// Numbas version: exam_results_page_options
{"name": "FY001 - 2021 MOCK A", "metadata": {"description": "<p>MOCK Exam for FY001-2021</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 5400, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", ""], "variable_overrides": [[], [], [], [], [], [], [], [], [], [], []], "questions": [{"name": "Ugur's copy of Simultaneous Equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Mark Patterson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5064/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Exercises in solving simultaneous equations with 2 variables.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<h2>Simultaneous Equations</h2>\n<p></p>\n<p>Solve the following simultaneous equations.</p>", "advice": "<p>To solve simultaneous equations, first&nbsp;rewrite the equations in matrix form of $Ab = C$, where the matrix $A$ will contain the coefficients of the equation, $b$ will contain the unknown values and $C$ will contain the constants.</p>\n<p>We rearrange&nbsp;$Ab = C$ to get $A^{-1}C = b$.</p>\n<p>So, we need to find the inverse of $A$ and multiple it with $C$. This will give us $b$, i.e. the values for $x$ and $y$.</p>\n<p></p>\n<p>For the first question:</p>\n<p>$A = \\simplify{{maA}}$&nbsp; $b = \\begin{pmatrix}x\\\\y\\end{pmatrix}$ $C = \\begin{pmatrix}\\var{a1}\\\\ \\var{a2}\\end{pmatrix}$</p>\n<p>To find the inverse of $A$, switch the elements on the leading diagonal, change the signs on the non-leading diagonal, and divide all elements by the determinant.</p>\n<p>The determinant of $A$ is $|A| = (\\var{maA[0][0]}\\times\\var{maA[1][1]}) -&nbsp;(\\var{maA[0][1]}\\times\\var{maA[1][0]}) = \\var{detA}$</p>\n<p>The inverse of $A$ is $A^{-1} = \\dfrac{1}{\\var{detA}}\\times\\begin{pmatrix}<br/>\\var{maAA[0][0]} &amp; \\var{maAA[0][1]} \\\\<br/>\\var{maAA[1][0]} &amp; \\var{maAA[1][1]}<br/>\\end{pmatrix} =<br/>\\begin{pmatrix}<br/>\\frac{\\var{maAA[0][0]}}{\\var{detA}} &amp; \\frac{\\var{maAA[0][1]}}{\\var{detA}} \\\\<br/>\\frac{\\var{maAA[1][0]}}{\\var{detA}} &amp;&nbsp;\\frac{\\var{maAA[1][1]}}{\\var{detA}}<br/>\\end{pmatrix}$</p>\n<p></p>\n<p>Rearranging for&nbsp;$A^{-1}C = b$:</p>\n<p>$A^{-1}\\times C =&nbsp;<br/>\\begin{pmatrix}<br/>\\frac{\\var{maAA[0][0]}}{\\var{detA}} &amp; \\frac{\\var{maAA[0][1]}}{\\var{detA}} \\\\<br/>\\frac{\\var{maAA[1][0]}}{\\var{detA}} &amp;&nbsp;\\frac{\\var{maAA[1][1]}}{\\var{detA}}<br/>\\end{pmatrix}<br/>\\times\\begin{pmatrix}\\var{a1}\\\\ \\var{a2}\\end{pmatrix} =<br/>\\begin{pmatrix}<br/>\\frac{\\var{maAA[0][0]}}{\\var{detA}}\\times\\var{a1} +&nbsp;\\frac{\\var{maAA[0][1]}}{\\var{detA}}\\times\\var{a2} \\\\<br/>\\frac{\\var{maAA[1][0]}}{\\var{detA}}\\times\\var{a1} +&nbsp;\\frac{\\var{maAA[1][1]}}{\\var{detA}}\\times\\var{a2}<br/>\\end{pmatrix} =<br/> \\begin{pmatrix}\\var{ax}\\\\ \\var{ay}\\end{pmatrix}$</p>\n<p>Therefore $x = \\var{ax}$ and&nbsp;$y = \\var{ay}$.</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ax": {"name": "ax", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,1,1,-1)", "description": "", "templateType": "anything", "can_override": false}, "ay": {"name": "ay", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,1,1,-1)", "description": "", "templateType": "anything", "can_override": false}, "maA": {"name": "maA", "group": "Ungrouped variables", "definition": "matrixrandomN(2,2)", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "maA[0][0]*ax + maA[0][1]*ay", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "maA[1][0]*ax + maA[1][1]*ay", "description": "", "templateType": "anything", "can_override": false}, "bx": {"name": "bx", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,1,1,-1)", "description": "", "templateType": "anything", "can_override": false}, "by": {"name": "by", "group": "Ungrouped variables", "definition": "random(1..9)*random(1,1,1,-1)", "description": "", "templateType": "anything", "can_override": false}, "maB": {"name": "maB", "group": "Ungrouped variables", "definition": "matrixrandomN(2,2)", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "maB[0][0]*bx + maB[0][1]*by", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "maB[1][0]*bx + maB[1][1]*by", "description": "", "templateType": "anything", "can_override": false}, "maAA": {"name": "maAA", "group": "Ungrouped variables", "definition": "matrix(\n  [maA[1][1], -maA[0][1]],\n  [-maA[1][0], maA[0][0]]\n)", "description": "", "templateType": "anything", "can_override": false}, "detA": {"name": "detA", "group": "Ungrouped variables", "definition": "det(maA)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["ax", "ay", "maA", "a1", "a2", "bx", "by", "maB", "b1", "b2", "maAA", "detA"], "variable_groups": [], "functions": {"matrixrandomN": {"parameters": [["rows", "integer"], ["columns", "integer"]], "type": "matrix", "language": "jme", "definition": "matrix(repeat(repeat(random(1..9)*(random(1,1,1,-1)),columns),rows))"}}, "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>$\\simplify{{maA[0][0]}}x + \\simplify{{maA[0][1]}}y = \\simplify{{a1}}$</p>\n<p>$\\simplify{{maA[1][0]}}x +\\simplify{{maA[1][1]}}y = \\simplify{{a2}}$</p>\n<p></p>\n<p>$x =$ [[0]]</p>\n<p>$y =$ [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ax", "maxValue": "ax", "correctAnswerFraction": false, "allowFractions": false, "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": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ay", "maxValue": "ay", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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>$\\simplify{{maB[0][0]}}x + \\simplify{{maB[0][1]}}y = \\simplify{{b1}}$</p>\n<p>$\\simplify{{maB[1][0]}}x +\\simplify{{maB[1][1]}}y = \\simplify{{b2}}$</p>\n<p></p>\n<p>$x =$ [[0]]</p>\n<p>$y =$ [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "bx", "maxValue": "bx", "correctAnswerFraction": false, "allowFractions": false, "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": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "by", "maxValue": "by", "correctAnswerFraction": false, "allowFractions": false, "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"}, {"name": "Functions: domain and range", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Wan Mekwi", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4058/"}], "tags": [], "metadata": {"description": "<p>Determine the largest possible domain of a rational function.</p>", "licence": "Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International"}, "statement": "<p>Consider the function \\[ f(x) = \\frac{\\var{a}}{(\\simplify{(x+{b})(x^2+{c})}} \\]</p>\n<p></p>", "advice": "<p>For this problem, the function will not be defined if the denominator is zero.</p>\n<p>a) So, $f(x)$ is not defined if $\\simplify{x+{b}}=0\\implies x=\\simplify{-{b}}$.</p>\n<p>b) The largest possible domain is the set of real numbers $\\mathbb{R}$ excluding any numbers where $f$ is not defined. Therefore, the largest possible domain is</p>\n<p>$\\mathbb{R}\\backslash \\{\\simplify{-{b}}\\}$.&nbsp;</p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..15)", "description": "<p>Non-zero numbers&nbsp;</p>", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-15..15 except 0)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..10)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "(a)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>State the value of $x$ for which $f(x)$ is NOT defined.</p>\n<p>$x=$&nbsp;[[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "ansA", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "-{b}", "maxValue": "-{b}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "useCustomName": true, "customName": "(b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The largest possible domain for $f$ is:</p>\n<p></p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["$\\mathbb{R}\\backslash\\{\\var{c}\\}$", "$\\mathbb{R}$", "$\\mathbb{R}\\backslash\\{\\simplify{-{b}}\\}$", "$\\{\\simplify{-{b}}\\}$"], "matrix": [0, 0, "1", 0], "distractors": ["Incorrect answer!", "Incorrect answer!", "", "Incorrect answer!"]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Rational functions: Domain, Range, Inverse", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(15..30)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(15..30 except a)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..9 except a)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..20 except b except c)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(15..30 except t except q)", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(15..30 except s)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(15..30 except 0)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(15..30 except s)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "k", "q", "s", "t"], "variable_groups": [], "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 $f(x) = \\frac{\\var{a}x \\ +\\ \\var{b}}{\\var{c}x\\ +\\ \\var{d}}$. Find the natural domain of $f$, $f^{-1}$ and the natural domain of $f^{-1}$.