\;\\;$ [[1]]</p>\n<p>when&nbsp; \\[z=i^2\\] $z=\\;\\;$ [[2]] &nbsp;</p>\n<p>when&nbsp; \\[z=i^3\\]$z=\\;\\;$ [[3]]</p>\n<p>Hence, or otherwise calculate</p>\n<p>\\[z=i^{\\var{n}}\\]<br/>$z=\\;\\;$[[4]]</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": "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": []}, {"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": "i", "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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-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": []}, {"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": "-i", "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": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{i^n}", "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", "type": "question"}, {"name": "Modulus and argument calculations", "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": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "<p><span style=\"color: #222222; font-family: sans-serif;\"><span style=\"line-height: 20px;\">Finding&nbsp;</span></span>the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the modulus and argument (in radians) of the following complex number, where the argument lies between $-\\pi$ and $\\pi$.</p>\n<p>When calculating the argument pay particular attention to the quadrant in which the complex number lies.</p>\n<p>Input final answers to 3 decimal places.</p>", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"arg3": {"name": "arg3", "group": "Ungrouped variables", "definition": "precround(arg(z3),3)", "description": "", "templateType": "anything", "can_override": true}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "c2+d2*i", "description": "", "templateType": "anything", "can_override": false}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "precround(abs(z3),3)", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.001", "description": "", "templateType": "anything", "can_override": false}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ans3", "arg3", "z3", "c2", "d2", "tol"], "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>$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians</p>\n<p>Input both answers to 3 decimal places.</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>HINT<br/><br/>Remember that</p>\n<p>$|\\var{z3}|=\\;\\;sqrt (\\var{c2}^2+\\var{d2}^2)$&nbsp; &nbsp;and&nbsp; &nbsp;$\\arg(\\var{z})=\\;\\;tan (\\var{d2}/\\var{c2})$</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": "ans3-tol", "maxValue": "ans3+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": 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": "arg3-tol", "maxValue": "arg3+tol", "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", "type": "question"}, {"name": "Harder practice Q - roots of a cubic ", "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": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1809/"}], "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Using either algebraic division or the factor theorem, choose which one of these complex numbers is a root of the given equation $f(z)=0$ , and hence find all the roots.</p>", "advice": "", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "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}, "tx2a2": {"name": "tx2a2", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(1..8)", "description": "", "templateType": "anything", "can_override": false}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "x3b2": {"name": "x3b2", "group": "Ungrouped variables", "definition": "xs1*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}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x1a1": {"name": "x1a1", "group": "Ungrouped variables", "definition": "if(tx1a1=a1,a1+1,tx1a1)", "description": "", "templateType": "anything", "can_override": false}, "x2a1": {"name": "x2a1", "group": "Ungrouped variables", "definition": "if(tx2a1=a1,a1+1,tx2a1)", "description": "", "templateType": "anything", "can_override": false}, "tx1a1": {"name": "tx1a1", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "x2a1+x2b1*i", "description": "", "templateType": "anything", "can_override": false}, "x2b2": {"name": "x2b2", "group": "Ungrouped variables", "definition": "xs3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "xs2": {"name": "xs2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x1a2": {"name": "x1a2", "group": "Ungrouped variables", "definition": "if(tx1a2=a2,a2+1,tx1a2)", "description": "", "templateType": "anything", "can_override": false}, "tx3a2": {"name": "tx3a2", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "a1+ b1*i", "description": "", "templateType": "anything", "can_override": false}, "tx1a2": {"name": "tx1a2", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "tx2a1": {"name": "tx2a1", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "if(abs(a1)=f1,f1+1,f1)", "description": "", "templateType": "anything", "can_override": false}, "z4": {"name": "z4", "group": "Ungrouped variables", "definition": "x3a1+x3b1*i", "description": "", "templateType": "anything", "can_override": false}, "x1b2": {"name": "x1b2", "group": "Ungrouped variables", "definition": "xs2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "x1a1+x1b1*i", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x2b1": {"name": "x2b1", "group": "Ungrouped variables", "definition": "xs2*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x2a2": {"name": "x2a2", "group": "Ungrouped variables", "definition": "if(tx2a2=a2,a2+1,tx2a2)", "description": "", "templateType": "anything", "can_override": false}, "x3a1": {"name": "x3a1", "group": "Ungrouped variables", "definition": "if(tx3a1=a1,a1+1,tx3a1)", "description": "", "templateType": "anything", "can_override": false}, "tx3a1": {"name": "tx3a1", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x3a2": {"name": "x3a2", "group": "Ungrouped variables", "definition": "if(tx3a2=a2,a2+1,tx3a2)", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "xs1": {"name": "xs1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "xs3": {"name": "xs3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "x3b1": {"name": "x3b1", "group": "Ungrouped variables", "definition": "xs3*random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "x1b1": {"name": "x1b1", "group": "Ungrouped variables", "definition": "xs1*random(1..9)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["tx3a1", "f1", "tx3a2", "b2", "tx2a2", "x1b2", "x1b1", "b1", "tx1a1", "tx2a1", "xs2", "xs3", "xs1", "x3a1", "x3a2", "s3", "s2", "s1", "s6", "s5", "s4", "x2a1", "x2a2", "x1a2", "x1a1", "a1", "a2", "z4", "c2", "c1", "z1", "z2", "z3", "x3b1", "x3b2", "x2b1", "x2b2", "tx1a2"], "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>Given &nbsp;$\\displaystyle f(z) = \\simplify[std]{z ^ 3 + {( -2) * a1 -c1}*z ^ 2 + {2 * a1 * c1 + a1 ^ 2 + b1 ^ 2} * z -{c1 * (a1 ^ 2 + b1 ^ 2)}}$, one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$.</p>\n        <p>Choose the correct value for $z_1$:[[0]]</p>", "gaps": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 4, "showCellAnswerState": true, "choices": ["<p>$\\simplify{{a1}+{b1}i}$</p>", "<p>$\\simplify{{x1a1}+{x1b1}i}$</p>", "<p>$\\simplify{{x2a1}+{x2b1}i}$</p>", "<p>$\\simplify{{x3a1}+{x3b1}i}$</p>"], "matrix": ["5", 0, 0, 0], "distractors": ["", "", "", ""]}], "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>The remaining roots of $f(z)$ are:</p>\n        <p>$z_2=\\;\\;$[[0]] (enter the complex root here)</p>\n        <p>$z_3=\\;\\;$[[1]] (enter the real root here)</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a1}-{b1}i", "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": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c1}", "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", "type": "question"}, {"name": "Considering efficiency - roots of a quartic real polynomial", "extensions": [], "custom_part_types": [], "resources": [["question-resources/wm175_method_1.png", "/srv/numbas/media/question-resources/wm175_method_1.png"], ["question-resources/20210608_165320.jpg", "/srv/numbas/media/question-resources/20210608_165320.jpg"], ["question-resources/wm175_method_1_jNN3Dz7.png", "/srv/numbas/media/question-resources/wm175_method_1_jNN3Dz7.png"], ["question-resources/method_1.jpg", "/srv/numbas/media/question-resources/method_1.jpg"], ["question-resources/method_2.jpg", "/srv/numbas/media/question-resources/method_2.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>Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Consider both methods shown to get the correct answer to the question below.<br/>After looking through both methods, choose which one you prefer and say why, giving a full explanation of your choice.</p>\n<p>Remember to compare efficiency, clarity of setting out, the reader's ability to follow the logic used easily in your answer.<br/><br/><strong>Question</strong></p>\n<p>Given &nbsp;$\\displaystyle f(z) = \\simplify[std]{z ^ 4+ 4*z ^ 3+ 28 * z^2 +240z+800}$, one of the following complex numbers is a root $z_1$ of the equation $f(z)=0$.