// Numbas version: exam_results_page_options
{"name": "Morteza's copy of Solve simultaneous equations by Crammer's method", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "maxValue": "a00*a11-a01*a10", "correctAnswerFraction": false, "type": "numberentry", "prompt": "<p>Find the determinant of $\\left[\\begin{array}{cc}a &amp; b\\\\c &amp; d \\end{array}\\right]$:&nbsp;</p>\n<p></p>", "marks": 1, "mustBeReduced": false, "unitTests": [], "minValue": "a00*a11-a01*a10", "extendBaseMarkingAlgorithm": true, "allowFractions": false, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "prompt": "<p>Finally, solve the equations.</p>\n<p>$x = $&nbsp;[[0]]</p>\n<p>$y = $&nbsp;[[1]]</p>", "marks": 0, "unitTests": [], "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "maxValue": "x", "correctAnswerFraction": true, "type": "numberentry", "marks": "0.5", "mustBeReduced": false, "unitTests": [], "minValue": "x", "extendBaseMarkingAlgorithm": true, "allowFractions": true, "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "maxValue": "y", "correctAnswerFraction": true, "type": "numberentry", "marks": "0.5", "mustBeReduced": false, "unitTests": [], "minValue": "y", "extendBaseMarkingAlgorithm": true, "allowFractions": true, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "tags": [], "variable_groups": [], "metadata": {"description": "<p>Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix.&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Solve the following system of equations by using the Crammer's method:</p>\n<p>$ x =\\frac{{\\left|\\begin{array}{cc}v_1 &amp; b\\\\v_2 &amp; d&nbsp;\\end{array}\\right|}}{\\left|\\begin{array}{cc}a &amp; b\\\\c &amp; d&nbsp;\\end{array}\\right|}$</p>\n<p>and&nbsp;</p>\n<p>$y =\\frac{{\\left|\\begin{array}{cc}a &amp; v_1\\\\c &amp; v_2 \\end{array}\\right|}}{\\left|\\begin{array}{cc}a &amp; b\\\\c &amp; d&nbsp;\\end{array}\\right|}$</p>\n<p></p>\n<p>for a matrix $\\mathbf{A}$ and column vectors $\\mathbf{v}$ and $\\mathbf{b}$.</p>\n<p>\\begin{align}<br/>\\simplify[std]{ {ma[0][0]}x + {ma[0][1]}y} &amp;= \\var{mb[0][0]} \\\\<br/>\\simplify[std]{ {ma[1][0]}x + {ma[1][1]}y} &amp;= \\var{mb[1][0]}<br/>\\end{align}</p>\n<p><em><span>Input all numbers as fractions or integers and not as decimals.</span></em></p>", "name": "Morteza's copy of Solve simultaneous equations by Crammer's method", "preamble": {"css": "", "js": ""}, "functions": {}, "extensions": [], "advice": "<h4>a)&nbsp;</h4>\n<p>$\\left|\\begin{array}{cc}a &amp; b\\\\c &amp; d \\end{array}\\right| =a d-bc = simplify[]{ {ma[0][0]}*{ma[1][1]} - {ma[0][1]}*{ma[1][0]}}$&nbsp;</p>\n<p></p>\n<p><strong>b)</strong></p>\n<p>\\[ x =\\frac{{\\left|\\begin{array}{cc}v_1 &amp; b\\\\v_2 &amp; d&nbsp;\\end{array}\\right|}}{\\left|\\begin{array}{cc}a &amp; b\\\\c &amp; d&nbsp;\\end{array}\\right|}\\]</p>\n<p>and&nbsp;</p>\n<p>\\[ y =\\frac{{\\left|\\begin{array}{cc}a &amp; v_1\\\\c &amp; v_2 \\end{array}\\right|}}{\\left|\\begin{array}{cc}a &amp; b\\\\c &amp; d&nbsp;\\end{array}\\right|}\\]</p>\n<p><strong>&nbsp;</strong></p>\n<h4><strong></strong></h4>", "variables": {"mb": {"definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "description": "", "group": "Ungrouped variables", "name": "mb", "templateType": "anything"}, "a10": {"definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "description": "", "group": "Ungrouped variables", "name": "a10", "templateType": "anything"}, "ma_inverse": {"definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "description": "", "group": "Ungrouped variables", "name": "ma_inverse", "templateType": "anything"}, "a01": {"definition": "random(-9..9 except 0)", "description": "", "group": "Ungrouped variables", "name": "a01", "templateType": "anything"}, "x": {"definition": "(ma_inverse*mb)[0][0]", "description": "", "group": "Ungrouped variables", "name": "x", "templateType": "anything"}, "ma": {"definition": "matrix([\n    [a00,a01],\n    [a10,a11]\n])", "description": "<p>Matrix A. a10 is picked so it's non-singular, and a11 is never $\\pm a01$.</p>\n<p>No entry is 0.</p>", "group": "Ungrouped variables", "name": "ma", "templateType": "anything"}, "a00": {"definition": "random(-9..9 except 0)", "description": "", "group": "Ungrouped variables", "name": "a00", "templateType": "anything"}, "a11": {"definition": "random(-9..9 except [0,a01,-a01])", "description": "", "group": "Ungrouped variables", "name": "a11", "templateType": "anything"}, "y": {"definition": "(ma_inverse*mb)[1][0]", "description": "", "group": "Ungrouped variables", "name": "y", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["ma", "a00", "a01", "a10", "a11", "mb", "ma_inverse", "x", "y"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Morteza Shahpari", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2886/"}]}]}], "contributors": [{"name": "Blathnaid Sheridan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/447/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Morteza Shahpari", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2886/"}]}