</p>\n<p>Domain of $f$ is all real numbers except [[0]]</p>\n<p>$f^{-1}(x) =$ [[1]]</p>\n<p>Domain of $f^{-1}$ is all real numbers except [[2]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(-{d})/{c}", "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": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(-{d}x + {b})/({c}x - {a})", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a}/{c}", "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": []}], "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>Let $f(x) = \\frac{\\var{s}x \\ -\\ \\var{t}}{\\var{q}x\\ -\\ \\var{k}}$. Find the natural domain of $f$, $f^{-1}$ and the natural domain of $f^{-1}$.</p>\n<p>Domain of $f$ is all real numbers except [[0]]</p>\n<p>$f^{-1} =$ [[1]]</p>\n<p>Domain of $f^{-1}$ is all real numbers except [[2]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{k}/{q}", "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": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({k}x - {t})/({q}x -{s})", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{s}/{q}", "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": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Domain of a rational function short version", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Given a randomised rational function&nbsp;select the possible ways of writing the domain of the function.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given the real functions below, you should be able to determine their domains.&nbsp;</p>", "advice": "<p></p>\n<p><g class=\"gr_ gr_20 gr-alert gr_gramm gr_inline_cards gr_run_anim Grammar multiReplace\" id=\"20\" data-gr-id=\"20\">Division</g> is defined for all real numbers except zero. So when a function involves division (such as a rational function), any input that will result in a division by zero is not allowed. Rational functions are simply a fraction/division of polynomials, so&nbsp;the only real numbers that are not in the domain of a rational function are the roots of the polynomial in the denominator. Sometimes, (for example part c above) you will need to factorise the denominator to determine its roots.</p>\n<p></p>\n<p>d) Even though 'something divided by itself is 1' division by zero is still undefined. So the domain of $h$ is not all of $\\mathbb{R}$, the domain does not include the number $\\var{c[0]}$. In other words, $h(\\var{c[0]})$ is undefined but for all other $r$, $h(r)=1$.</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-12..-1)", "description": "", "templateType": "anything", "can_override": false}, "inp": {"name": "inp", "group": "Ungrouped variables", "definition": "expression(random('x','r','s','t','w'))", "description": "", "templateType": "anything", "can_override": false}, "set_of_holes": {"name": "set_of_holes", "group": "Ungrouped variables", "definition": "set(c)", "description": "", "templateType": "anything", "can_override": false}, "out": {"name": "out", "group": "Ungrouped variables", "definition": "expression(random('f','h','g','p','q','y'))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "a+random(1..12)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "[random(-12..12),random(-12..12 except 0)]", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "sort(shuffle(-12..12)[0..n])", "description": "", "templateType": "anything", "can_override": false}, "rat": {"name": "rat", "group": "Ungrouped variables", "definition": "if(n=3,'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]})*({inp}-{c[2]}))}\\\\]',\nif(n=2,'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]}))}\\\\]',\n'\\\\[\\\\simplify{{out}({inp})=({num[0]}{inp}+{num[1]})/(({inp}-{c[0]})({inp}-{c[1]})*({inp}-{c[2]})*({inp}-{c[3]}))}\\\\]'))\n    \n  \n", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["out", "inp", "num", "n", "c", "rat", "a", "b", "set_of_holes"], "variable_groups": [], "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>There are&nbsp;[[0]]&nbsp;real numbers that are not in the domain&nbsp;of&nbsp;{rat} these are&nbsp;[[1]].