</p>\n<p>Choose the correct value for $z_1$:&nbsp; &nbsp; &nbsp;(i)&nbsp; $1-4i$ &nbsp; &nbsp; &nbsp;or&nbsp; &nbsp; &nbsp;(ii)&nbsp; $-4+2i$</p>\n<p><br/><img src=\"resources/question-resources/method_1.jpg\" width=\"545\" height=\"726\"/>&nbsp; &nbsp; <img src=\"resources/question-resources/method_2.jpg\" width=\"543\" height=\"725\"/></p>\n<p></p>\n<p></p>\n<p><span style=\"text-decoration: underline;\"></span></p>", "advice": "<p></p>\n<p>&nbsp;</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {"ch": {"parameters": [["a", "number"], ["s", "number"], ["b", "number"], ["r", "number"], ["t", "number"]], "type": "number", "language": "jme", "definition": "if(round(a^2+b^2)|round(s),ch(a,s,random(r..t except 0),r,t),b)"}}, "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>Please give your full reasoning for which of the methods shown above you prefer.<br/><br/></p>\n<p>You should aim to write about 50 words in your answer.</p>\n<p>Then send it to Shaheen Charlwood via Teams direct chat before the submission time given (usually the end time of the exam).</p>\n<p></p>", "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}]}, {"name": "Partial derivatives", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Shaheen's copy of Functions of two variables: Locate stationary point", "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": ["Calculus", "calculus", "derivative", "differentiation", "Differentiation", "function of 2 variables", "functions of 2 variables", "functions of two variables", "partial derivatives", "partial differentiation", "stationary points", "stationary points of functions of two variables"], "metadata": {"description": "<p>Find the stationary point $(p,q)$ of the function:&nbsp;$f(x,y)=ax^2+bxy+cy^2+dx+gy$. Calculate $f(p,q)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b)$ of the function</p>\n<p>\\[f(x,y)=\\simplify[std]{{a}*x^2+{b}*x*y+{c1}*y^2+{d}*x+{f}*y}\\]</p>", "advice": "<p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$:</p>\n<p>\\[\\begin{eqnarray*} \\partial f \\over \\partial x &amp;=&amp;0\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;0 \\end{eqnarray*} \\]</p>\n<p>In this case you get two linear equations to solve for $x$ and $y$:</p>\n<p>\\[\\begin{eqnarray*} \\simplify[std]{{2*a}x+{b}y+{d}}&amp;=&amp;0\\\\ \\\\ \\simplify[std]{{b}x+{2*c1}y+{f}}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>On solving these we get \\[ x = \\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\\;\\;\\;y=\\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\\]<br/>On substituting these values into $f(x,y)$ we get:</p>\n<p>\\[f\\left(\\simplify[std]{{2*c1*d-b*f}/{b^2-4*a*c1}},\\simplify[std]{{2*a*f-b*d}/{b^2-4*a*c1}}\\right) = \\var{rawstatval}\\approx\\var{statval}\\]<br/>to 2 decimal places.</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"rawstatval": {"name": "rawstatval", "group": "Ungrouped variables", "definition": "a*x^2+b*x*y+c1*y^2+d*x+f*y", "description": "", "templateType": "anything", "can_override": false}, "statval": {"name": "statval", "group": "Ungrouped variables", "definition": "precround(rawstatval,2)", "description": "", "templateType": "anything", "can_override": false}, "sol": {"name": "sol", "group": "Ungrouped variables", "definition": "switch(b=1,'by multiplying the second equation by '+2*a,b=-1,'by multiplying the second equation by '+2*a,'by multiplying the second equation by '+2*a+' and multiplying the first equation by '+b)", "description": "", "templateType": "anything", "can_override": false}, "lmin": {"name": "lmin", "group": "Ungrouped variables", "definition": "'Local minimum'", "description": "", "templateType": "anything", "can_override": false}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}, "dvalue": {"name": "dvalue", "group": "Ungrouped variables", "definition": "4*a*c1-b^2", "description": "", "templateType": "anything", "can_override": false}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything", "can_override": false}, "neither": {"name": "neither", "group": "Ungrouped variables", "definition": "'Saddle point'", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "if(b^2=4*a*c,c+1,c)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "s1*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "s3*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "s2*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "s4*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "s5*random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "lmax": {"name": "lmax", "group": "Ungrouped variables", "definition": "'Local maximum'", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "(2*a*f-b*d)/(b^2-4*a*c1)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(2*c1*d-b*f)/(b^2-4*a*c1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["rawstatval", "statval", "sol", "lmin", "q1", "dvalue", "s3", "s2", "s1", "s5", "s4", "neither", "b", "c1", "a", "c", "p1", "d", "f", "lmax", "y", "x"], "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>Input both cooordinates as fractions or integers and not decimals.