// Numbas version: exam_results_page_options
{"name": "Andrew's copy of Matrices:  Cramers Rule 3x3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "preamble": {"css": "", "js": ""}, "statement": "<p>Using Cramer's rule , solve the system of equations:</p>\n<p>$\\var{a11}x_1+\\var{a12}x_2+\\var{a13}x_3=\\var{c1}$</p>\n<p>$\\var{a21}x_1+\\var{a22}x_2+\\var{a23}x_3=\\var{c2}$</p>\n<p>$\\var{a31}x_1+\\var{a32}x_2+\\var{a33}x_3=\\var{c3}$</p>\n<p></p>\n<p></p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Cramers Rule applied to 3 simultaneous equations</p>"}, "rulesets": {}, "variables": {"a13": {"name": "a13", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a33": {"name": "a33", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "c2": {"name": "c2", "definition": "a21*x1+a22*x2+a23*x3", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a12": {"name": "a12", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "matrixA1": {"name": "matrixA1", "definition": "matrix([c1,a12,a13],[c2,a22,a23],[c3,a32,a33])", "description": "", "group": "Cramer determinants", "templateType": "anything"}, "matrixA": {"name": "matrixA", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "c3": {"name": "c3", "definition": "a31*x1+a32*x2+a33*x3", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a31": {"name": "a31", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "x3": {"name": "x3", "definition": "random(-100..100 except 0)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a11": {"name": "a11", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a21": {"name": "a21", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a22": {"name": "a22", "definition": "random(0..100 except(a21*a12/a11))/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "x1": {"name": "x1", "definition": "random(-100..100 except 0)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "c1": {"name": "c1", "definition": "a11*x1+a12*x2+a13*x3", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "x2": {"name": "x2", "definition": "random(-100..100 except 0)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "matrixA2": {"name": "matrixA2", "definition": "matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])", "description": "", "group": "Cramer determinants", "templateType": "anything"}, "a23": {"name": "a23", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "matrixA3": {"name": "matrixA3", "definition": "matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])", "description": "", "group": "Cramer determinants", "templateType": "anything"}, "a32": {"name": "a32", "definition": "random(0..100)/10", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "advice": "<p>If \\[&nbsp; A=\\left( \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right),\\]</p>\n<p>\\[&nbsp; C=\\left( \\begin{array}{ccc}<br/>c_{1} \\\\&nbsp;c_{2}&nbsp;\\\\c_{3} \\end{array} \\right),\\]</p>\n<p>Cramer's Rule : ${x_1}=\\frac{\\Delta_1}{\\Delta_0}$ ,&nbsp; ${x_2}=\\frac{\\Delta_2}{\\Delta_0}$ , ${x_3}=\\frac{\\Delta_3}{\\Delta_0}$</p>\n<p>Where:\\[ \\Delta_0=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; a_{13} \\\\a_{21} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_1=\\left| \\begin{array}{ccc}<br/>c_{1} &amp;&nbsp;a_{12} &amp; a_{13} \\\\c_{2} &amp;&nbsp;a_{22} &amp; a_{23}\\\\ c_{3} &amp;&nbsp;a_{32} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_2=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;c_{1} &amp; a_{13} \\\\a_{21} &amp;&nbsp;c_{2} &amp; a_{23}\\\\ a_{31} &amp;&nbsp;c_{3} &amp; a_{33}\\end{array} \\right|\\]</p>\n<p>\\[ \\Delta_3=\\left| \\begin{array}{ccc}<br/>a_{11} &amp;&nbsp;a_{12} &amp; c_{1} \\\\a_{21} &amp;&nbsp;a_{22} &amp; c_{2}\\\\ a_{31} &amp;&nbsp;a_{32} &amp; c_{3}\\end{array} \\right|\\]</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>\n<p><span class=\"nb\"><span class=\"k\"><span class=\"s\">&nbsp;</span></span></span></p>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "x1", "x2", "x3", "c1", "c2", "c3"], "parts": [{"marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>What is the determinant of A=$\\var{matrixA}$? i.e $\\Delta_0$</p>\n<p>[[0]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "det(matrixA)", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "det(matrixA)", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}], "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Calculate&nbsp;$\\Delta_1$ [[0]]</p>\n<p>Hence, calculate ${x_1}$&nbsp; [[1]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "det(matrixA1)", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "det(matrixA1)", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}, {"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{x1}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "{x1}", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}], "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Calculate&nbsp;$\\Delta_2$[[0]]</p>\n<p>Hence, calculate ${x_2}$&nbsp; [[1]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "det(matrixA2)", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "det(matrixA2)", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}, {"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{x2}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "{x2}", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}], "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Calculate&nbsp;$\\Delta_3$[[0]]</p>\n<p>Hence, calculate ${x_3}$&nbsp; [[1]]</p>", "type": "gapfill", "variableReplacements": [], "gaps": [{"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "det(matrixA3)", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "det(matrixA3)", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}, {"marks": 1, "showFeedbackIcon": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "minValue": "{x3}", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "{x3}", "variableReplacements": [], "type": "numberentry", "correctAnswerFraction": false, "allowFractions": false}], "variableReplacementStrategy": "originalfirst"}], "name": "Andrew's copy of Matrices:  Cramers Rule 3x3", "variable_groups": [{"name": "Cramer determinants", "variables": ["matrixA1", "matrixA2", "matrixA3"]}], "functions": {}, "type": "question", "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}]}]}], "contributors": [{"name": "Andrew Dunbar", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/770/"}]}