// Numbas version: exam_results_page_options
{"name": "Sample Paper Q1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Sample Paper Q1", "tags": [], "metadata": {"description": "<p>Adapted from Q1 of the sample paper for Unit 1</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "<hr/>\n<p><strong>a)&nbsp;&nbsp;</strong>Using \"order of operations\" we first need to expand, or multiply out, the brackets.</p>\n<p>We have:</p>\n<h4 style=\"text-align: center;\">\\( \\simplify{{C1} *(x+{C2})+{C3}} \\)</h4>\n<p>so EVERYTHING inside the bracket must get multiplied by \\( \\var{C1}\\)</p>\n<h4 style=\"text-align: center;\">\\( \\simplify{{C1} *(x+{C2})+{C3}} \\)</h4>\n<p>becomes:</p>\n<h4 style=\"text-align: center;\">\\( \\simplify{{C1}*x+{C1}*{C2}} \\simplify{{C3}}&nbsp; \\)</h4>\n<p>Finally, we can collect \"like terms\" to give:</p>\n<h4 style=\"text-align: center;\">\\(&nbsp;\\simplify{{C1}*x+{C1}*{C2} +{C3}}&nbsp; \\)</h4>\n<p>&nbsp;</p>\n<hr/>\n<p><strong>b)</strong>&nbsp;&nbsp;When an expression can be viewed as the difference of two perfect squares, i.e. \\(a^2-b^2\\), then we can factor it as \\((a+b)(a-b)\\).</p>\n<p>We have:</p>\n<h4 style=\"text-align: center;\">\\(&nbsp; x^2-\\var{Sq} \\)</h4>\n<p>So this can be factorised as:</p>\n<h4 style=\"text-align: center;\">\\(&nbsp; (x+\\var{root})(x-\\var{root})&nbsp; &nbsp;\\)</h4>\n<p><strong>Note:</strong> This relies on you being able to recognise square numbers.</p>\n<p>&nbsp;</p>\n<hr/>\n<p><strong>c)&nbsp;</strong> &nbsp;Rearrange (Transpose):</p>\n<h4 style=\"text-align: center;\">\\( R_1=R_0&nbsp; (1+\\alpha t)&nbsp; &nbsp; &nbsp; &nbsp;\\)</h4>\n<h4 style=\"text-align: center;\">\\( R_1=R_0 +R_0 \\alpha t&nbsp; &nbsp; &nbsp; &nbsp;\\)</h4>\n<h4 style=\"text-align: center;\">\\(&nbsp;R_0&nbsp;\\alpha t = R_0 - R_1&nbsp; &nbsp; &nbsp; \\)</h4>\n<h4 style=\"text-align: center;\">\\(&nbsp; \\alpha = \\frac{R_0 - R_1}{R_0 t }&nbsp; &nbsp; &nbsp; \\)</h4>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"C1": {"name": "C1", "group": "Q1a", "definition": "random(2..10)", "description": "<p>Constant #1</p>", "templateType": "anything"}, "C2": {"name": "C2", "group": "Q1a", "definition": "random(-9..9 except 0 except C1)", "description": "<p>Constant #2</p>", "templateType": "anything"}, "C3": {"name": "C3", "group": "Q1a", "definition": "random(-9..-1 except 0 except C1 except C2)", "description": "<p>Constant #3</p>", "templateType": "anything"}, "Squares": {"name": "Squares", "group": "Q1b", "definition": "[ 4, 9, 16, 25, 36, 49, 64, 81, 100 ]", "description": "<p>List of square numbers to be used in \"difference of two squares\"</p>", "templateType": "list of numbers"}, "Sq": {"name": "Sq", "group": "Q1b", "definition": "random(Squares)", "description": "<p>The random choice of square number to use</p>", "templateType": "anything"}, "root": {"name": "root", "group": "Q1b", "definition": "sqrt(Sq)", "description": "<p>root of the chosen Sq</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Q1a", "variables": ["C1", "C2", "C3"]}, {"name": "Q1b", "variables": ["Squares", "Sq", "root"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"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, "prompt": "<p>Remove the brackets and simplify:</p>\n<h4 style=\"text-align: center;\">\\( \\simplify{{C1} *(x+{C2})+{C3}} \\)</h4>", "answer": "{C1}x+{C1*C2}+{C3}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "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, "prompt": "<p>Factorise the expression:</p>\n<h4 style=\"text-align: center;\">\\(&nbsp; x^2-\\var{Sq} \\)</h4>", "answer": "(x+{root})(x-{root})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}, {"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>Transpose the formula to make \\( \\alpha&nbsp; \\) the subject.</p>\n<h4 style=\"text-align: center;\">\\( R_1=R_0&nbsp; (1+\\alpha t)&nbsp; &nbsp; &nbsp; &nbsp;\\)</h4>\n<p>\\( \\alpha = \\) [[0]]</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>This is algebraically correct but is a much clumsier transposition.</p>", "useAlternativeFeedback": false, "answer": "(1/t)(R_1/R_0-1)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "r_0", "value": ""}, {"name": "r_1", "value": ""}, {"name": "t", "value": ""}]}], "answer": "(R_1-R_0)/(R_0*t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "r_0", "value": ""}, {"name": "r_1", "value": ""}, {"name": "t", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}