</p>\n<p>$x$&ndash;coordinate, $a=$&nbsp;[[0]].</p>\n<p>$y$&ndash;coordinate, $b=$&nbsp;[[1]].</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{2*c1*d-b*f}/{b^2-4*a*c1}", "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, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input answer as a fraction or an integer, not a decimal</p>"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{2*a*f-b*d}/{b^2-4*a*c1}", "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, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input answer as a fraction or an integer, not a decimal</p>"}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "** Functions of two variables: Stationary points Q1", "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": ["Calculus", "calculus", "derivative", "differentiation", "Differentiation", "functions of two variables", "partial derivative", "partial differentiation", "stationary points", "stationary points of functions of two variables"], "metadata": {"description": "<p>Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n  <p>Answer the following questions about the function:</p>\n  <p>\\[f(x,y)=\\simplify[std]{ ({a} / 3) * x ^ 3 + ({b} / 2) * x ^ 2 * y + {c} * y ^ 2 * x + {d} * y}\\]</p>\n  ", "advice": "<h4>a)</h4>\n<p>\\[\\begin{eqnarray*} {\\partial f \\over \\partial x} &amp;=&amp;\\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))}\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;\\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})} \\end{eqnarray*}\\]</p>\n<h4>b)</h4>\n<p>$(a,b)$ is a stationary point for the function $f(x,y)$ if $f_x=0,\\;\\;f_y=0$,where the partial derivatives are evaluated at $x=a,\\;\\;y=b$. <br/>So you have to make sure that both these partial derivatives are $0$ at the stationary point.</p>\n<p>For this example we have from the above equations that:<br/>\\[\\begin{eqnarray*} \\simplify[std]{(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))}&amp;=&amp;0,\\qquad &amp;\\mathbf{(1)}&amp;\\\\ \\\\ \\simplify[std]{((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})}&amp;=&amp;0, \\qquad &amp;\\mathbf{(2)}&amp; \\end{eqnarray*}\\]</p>\n<p>The left hand side of equation (1) can be factorised as:</p>\n<p>\\[\\simplify[std]{({a1}x+{b1}y)*({c1}x+{d1}y)=0}\\]</p>\n<p>and so we have:<br/>\\[y=\\simplify[std]{{-a1}/{b1}*x},\\mbox{ or } y= \\simplify[std]{{-c1}/{d1}*x}\\]</p>\n<h5>First case: $y= \\simplify[std]{{-a1}/{b1}*x}$</h5>\n<p>Substituting this into equation (2) gives:<br/>\\[\\simplify[std]{{b}/2*x^2-{2c*a1}/{b1}*x^2+{d}}=0 \\Rightarrow \\simplify[std]{{-b*b1+4*c*a1}/{2*b1}*x^2={d}}\\]</p>\n<p>Hence $x=\\var{m}\\mbox{ or } x = \\var{-m}$ and the stationary points which are on the list and which you had to choose are:<br/>\\[\\left(\\var{m},\\simplify[std]{-{a1*m}/{b1}}\\right)\\mbox{ and }\\left(\\var{-m},\\simplify[std]{{a1*m}/{b1}}\\right)\\]</p>\n<h5>Second case: $y= \\simplify[std]{{-c1}/{d1}*x}$</h5>\n<p>{check}</p>\n<p>Substituting this into equation (2) gives:<br/>\\[\\simplify[std]{{b}/2*x^2-{2c*c1}/{d1}*x^2+{d}}=0 \\Rightarrow \\simplify[std]{{-b*d1+4*c*c1}/{2*d1}*x^2={d}}\\]</p>\n<p>{other}</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "a1*c1", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "b1*d1", "description": "", "templateType": "anything", "can_override": false}, "ch": {"name": "ch", "group": "Ungrouped variables", "definition": "if(a1*d1=b1*c1,0,1)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "-(b1*c1-3*a1*d1)/2*m^2", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "if(d*(-b*d1+4*c*c1)<=0,-1,1)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "other": {"name": "other", "group": "Ungrouped variables", "definition": "if(s5=-1,'There can be no more stationary points as we cannot solve this.', if(ch=0,' we get the same stationary points',' You solve this getting 2 values of $x$ and then you obtain two more stationary points (which are not on the list you choose from).'))", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2,4,6)", "description": "", "templateType": "anything", "can_override": false}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "if(b1*c2=3*a1*d1,c2+1,c2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "b1*c1+a1*d1", "description": "", "templateType": "anything", "can_override": false}, "check": {"name": "check", "group": "Ungrouped variables", "definition": "if(ch=0, 'The two expressions for y in terms of x are the same and we will get exactly the same so ignore the rest!',' ')", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(2,4,6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "ch", "d", "m", "s5", "a1", "other", "b1", "c2", "c1", "b", "check", "d1"], "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>Enter the partial derivatives here. Note if you want to enter a product of unknowns, such as $xy$ then you input the expression in the form <code>x*y</code>.