</p>\n<p><strong>Note:&nbsp;</strong>If the numbers were $-2,1$ and $4$ you would enter&nbsp;<strong>set(-2,1,4)</strong></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{length(c)}", "maxValue": "{length(c)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{set_of_holes}", "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": []}], "sortAnswers": false}, {"type": "m_n_2", "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>Which of the following represents the domain of \\[\\simplify{g(x)=(x^2-{c[0]+c[1]}x+{c[0]*c[1]})/(x^2-{a+b}x+{a*b})} \\ ?\\]</p>\n<p>(There could be more than one correct choice).</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": "1", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$\\mathbb{R}$</p>", "<p>$\\{x\\in \\mathbb{R}:\\,\\var{a+b}&lt;x&lt;\\var{a*b}\\}$</p>", "<p>$\\{x\\in \\mathbb{R}:\\,x\\ne \\var{c[0]},\\var{c[1]}\\}$</p>", "<p>$\\{x\\in \\mathbb{R}:\\,x\\ne \\var{a},\\var{b}\\}$</p>", "<p>$\\mathbb{R}\\setminus \\{\\var{a},\\var{b}\\}$</p>", "<p>$\\mathbb{R}\\setminus \\{\\var{c[0]},\\var{c[1]}\\}$</p>", "<p>$\\{x\\in \\mathbb{R}:\\,x\\ne \\var{-a},\\var{-b}\\}$</p>"], "matrix": [0, 0, 0, "1", "1", 0, 0], "distractors": ["", "", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Exponential population growth fully randomised", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Students are given an exponential equation and asked to evaluate it at two points.</p>\n<p>The constants in the exponential are randomised.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A population of bacteria has a population, $P$, that can be described by the function $P = \\var{k} \\times \\var{a}^t $, where $t$ is the time in {timeperiod}.</p>", "advice": "<p>a) When $t=0$,</p>\n<p>$&nbsp; P = \\var{k} \\times \\var{a}^t = \\var{k} \\times \\var{a}^0 = 1$</p>\n<p></p>\n<p>b) When $t = \\var{time}$,</p>\n<p>$P = \\var{k} \\times \\var{a}^\\var{time} = \\var{answer_raw}$</p>\n<p>When this is rounded to the nearest whole number, we get</p>\n<p>$P = \\var{answer}$</p>\n<p></p>\n<p>c) We need to take $t = -2$,</p>\n<p>$P = \\var{k} \\times \\var{a}^{-2} = \\var{ansc}$,</p>\n<p>When this is rounded to the nearest whole number, we get</p>\n<p>$P = \\var{ansc_r}$.</p>\n<p></p>\n<p>d) We need to solve the following equation:</p>\n<p>\\[\\var{l} = \\var{k} \\times \\var{a}^t\\]</p>\n<p>for $t$. Take the appropriate logarithm on both sides&nbsp;</p>\n<p>\\[log_{\\var{a}}(\\var{l}/\\var{k}) = \\var{ansd}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"objectchoice": {"name": "objectchoice", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "timeperiodlist": {"name": "timeperiodlist", "group": "Ungrouped variables", "definition": "['days','weeks','months','years']", "description": "", "templateType": "anything", "can_override": false}, "timeperiodchoice": {"name": "timeperiodchoice", "group": "Ungrouped variables", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "timeperiod": {"name": "timeperiod", "group": "Ungrouped variables", "definition": "timeperiodlist[timeperiodchoice]", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(1..10000)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "1 + random(10..70)/100", "description": "", "templateType": "anything", "can_override": false}, "time": {"name": "time", "group": "Ungrouped variables", "definition": "random(2..15)", "description": "", "templateType": "anything", "can_override": false}, "answer_raw": {"name": "answer_raw", "group": "Ungrouped variables", "definition": "k*a^time", "description": "", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "round(answer_raw)", "description": "", "templateType": "anything", "can_override": false}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "random(k+2000, k+2500, k+3000, k+3500)", "description": "", "templateType": "anything", "can_override": false}, "ansc": {"name": "ansc", "group": "Ungrouped variables", "definition": "k*(a^(-2))", "description": "", "templateType": "anything", "can_override": false}, "ansc_r": {"name": "ansc_r", "group": "Ungrouped variables", "definition": "round(ansc)", "description": "", "templateType": "anything", "can_override": false}, "ansd": {"name": "ansd", "group": "Ungrouped variables", "definition": "log(l/k,a)", "description": "", "templateType": "anything", "can_override": false}, "time1": {"name": "time1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["objectchoice", "timeperiodlist", "timeperiodchoice", "timeperiod", "k", "a", "time", "answer_raw", "answer", "l", "ansc", "ansc_r", "ansd", "time1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the initial population, $P$, when $t$ = 0?