// Numbas version: finer_feedback_settings
{"name": "M2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/M2.JPG", "M2.JPG"], ["question-resources/M2a.JPG", "M2a.JPG"], ["question-resources/M2b.JPG", "M2b.JPG"], ["question-resources/CaptureM2c.JPG", "CaptureM2c.JPG"], ["question-resources/M2d.JPG", "M2d.JPG"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "M2", "tags": [], "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>Rewrite the following system of equations as a matrix equation</p>\n<p></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>", "advice": "<h4>a)</h4>\n<p>The equations can be written in the matrix form</p>\n<p>\\[ \\var{ma}\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\var{mb} \\]</p>\n<h4><strong></strong>b)</h4>\n<p>$\\mathrm{det}(\\mathbf{A}) = \\simplify[]{ {ma[0][0]}*{ma[1][1]} - {ma[0][1]}*{ma[1][0]}} = \\var{det(ma)} \\neq 0$, so $\\mathbf{A}$ is invertible.</p>\n<p>\\[ \\mathbf{A}^{-1} = \\simplify[fractionnumbers]{{ma_inverse}} \\]</p>\n<h4>c)</h4>\n<p>We have&nbsp;</p>\n<p>\\begin{align}<br/>\\mathbf{A}^{-1}\\mathbf{b} &amp;= \\simplify[fractionnumbers]{{ma_inverse}*{mb}} \\\\<br/>&amp;= \\simplify[fractionnumbers]{{ma_inverse*mb}}<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>Rearrange the equation $\\mathbf{Av}=\\mathbf{b}$ to make $\\mathbf{v}$ the subject:</p>\n<p>\\begin{align}<br/>\\mathbf{A}^{-1}\\mathbf{A}\\mathbf{v} &amp;= \\mathbf{A}^{-1}\\mathbf{b} \\\\<br/>\\mathbf{v} &amp;= \\mathbf{A}^{-1}\\mathbf{b} \\\\ \\\\<br/>\\end{align}</p>\n<p>Hence,</p>\n<p>\\[ \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\simplify[fractionnumbers]{{ma_inverse*mb}} \\]</p>\n<p>That is,</p>\n<p>\\begin{align}<br/>x &amp;= \\simplify[fractionnumbers]{{x}}, \\\\ \\\\<br/>y &amp;= \\simplify[fractionnumbers]{{y}}<br/>\\end{align}</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"mb": {"name": "mb", "group": "Ungrouped variables", "definition": "matrix([\n    [random(-9..9 except 0)],\n    [random(-9..9 except 0)]\n])", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(ma_inverse*mb)[0][0]", "description": "", "templateType": "anything", "can_override": false}, "ma": {"name": "ma", "group": "Ungrouped variables", "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>", "templateType": "anything", "can_override": false}, "a10": {"name": "a10", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a00,-a00,a00*a11/a01])", "description": "", "templateType": "anything", "can_override": false}, "ma_inverse": {"name": "ma_inverse", "group": "Ungrouped variables", "definition": "matrix([\n    [ma[1][1], -ma[0][1]],\n    [-ma[1][0], ma[0][0]]\n])/det(ma)", "description": "", "templateType": "anything", "can_override": false}, "a01": {"name": "a01", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "(ma_inverse*mb)[1][0]", "description": "", "templateType": "anything", "can_override": false}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a01,-a01])", "description": "", "templateType": "anything", "can_override": false}, "a00": {"name": "a00", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ma", "a00", "a01", "a10", "a11", "mb", "ma_inverse", "x", "y"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p style=\"text-align: center;\">$A = $&nbsp;[[0]]</p>\n<div style=\"text-align: center;\">$X= $&nbsp;\n<div class=\"matrix-input\">\n<div class=\"matrix-wrapper\"><span class=\"left-bracket\"></span>\n<table class=\"matrix\">\n<tbody>\n<tr>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<span class=\"right-bracket\"></span></div>\n</div>\n<p style=\"text-align: center;\">$b = $ [[3]]</p>\n</div>", "stepsPenalty": "3", "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>Hints:</p>\n<p><img src=\"resources/question-resources/M2a.JPG\" width=\"611\" height=\"262\"/></p>"}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": true, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"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, 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"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the inverse of $A$.</p>\n<p>$A^{-1} = $ [[0]]&nbsp; &nbsp; <em>use fractions or integers in your answer</em></p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Remember that:</p>\n<p><img src=\"resources/question-resources/M2b.JPG\" width=\"627\" height=\"68\"/></p>"}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma_inverse", "correctAnswerFractions": true, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Now find $A^{-1} b$.</p>\n<p><span>$A^{-1}b = $ [[0]]</span></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><img src=\"resources/question-resources/CaptureM2c.JPG\" width=\"568\" height=\"412\"/></p>"}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "ma_inverse*mb", "correctAnswerFractions": true, "numRows": "2", "numColumns": 1, "allowResize": false, "tolerance": 0, "markPerCell": true, "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": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Finally, solve the equations.</p>\n<p>$x = $&nbsp;[[0]]</p>\n<p>$y = $&nbsp;[[1]]</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><img src=\"resources/question-resources/M2d.JPG\" width=\"283\" height=\"169\"/></p>\n<p></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": "x", "maxValue": "x", "correctAnswerFraction": true, "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": "y", "maxValue": "y", "correctAnswerFraction": true, "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", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Shaheen 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