</p>\n<p>$\\displaystyle { \\partial f \\over \\partial x}=$&nbsp;[[0]]</p>\n<p>$\\displaystyle {\\partial f \\over \\partial y}=$&nbsp;[[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(({a} * (x ^ 2)) + ({b} * x * y) + ({c} * (y ^ 2)))", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "((({b} / 2) * (x ^ 2)) + ({(2 * c)} * x * y) + {d})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}], "sortAnswers": false}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "\n      <h4>Finding Stationary Points.</h4>\n        <p>Tick the two choices which give stationary points for $f(x,y)$.</p>\n        <p>Note that the easiest way to do this question is to substitute the values for $x$ and for $y$ into the expressions for $\\displaystyle {\\partial f \\over \\partial x}$ and $\\displaystyle{\\partial f \\over \\partial y}$ and see if you get $0$ for both.</p>\n      ", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 3, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$x=\\var{m},\\;\\;y=\\simplify[std]{-{a1*m}/{b1}}$</p>", "<p>$x=\\var{-m},\\;\\;y=\\simplify[std]{{a1*m}/{b1}}$</p>", "<p>$x=\\var{m+1},\\;\\;y=\\simplify[std]{-{c1*(m+1)}/{d1}}$</p>", "<p>$x=\\var{-m-1},\\;\\;y=\\simplify[std]{{c1*(m+1)}/{d1}}$</p>", "<p>$x=\\var{m-1},\\;\\;y=\\simplify[std]{-{a1+2*b1}/{b1}}$</p>", "<p>$x=\\var{-m+1},\\;\\;y=\\simplify[std]{{a1+2*b1}/{b1}}$</p>"], "matrix": [2, 2, 0, 0, 0, 0], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Shaheen's copy of Functions of two variables: Stationary points 3", "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/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d", "s3", "s2", "s1", "s5", "s4", "r"], "tags": ["Calculus", "Differentiation", "Simultaneous equations", "calculus", "differentiate", "differentiation", "functions of two variables", "partial derivative", "partial differentiation", "simultaneous equations", "stationary points"], "preamble": {"css": "", "js": ""}, "advice": "\n    <p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$</p>\n    <p>\\[\\begin{eqnarray*} \\partial f \\over \\partial x &amp;=&amp;0\\\\ \\\\ \\partial f \\over \\partial y &amp;=&amp;0 \\end{eqnarray*} \\]</p>\n    <p>In this case you get two equations to solve for $x$ and $y$</p>\n    <p>\\[\\begin{eqnarray*} \\simplify[std]{{-2*b}*(x-{c})*e^(-(x-{c})^2-(y-{d})^2)}&amp;=&amp;0\\\\ \\\\ \\simplify[std]{{-2*b}*(y-{d})*e^(-(x-{c})^2-(y-{d})^2)}&amp;=&amp;0 \\end{eqnarray*} \\]<br/>We can cancel off the term $\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)}$ in both equations as&nbsp; &nbsp;$\\simplify[std]{e^(-(x-{c})^2-(y-{d})^2)} \\neq 0,\\;\\forall x,\\;y$.&nbsp;&nbsp;</p>\n    <p>On solving these we get \\[ x = \\var{c},\\;\\;\\;y=\\var{d}\\]</p>\n    <p>So the stationary point is $(\\var{c},\\var{d}) \\in D$.</p>\n    <p>On substituting these values into $f(x,y)$ we get:</p>\n    <p>\\[f(\\var{c},\\var{d})=\\simplify[std]{{a}+{b}*e^0={a+b}}\\]</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 0, "prompt": "<p>$x$&ndash;coordinate, $a=$&nbsp;[[0]]</p>\n<p>$y$&ndash;coordinate, $b=$&nbsp;[[1]]</p>\n<p>Input the value of $f(x,y)$ at $(a,b)$:</p>\n<p>$f(a,b)=$&nbsp;[[2]]</p>\n<p>If you want some help, click on <em>Show steps.</em>&nbsp;You will not lose any marks if you do so.