// Numbas version: finer_feedback_settings
{"name": "Linear Algebra In-class test.", "metadata": {"description": "<p>In class test contributes about 30% of your assesment work. Make sure you write your REG number.</p>", "licence": "None specified"}, "duration": 1800, "percentPass": "50", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-shuffled", "pickQuestions": 1, "questionNames": ["", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], []], "questions": [{"name": "Ma planes ne ma lines acho", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Find the line of intersection of the plane given by $3x+6y-5z=-3$ and the plane given by $-2x+7y-z=24$</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Apa we want you to format you answer as, eg the equation you found is $\\vec r (t)&nbsp; = (5, 5, 5) + \\lambda (44, 13, 44) $ just write [5,5,5] + [44t, 13t, 44t]. Note that use $t$ and also square brackets [] for the vector to replace lambda since you can't write it on your keyboard. Unless you know Tex (laTeX) \ud83e\udd2a</p>", "answer": "[-5, 2, 0] +  [29t, 13t, 33t]", "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": "t", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Inverse of a 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Clodagh Carroll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/384/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}, {"name": "Tamsin Smith", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14108/"}, {"name": "Fraser Buxton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/24224/"}, {"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "tags": [], "metadata": {"description": "<p>Cofactors Determinant and inverse of a 3x3 matrix.</p>", "licence": null}, "statement": "<h1>Inverse of a $3 \\times 3$ Matrix</h1>\n<p>Follow the steps in the questions to find the inverse of a $3 \\times 3$ matrix</p>", "advice": "<p><strong>a)</strong></p>\n<p>For simplicity, we will use the expansion of the first row to find the determinant</p>\n<p>$$<br/>\\begin{aligned}<br/>\\det{A} &amp;= a_1A_1 + b_1B_1+ c_1C_1 \\\\<br/>&amp;= \\var{matrixA[0][0]} <br/>\\begin{vmatrix} <br/>\\var{matrixA[1][1]} &amp; \\var{matrixA[1][2]} \\\\ <br/>\\var{matrixA[2][1]} &amp; \\var{matrixA[2][2]} \\\\<br/>\\end{vmatrix} - \\var{matrixA[0][1]} \\begin{vmatrix}<br/>\\var{matrixA[1][0]} &amp; \\var{matrixA[1][2]} \\\\<br/>\\var{matrixA[2][0]} &amp; \\var{matrixA[2][2]} \\\\<br/>\\end{vmatrix} + \\var{matrixA[0][2]} \\begin{vmatrix}<br/>\\var{matrixA[1][0]} &amp; \\var{matrixA[1][1]} \\\\<br/>\\var{matrixA[2][0]} &amp; \\var{matrixA[2][1]} \\\\<br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{matrixA[0][0]} \\times \\var{cof11} - \\left(\\var{matrixA[0][1]} \\times \\var{cof12}\\right) + \\var{matrixA[0][2]} \\times \\var{cof13} \\\\<br/>&amp;= \\var{deta}<br/>\\end{aligned}<br/>$$</p>\n<p><strong>b)</strong></p>\n<p>Given arbitrary matrix</p>\n<p>$$<br/>A = \\begin{pmatrix}<br/>a &amp; b &amp; c \\\\<br/>d &amp; e &amp; f \\\\<br/>g &amp; h &amp; j \\\\<br/>\\end{pmatrix}<br/>$$</p>\n<p>It's cofactors are given by</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{11} &amp;=&nbsp; +\\begin{vmatrix}<br/>e &amp; f \\\\<br/>h &amp; j \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= +\\begin{vmatrix}<br/>\\var{a22} &amp; \\var{a23} \\\\<br/>\\var{a32} &amp; \\var{a33} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof11}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{12} &amp;=&nbsp; -\\begin{vmatrix}<br/>d &amp; f \\\\<br/>g &amp; j \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= -\\begin{vmatrix}<br/>\\var{a21} &amp; \\var{a23} \\\\<br/>\\var{a31} &amp; \\var{a33} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof12}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{13} &amp;=&nbsp; +\\begin{vmatrix}<br/>d &amp; e \\\\<br/>g &amp; h \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= +\\begin{vmatrix}<br/>\\var{a21} &amp; \\var{a22} \\\\<br/>\\var{a31} &amp; \\var{a32} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof13}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{21} &amp;=&nbsp; -\\begin{vmatrix}<br/>b &amp; c \\\\<br/>h &amp; j \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= -\\begin{vmatrix}<br/>\\var{a12} &amp; \\var{a13} \\\\<br/>\\var{a32} &amp; \\var{a33} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof21}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{22} &amp;=&nbsp; +\\begin{vmatrix}<br/>a &amp; c \\\\<br/>g &amp; j \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= +\\begin{vmatrix}<br/>\\var{a11} &amp; \\var{a13} \\\\<br/>\\var{a31} &amp; \\var{a33} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof22}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{23} &amp;=&nbsp; -\\begin{vmatrix}<br/>a &amp; b \\\\<br/>g &amp; h \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= -\\begin{vmatrix}<br/>\\var{a11} &amp; \\var{a12} \\\\<br/>\\var{a31} &amp; \\var{a32} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof23}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{31} &amp;=&nbsp; +\\begin{vmatrix}<br/>b &amp; c \\\\<br/>e &amp; f \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= +\\begin{vmatrix}<br/>\\var{a12} &amp; \\var{a13} \\\\<br/>\\var{a22} &amp; \\var{a23} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof31}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{32} &amp;=&nbsp; -\\begin{vmatrix}<br/>a &amp; c \\\\<br/>d &amp; f \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= -\\begin{vmatrix}<br/>\\var{a11} &amp; \\var{a13} \\\\<br/>\\var{a21} &amp; \\var{a23} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof32}<br/>\\end{aligned}<br/>$$</p>\n<p>$$<br/>\\begin{aligned}<br/>A_{33} &amp;=&nbsp; +\\begin{vmatrix}<br/>a &amp; b \\\\<br/>d &amp; e \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= +\\begin{vmatrix}<br/>\\var{a11} &amp; \\var{a12} \\\\<br/>\\var{a21} &amp; \\var{a22} \\\\ <br/>\\end{vmatrix} \\\\<br/>&amp;= \\var{cof33}<br/>\\end{aligned}<br/>$$</p>\n<p><strong>c)</strong></p>\n<p>Using our answer from the previous question, we simply write the cofactors in the form</p>\n<p>$$<br/>\\begin{pmatrix}<br/>A_{11} &amp; A_{12} &amp; A_{13} \\\\<br/>A_{21} &amp; A_{22} &amp; A_{23} \\\\<br/>A_{31} &amp; A_{32} &amp; A_{33} \\\\<br/>\\end{pmatrix}<br/>$$</p>\n<p>Giving us our matrix of cofactors</p>\n<p>$$<br/>\\begin{pmatrix}<br/>\\var{cof11} &amp; \\var{cof12} &amp; \\var{cof13} \\\\<br/>\\var{cof21} &amp; \\var{cof22} &amp; \\var{cof23} \\\\<br/>\\var{cof31} &amp; \\var{cof32} &amp; \\var{cof33} \\\\<br/>\\end{pmatrix}<br/>$$</p>\n<p><strong>d)</strong></p>\n<p>The transposition process turns rows into columns and columns into rows</p>\n<p>Carrying out this process on our matrix of cofactors gives us the adjugate</p>\n<p>$$<br/>\\begin{pmatrix}<br/>\\var{cof11} &amp; \\var{cof21} &amp; \\var{cof31} \\\\<br/>\\var{cof12} &amp; \\var{cof22} &amp; \\var{cof32} \\\\<br/>\\var{cof13} &amp; \\var{cof23} &amp; \\var{cof33} \\\\<br/>\\end{pmatrix}<br/>$$</p>\n<p><strong>e)</strong></p>\n<p>We can find the inverse of $A$ using our determinant and adjugate, using the formula</p>\n<p>$$<br/>A^{-1} = \\frac{1}{\\det A}(adj \\; A)<br/>$$</p>\n<p>Therefore, we can calculate $A^{-1}$ by</p>\n<p>$$<br/>\\begin{aligned}<br/>A^{-1} &amp;= \\frac{1}{\\var{deta}} \\begin{pmatrix}<br/>\\var{cof11} &amp; \\var{cof21} &amp; \\var{cof31} \\\\<br/>\\var{cof12} &amp; \\var{cof22} &amp; \\var{cof32} \\\\<br/>\\var{cof13} &amp; \\var{cof23} &amp; \\var{cof33} \\\\<br/>\\end{pmatrix} \\\\<br/>&amp;= \\var{inverseA}<br/>\\end{aligned}<br/>$$</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"inverseA": {"name": "inverseA", "group": "Ungrouped variables", "definition": "precround(matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/detA,2)", "description": "", "templateType": "anything", "can_override": false}, "cof22": {"name": "cof22", "group": "cofactors", "definition": "a11*a33-a31*a13", "description": "", "templateType": "anything", "can_override": false}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof11": {"name": "cof11", "group": "cofactors", "definition": "a22*a33-a32*a23", "description": "", "templateType": "anything", "can_override": false}, "a32": {"name": "a32", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof23": {"name": "cof23", "group": "cofactors", "definition": "a12*a31-a11*a32", "description": "<p>cof23</p>", "templateType": "anything", "can_override": false}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(0..5 except(a21*a12/a11))", "description": "", "templateType": "anything", "can_override": false}, "cof32": {"name": "cof32", "group": "cofactors", "definition": "a13*a21-a11*a23", "description": "", "templateType": "anything", "can_override": false}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof13": {"name": "cof13", "group": "cofactors", "definition": "a21*a32-a31*a22", "description": "", "templateType": "anything", "can_override": false}, "matrixA": {"name": "matrixA", "group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything", "can_override": false}, "a31": {"name": "a31", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything", "can_override": false}, "cof21": {"name": "cof21", "group": "cofactors", "definition": "a32*a13-a12*a33", "description": "", "templateType": "anything", "can_override": false}, "a13": {"name": "a13", "group": "Ungrouped