</p>", "minValue": "{k}", "maxValue": "{k}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "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>What is the population, $P$, when $t = \\var{time}$ {timeperiod}</p>\n<p>Round your answer to the nearest whole number.</p>", "minValue": "round({k}*{a}^{time})", "maxValue": "round({k}*{a}^{time})", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"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>What was the population, $P$, {time1} {timeperiod} ago</p>\n<p>Round your answer to the nearest whole number.</p>\n<p>[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap 0", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "round({k}*{a}^{-time1})", "maxValue": "round({k}*{a}^{-time1})", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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>When the population will reach $P=\\var{l}$. Give your answer accurate to 3 decimal places.&nbsp;</p>\n<p>$t =$[[0]] {timeperiod}</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap 1", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "log({l}/{k}, a)", "maxValue": "log({l}/{k}, a)", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "NC Math 3 Solve an exponential decay problem -- Depreciation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Terry Young", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3130/"}], "ungrouped_variables": ["A", "r", "rPercent", "dFactor", "tDepreciate", "valueNow"], "parts": [{"customMarkingAlgorithm": "", "displayAnswer": "Decay", "scripts": {}, "unitTests": [], "showFeedbackIcon": true, "prompt": "<p>Is the value of the truck exponential growth or decay?</p>", "answer": "decay", "showCorrectAnswer": true, "type": "patternmatch", "customName": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "matchMode": "exact", "adaptiveMarkingPenalty": 0, "variableReplacements": []}, {"customMarkingAlgorithm": "", "scripts": {}, "unitTests": [], "gaps": [{"correctAnswerFraction": false, "showFractionHint": true, "type": "numberentry", "maxValue": "A", "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "mustBeReduced": false, "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "minValue": "A", "scripts": {}, "mustBeReducedPC": 0, "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "allowFractions": false}], "showFeedbackIcon": true, "prompt": "<p>What is the original value of the truck?</p>\n<p>$\\$$[[0]]</p>", "showCorrectAnswer": true, "type": "gapfill", "customName": "", "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "adaptiveMarkingPenalty": 0, "variableReplacements": []}, {"customMarkingAlgorithm": "", "displayAnswer": "{rPercent}%", "scripts": {}, "unitTests": [], "showFeedbackIcon": true, "prompt": "<p>What is the growth or decay rate as a percent?</p>", "answer": "{rPercent}%", "showCorrectAnswer": true, "type": "patternmatch", "customName": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "matchMode": "exact", "adaptiveMarkingPenalty": 0, "variableReplacements": []}, {"customMarkingAlgorithm": "", "scripts": {}, "unitTests": [], "gaps": [{"correctAnswerFraction": false, "precisionPartialCredit": "50", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "precisionType": "dp", "type": "numberentry", "maxValue": "valueNow", "correctAnswerStyle": "plain", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "precision": "2", "adaptiveMarkingPenalty": 0, "showPrecisionHint": false, "customMarkingAlgorithm": "", "minValue": "valueNow", "scripts": {}, "mustBeReducedPC": 0, "precisionMessage": "Money always is expressed with two decimal places.", "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "strictPrecision": true, "marks": 1, "unitTests": [], "variableReplacementStrategy": "originalfirst", "customName": "", "allowFractions": false}], "showFeedbackIcon": true, "prompt": "<p>What is the value of the truck $\\var{tDepreciate}$ years after it was built?