</p>", "marks": 0, "gaps": [{"allowFractions": false, "marks": 2, "maxValue": "{c}", "minValue": "{c}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 2, "maxValue": "{d}", "minValue": "{d}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "marks": 1, "maxValue": "{a+b}", "minValue": "{a+b}", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "steps": [{"type": "information", "showCorrectAnswer": true, "scripts": {}, "prompt": "\n          \n          \n          <p>The $(x,y)$ coordinates of the stationary point of a function of 2 variables $f(x,y)$ are given by solving<br/>the following 2 equations for $x$ and $y$</p>\n          \n          \n          \n          <p>\\[\\begin{eqnarray*}\n          \n          \\partial f \\over \\partial x &=&0\\\\\n          \n          \\\\\n          \n          \\partial f \\over \\partial y &=&0\n          \n          \\end{eqnarray*}\n          \n          \\]</p>\n          \n          \n          \n          <p>In this case you get two equations to solve for $x$ and $y$</p>\n          \n          \n            ", "marks": 0}], "type": "gapfill"}], "statement": "<p>In the following question find the $(x,y)$ coordinates of the single stationary point $(a,b) \\in D$ of the continuous function $f: D \\rightarrow \\mathbb{R}$:</p>\n<p>\\[f(x,y) = \\simplify[std]{{a} + {b}*e^(-(x-{c})^2-(y-{d})^2)}\\]</p>\n<p>where \\[D = \\{(x,y): \\simplify[std]{(x-{c})^2+(y-{d})^2}\\} \\le \\var{r}\\]</p>\n<p>That is, $D$ is a disk of radius $\\simplify[std]{sqrt({r})}$ and centre $(\\var{c},\\var{d})$.</p>\n<p>Input both cooordinates as fractions or integers and not decimals.</p>", "type": "question", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "s1*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "s3*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "s2*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "s4*random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "s3": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "s2": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s1": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "s5": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "s4": {"definition": "random(1,-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s4", "description": ""}, "r": {"definition": "random(2,3,5,6,7)", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}}, "metadata": {"notes": "\n        \t\t<p><strong>10/07/2012:</strong></p>\n        \t\t<p>Added tags.<strong><br /></strong></p>\n        \t\t<p>Question appears to be working correctly.</p>\n        \t\t<p><strong>&nbsp;</strong></p>\n        \t\t", "description": "<p>Find the coordinates of the stationary point for $f: D \\rightarrow \\mathbb{R}$:&nbsp;$f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$,&nbsp;$D$ is a disk centre $(c,d)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "A1", "autoSubmit": true, "allowAttemptDownload": false, "downloadEncryptionKey": "", "showresultspage": "never"}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "<p>This assessment has now finished.&nbsp;<br/><br/></p>"}, "timedwarning": {"action": "warn", "message": "<p>You have 5 minutes left.</p>\n<p>Please ensure you have submitted all answers and sent your explanation for Question 15 to Shaheen via Teams direct chat now!</p>"}}, "feedback": {"enterreviewmodeimmediately": false, "showactualmarkwhen": "never", "showtotalmarkwhen": "always", "showanswerstatewhen": "never", "showpartfeedbackmessageswhen": "never", "showexpectedanswerswhen": "never", "showadvicewhen": "never", "allowrevealanswer": false, "intro": "<ul>\n<li>You have 2 hours to complete all the questions.</li>\n<li>Make sure you submit your answers as you complete them and ensure all anwers are submitted before the end of the exam - you will see a reminder when there are 5 minutes left.</li>\n<li>One question requires you to write an explanation of no more than 50 words to be sent to Shaheen Hassan via Teams direct chat before the end of the exam.</li>\n<li>You may use a scientific calculator throughout this exam - if it not a casio fx series calculator then please ensure it has been verified as acceptable by the invigilator.</li>\n<li>You will need paper for your rough work - this is not usully required to be handed in although you should keep it safe in case the marker wishes to see it.&nbsp;</li>\n<li>You may take a comfort break if the invigilator knows and has allowed it (via Teams chat).</li>\n<li>If you have any questions during the exam please use the Teams chat feature to send a message to Shaheen.</li>\n</ul>\n<p>&nbsp; &nbsp;</p>", "end_message": "<p>Thank you - the exam is now finished.</p>", "results_options": {"printquestions": false, "printadvice": false}, "feedbackmessages": [], "reviewshowexpectedanswer": false, "showanswerstate": false, "reviewshowfeedback": false, "showactualmark": false, "showtotalmark": true, "reviewshowscore": false, "reviewshowadvice": false}, "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": [], "resources": [["question-resources/chain_rule_example_mathcentre.JPG", "/srv/numbas/media/question-resources/chain_rule_example_mathcentre.JPG"], ["question-resources/wm175_method_1.png", 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