variables", "definition": "random(-5..10)", "description": "", "templateType": "anything", "can_override": false}, "cof12": {"name": "cof12", "group": "cofactors", "definition": "a23*a31-a21*a33", "description": "", "templateType": "anything", "can_override": false}, "cof33": {"name": "cof33", "group": "cofactors", "definition": "a11*a22-a12*a21", "description": "", "templateType": "anything", "can_override": false}, "cof31": {"name": "cof31", "group": "cofactors", "definition": "a12*a23-a22*a13", "description": "", "templateType": "anything", "can_override": false}, "detA": {"name": "detA", "group": "Ungrouped variables", "definition": "a11*cof11+a12*cof12+a13*cof13", "description": "", "templateType": "anything", "can_override": false}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(-4..4)", "description": "", "templateType": "anything", "can_override": false}, "a33": {"name": "a33", "group": "Ungrouped variables", "definition": "random(0..20)", "description": "", "templateType": "anything", "can_override": false}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "deta <> 0", "maxRuns": 100}, "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "inverseA", "detA"], "variable_groups": [{"name": "cofactors", "variables": ["cof11", "cof12", "cof13", "cof21", "cof22", "cof23", "cof31", "cof32", "cof33"]}, {"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Let:</p>\n<p>$$<br/>A=\\var{matrixA}<br/>$$</p>\n<p>Find the determinant of $A$</p>\n<p style=\"text-align: center;\">$\\det A =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "det a", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "det(matrixA)", "maxValue": "det(matrixA)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the nine cofactors of $A$</p>\n<p><em>The cofactor $A_{ij}$ denotes the cofactor in row $i$ and column $j$</em></p>\n<p style=\"text-align: center;\">$A _{11}=$ [[0]]</p>\n<p style=\"text-align: center;\">$A_{12}=$ [[1]]</p>\n<p style=\"text-align: center;\">$A_{13}=$ [[2]]</p>\n<p style=\"text-align: center;\">$A_{21}=$ [[3]]</p>\n<p style=\"text-align: center;\">$A_{22}=$ [[4]]</p>\n<p style=\"text-align: center;\">$A_{23}=$ [[5]]</p>\n<p style=\"text-align: center;\">$A_{31}=$ [[6]]</p>\n<p style=\"text-align: center;\">$A_{32}=$ [[7]]</p>\n<p style=\"text-align: center;\">$A_{33}=$ [[8]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof11}", "maxValue": "{cof11}", "correctAnswerFraction": false, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof12}", "maxValue": "{cof12}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof13}", "maxValue": "{cof13}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof21}", "maxValue": "{cof21}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof22}", "maxValue": "{cof22}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof23}", "maxValue": "{cof23}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof31}", "maxValue": "{cof31}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof32}", "maxValue": "{cof32}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{cof33}", "maxValue": "{cof33}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using your answer to part $b)$, state the matrix of cofactors</p>\n<p style=\"text-align: center;\">[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([cof11,cof12,cof12],[cof21,cof22,cof32],[cof31,cof32,cof33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the transpose of your matrix from part $c)$, giving us the adjugate&nbsp;</p>\n<p style=\"text-align: center;\">[[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using your answers from all the previous parts, find the inverse of $A$</p>\n<p><em>Elements will be accepted as fractions or correct to 2 decimal places</em></p>\n<p style=\"text-align: center;\">$A^{-1}=$ [[0]]</p>", "gaps": [{"type": "matrix", "useCustomName": true, "customName": "inv a", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "cof11", "part": "p1g0", "must_go_first": true}, {"variable": "cof12", "part": "p1g1", "must_go_first": true}, {"variable": "cof13", "part": "p1g2", "must_go_first": true}, {"variable": "cof21", "part": "p1g3", "must_go_first": true}, {"variable": "cof22", "part": "p1g4", "must_go_first": true}, {"variable": "cof23", "part": "p1g5", "must_go_first": true}, {"variable": "cof31", "part": "p1g6", "must_go_first": true}, {"variable": "cof32", "part": "p1g7", "must_go_first": true}, {"variable": "cof33", "part": "p1g8", "must_go_first": true}, {"variable": "detA", "part": "p0g0", "must_go_first": true}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([cof11,cof21,cof31],[cof12,cof22,cof32],[cof13,cof23,cof33])/det(matrixA)", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": "0.005", "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": "", "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Use De Moivre's theorem to evaluate complex expressions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>If $z = \\cos \\theta + i \\sin \\theta$, what is the value of $z^n + \\frac{1}{z^n}$ in terms of $\\cos \\theta$</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write your answer as a simple algebraic expression. Use theta to represent the angle theta eg if the answer is $\\tan 3 \\theta$ write tan 3 theta in the answer box</p>", "answer": "2 cos n theta", "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": "cos", "value": ""}, {"name": "n", "value": ""}, {"name": "theta", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Simplify expressions using De Moivre's theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Simplify $ \\big (&nbsp; \\frac{1 + \\cos 2\\theta + i\\sin \\theta}{1 + \\cos 2 \\theta - i \\sin 2 \\theta} \\big )^{30}$.</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Write your expression in this form, for example if answer is $\\cos 3 \\theta + i \\sin 10 \\theta$, you write: cos (3 theta )+ i sin (10 theta). Y'all do computer programming so take care of the syntax!!!</p>", "answer": "cos (60 theta) + i sin (60 theta)", "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": "theta", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Simplify expressions using De Moivre's theorem", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Simplify $\\big (&nbsp; &nbsp;\\sin \\frac{\\pi}{6} + i \\cos \\frac{\\pi}{6}&nbsp; &nbsp; &nbsp; &nbsp;\\big )^{18}$</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Use either manual calculation, or shortcuts as we did in class\ud83d\ude02</p>", "minValue": "1", "maxValue": "1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Planes and stuff", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>Find the plane containing the point $ (-8, 3, 7)$ and parallel to the plane given by $4x + 8y - 2z =45$</p>", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Normal stuff. But format your answer like so, for example if your solution is $3x - 5y +2222222 z = 9$, you just write 3x - 5y + 2222222z = 9</p>", "answer": "4x + 8y -2z = -22", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}, {"name": "z", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": false, "browse": false, "allowsteps": true, "showfrontpage": true, "navigatemode": "sequence", "onleave": {"action": "warnifunattempted", "message": ""}, "preventleave": true, "typeendtoleave": false, "startpassword": "fnechz", "autoSubmit": true, "allowAttemptDownload": true, "downloadEncryptionKey": "algebruh", "showresultspage": "oncompletion"}, "timing": {"allowPause": true, "timeout": {"action": "warn", "message": "<p>Time yakupera ka, make sure you complete whats completable</p>"}, "timedwarning": {"action": "warn", "message": "<p>5 mins left \ud83d\ude29</p>"}}, "feedback": {"enterreviewmodeimmediately": true, "showactualmarkwhen": "never", "showtotalmarkwhen": "always", "showanswerstatewhen": "never", "showpartfeedbackmessageswhen": "always", "showexpectedanswerswhen": "never", "showadvicewhen": "never", "allowrevealanswer": true, "intro": "<p>Inclass test covering the stuff that we have covered already. Its timed, so make sure to be timely otherwise you get zeros. Sort out internet issues beforehand. For your name enter your REG number only</p>", "end_message": "<p>bon chance, on ce voit</p>", "results_options": {"printquestions": true, "printadvice": true}, "feedbackmessages": [], "reviewshowexpectedanswer": false, "showanswerstate": true, "reviewshowfeedback": false, "showactualmark": false, "showtotalmark": true, "reviewshowscore": false, "reviewshowadvice": false}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Farai Nechikwira", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/27403/"}], "extensions": [], "custom_part_types": [], "resources": []}