</p>\n<p>$\\$$[[0]]</p>", "showCorrectAnswer": true, "type": "gapfill", "customName": "", "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "adaptiveMarkingPenalty": 0, "variableReplacements": []}], "functions": {}, "advice": "", "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"tDepreciate": {"group": "Ungrouped variables", "definition": "random(2..6)", "name": "tDepreciate", "description": "", "templateType": "anything"}, "dFactor": {"group": "Ungrouped variables", "definition": "1-r", "name": "dFactor", "description": "", "templateType": "anything"}, "r": {"group": "Ungrouped variables", "definition": "random(0.1..0.25#0.01)", "name": "r", "description": "", "templateType": "anything"}, "rPercent": {"group": "Ungrouped variables", "definition": "100*r", "name": "rPercent", "description": "", "templateType": "anything"}, "valueNow": {"group": "Ungrouped variables", "definition": "precround(A*(dFactor)^tDepreciate,2)", "name": "valueNow", "description": "", "templateType": "anything"}, "A": {"group": "Ungrouped variables", "definition": "random(40000..80000)", "name": "A", "description": "", "templateType": "anything"}}, "tags": [], "statement": "<p>The value, $y$, of a truck can be modeled by $ y = \\var{A}(\\var{dFactor})^t $, where $t$ is the number of years since the truck was built.</p>", "rulesets": {}, "variable_groups": [], "metadata": {"licence": "All rights reserved", "description": ""}}, {"name": "Logarithms:  Using the Rules", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Practice using the log rules to add and subtract logarithms</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Express each of the following as a single logarithm</p>\n<p>(Remember that when you see $\\log(x)$ we are talking about $\\log_{10}(x)$)</p>\n<p></p>", "advice": "<p>Use of the laws of logarithms is crucial here:</p>\n<p>$\\log{a} + \\log{b} = \\log{ab}$</p>\n<p>$\\log{a} - \\log{b} = \\log{\\frac{a}{b}}$</p>\n<p>$\\log{a^n} = n\\log{a}$</p>\n<p></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"f2": {"name": "f2", "group": "Ungrouped variables", "definition": "random(3..5 except f1)", "description": "", "templateType": "anything", "can_override": false}, "f3": {"name": "f3", "group": "Ungrouped variables", "definition": "f1*f2", "description": "", "templateType": "anything", "can_override": false}, "f4": {"name": "f4", "group": "Ungrouped variables", "definition": "f2-1", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "f5": {"name": "f5", "group": "Ungrouped variables", "definition": "f1^2", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "precround(log(c),4)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "f1*f3/f5", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.0001", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "ans", "c", "tol"], "variable_groups": [], "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>If $\\log{\\var{f1}} + \\log{\\var{f3}} - \\log{\\var{f5}} = \\log(c)$, find c</p>\n<p>$c = $ [[1]]. Input your answer as an integer or a fraction.</p>\n<p>Now, compute $\\log(c)$ to 4 decimal places</p>\n<p>$\\log(c) = $ [[0]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ans-tol}", "maxValue": "{ans+tol}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{f1*f3}/{f5}", "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": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Using the Logarithm Equivalence  $\\log_ba=c \\Longleftrightarrow a=b^c$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Hannah Aldous", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1594/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Rearrange some expressions involving logarithms by applying the relation&nbsp;$\\log_b(a) = c \\iff a = b^c$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "", "advice": "<h4>a)</h4>\n<p><em>i)</em></p>\n<p>We can rearrange logarithms using indices.&nbsp;</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]</p>\n<p>Using this equivalence we can rewrite&nbsp;$\\log_\\var{f}x=\\var{f1}$.</p>\n<p>\\[\\begin{align}<br/>x&amp;= \\var{f}^\\var{f1} \\\\<br/>&amp;=\\var{f^f1}<br/>\\end{align}\\]</p>\n<p><em></em></p>\n<h4>b)</h4>\n<p><em>i)</em></p>\n<p>We can use the equivalence to rewrite our equation.</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]</p>\n<p>We can write out our values to&nbsp;makes it easier.</p>\n<p>\\[\\begin{align}<br/>a&amp;=x \\\\<br/>b&amp;=\\var{g1}\\\\<br/>c&amp;=y+\\var{g2}<br/>\\end{align}\\]</p>\n<p>Then we can write out our equation in the required form.</p>\n<p>\\[x=\\var{g1}^{y+\\var{g2}}\\]</p>\n<p><em></em></p>\n<h4>c)</h4>\n<p>We can&nbsp;use the same equivalence as in part&nbsp;<strong>b)</strong>.&nbsp;</p>\n<p>\\[\\log_ba=c \\Longleftrightarrow a=b^c\\]</p>\n<p>We have</p>\n<p>\\begin{align}<br/>a&amp;=y+\\var{h1} \\\\<br/>b&amp;=x\\\\<br/>c&amp;=\\var{h2}\\text{.} \\\\ \\\\<br/>\\log_{x}(y+\\var{h1}) &amp;= \\var{h2} \\\\<br/>\\implies y+\\var{h1} &amp;= x^{\\var{h2}} \\\\<br/>x &amp;= (y+\\var{h1})^{\\frac{1}{\\var{h2}}}<br/>\\end{align}</p>\n<p></p>\n<h4>d)&nbsp;</h4>\n<p>The two in this list that don't equal $x$ are&nbsp;$\\log_e(x)$ and&nbsp;$\\log_{10}(x)$.</p>\n<p>\\[\\begin{align}<br/>\\log_e(x)&amp;=\\ln(x)\\\\<br/>\\log_{10}(x)&amp;=\\log(x)\\text{.}<br/>\\end{align}\\]</p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"h2": {"name": "h2", "group": "part3", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "f3": {"name": "f3", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "", "templateType": "anything", "can_override": false}, "h1": {"name": "h1", "group": "part3", "definition": "random(1..10 except h2)", "description": "", "templateType": "anything", "can_override": false}, "g2": {"name": "g2", "group": "part 2", "definition": "random(2..10except g1)", "description": "", "templateType": "anything", "can_override": false}, "f2": {"name": "f2", "group": "Ungrouped variables", "definition": "random(2..10 except f3 f)", "description": "", "templateType": "anything", "can_override": false}, "f5": {"name": "f5", "group": "Ungrouped variables", "definition": "random(2..6 except f1)", "description": "", "templateType": "anything", "can_override": false}, "f4": {"name": "f4", "group": "Ungrouped variables", "definition": "random(5..12 except f2 f)", "description": "", "templateType": "anything", "can_override": false}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(2..5 except f)", "description": "", "templateType": "anything", "can_override": false}, "g1": {"name": "g1", "group": "part 2", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "g3": {"name": "g3", "group": "part 2", "definition": "random(2..10except g1 g2)", "description": "", "templateType": "anything", "can_override": false}, "g4": {"name": "g4", "group": "part 2", "definition": "random(2..10except g1 g2 g3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f", "f2", "f1", "f3", "f4", "f5"], "variable_groups": [{"name": "part 2", "variables": ["g3", "g2", "g4", "g1"]}, {"name": "part3", "variables": ["h1", "h2"]}], "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>Rearrange the equation to&nbsp;find $x$.</p>\n<p>$\\log_\\var{f}(x)=\\var{f1}$&nbsp;</p>\n<p>$x=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{f^f1}", "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": []}], "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>Make $x$ the subject of the following equation.</p>\n<p>$\\log_\\var{g1}(x)=y+\\var{g2}$</p>\n<p>$x=$&nbsp;[[0]]<em><br/></em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{g1}^(y+{g2})", "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": "y", "value": ""}]}], "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>Make $x$ the subject of the equation, leaving your answer in the form $a^{\\frac{1}{b}}$.</p>\n<p>$\\log_x(y+\\var{h1})=\\var{h2}$</p>\n<p>$x=$&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(y+{h1})^(1/{h2})", "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": "y", "value": ""}]}], "sortAnswers": false}, {"type": "m_n_2", "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>Which of the following expressions are equivalent to $x$?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "checkbox", "displayColumns": 0, "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>$\\log_a(a^x)$</p>", "<p>$a^{\\log_a(x)}$</p>", "<p>$e^{\\ln(x)}$</p>", "<p>$\\log_{10}(x)$</p>", "<p>$\\log_e(x)$</p>", "<p>$\\ln(e^x)$</p>"], "matrix": ["1", "1", "1", "-5", "-5", "1"], "distractors": ["", "", "", "", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ugur's copy of Write down and apply the formula for an arithmetic sequence.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Given the first few terms of an arithmetic sequence, write down its formula, then find a couple of particular terms.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><span color=\"#000000\" style=\"color: #000000;\">In this question, consider the sequence</span></p>\n<p><span color=\"#000000\" style=\"color: #000000;\">\\[ a = \\var{a1}, \\; \\var{a1+d}, \\; \\var{a1+d*2}, \\; \\var{a1+d*3}, \\; \\ldots \\]</span></p>\n<p><span color=\"#000000\" style=\"color: #000000;\"></span>A helpful person has drawn out a table of the terms so far.</p>\n<table>\n<tbody>\n<tr>\n<th>$\\boldsymbol{n}$</th>\n<td>$1$</td>\n<td>$2$</td>\n<td>$3$</td>\n<td>$4$</td>\n<td>$\\ldots$</td>\n</tr>\n<tr>\n<th>$\\boldsymbol{a_n}$</th>\n<td>$\\var{a1}$</td>\n<td>$\\var{a1+d}$</td>\n<td>$\\var{a1+2d}$</td>\n<td>$\\var{a1+3d}$</td>\n<td>$\\ldots$</td>\n</tr>\n</tbody>\n</table>", "advice": "<p>The formula for the $n^\\text{th}$ term, $a_n$, of an arithmetic sequence is</p>\n<p><span color=\"#000000\">\\[ a_n=a_1+(n-1)d \\text{.} \\]</span></p>\n<p><span color=\"#000000\">$a_1$ is the first term, and $d$ is the common difference between adjacent terms.</span></p>\n<h4><span color=\"#000000\">a)</span></h4>\n<p><span color=\"#000000\">In the given sequence, the common difference is $\\var{a1+d} - \\var{a1} = \\var{d}$, and the first term is $\\var{a1}$.</span></p>\n<p><span color=\"#000000\">So, the formula for this sequence is</span></p>\n<p><span color=\"#000000\">\\[ a_n = \\var{a1} + (n-1) \\times \\var{d} \\text{.} \\]</span></p>\n<h4><span color=\"#000000\">b)</span></h4>\n<p>\\[ a_\\var{small} = \\var{a1} + (\\var{small}-1) \\times \\var{d} = \\var{a1+(small-1)*d} \\text{.} \\]</p>\n<h4><span color=\"#000000\">c)</span></h4>\n<p>\\[ a_\\var{large} = \\var{a1} + (\\var{large}-1) \\times \\var{d} = \\var{a1+(large-1)*d} \\text{.} \\]</p>\n<h4><em></em>d)&nbsp;</h4>\n<p>\\[S_n = \\frac{n(a_1 + a_n)}{2}\\]&nbsp;</p>\n<p>Plug in the values to get $S_{\\var{medium}} = \\frac{\\var{medium}(\\var{a1} + \\var{amed})}{2} = \\var{ansd} $</p>\n<h4></h4>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"large": {"name": "large", "group": "Ungrouped variables", "definition": "random(10..50#5)*10", "description": "<p>A large index to compute</p>", "templateType": "anything", "can_override": false}, "small": {"name": "small", "group": "Ungrouped variables", "definition": "random(6..10)", "description": "<p>A&nbsp;small index to compute</p>", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..90)", "description": "<p>The first&nbsp;term in the sequence</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(3..13 except 10)", "description": "", "templateType": "anything", "can_override": false}, "medium": {"name": "medium", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "templateType": "anything", "can_override": false}, "amed": {"name": "amed", "group": "Ungrouped variables", "definition": "a1+(medium-1)*d", "description": "", "templateType": "anything", "can_override": false}, "ansd": {"name": "ansd", "group": "Ungrouped variables", "definition": "(medium*(a1 + amed))/2", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "d", "small", "large", "medium", "amed", "ansd"], "variable_groups": [], "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>Write out an expression for&nbsp;$a_n$, the $n^{\\text{th}}$ term of the sequence, in terms of $n$.</p>\n<p>$a_n =$&nbsp;[[0]]</p>\n<p><em><span style=\"color: #000000;\" color=\"#000000\"><span style=\"color: #000000;\" color=\"#000000\"></span></span></em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a1}+(n-1){d}", "answerSimplification": "basic", "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": "n", "value": ""}]}], "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></p>\n<p><span style=\"color: #000000;\" color=\"#000000\"><span style=\"color: #000000;\" color=\"#000000\">Find the $\\var{small}^{\\text{th}}$ term</span></span></p>\n<p><span style=\"color: #000000;\" color=\"#000000\"><span style=\"color: #000000;\" color=\"#000000\">$a_{\\var{small}} = $&nbsp;[[0]]</span></span></p>\n<p><em></em></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1+(small-1)*d", "maxValue": "a1+(small-1)*d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the&nbsp;<span color=\"#000000\">$\\var{large}^{\\text{th}}$ term</span></p>\n<p><span color=\"#000000\">$a_{\\var{large}} = $[[0]]</span></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a1+(large-1)*d", "maxValue": "a1+(large-1)*d", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "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>Find the sum of first $\\var{medium}$ terms.</p>\n<p>$S_{\\var{medium}} =$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "gap sum", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "({medium}*({a1} + {amed}))/2", "maxValue": "({medium}*({a1} + {amed}))/2", "correctAnswerFraction": false, "allowFractions": false, "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"}, {"name": "Ugur's copy of Percentages and ratios - box of chocolates", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Adelle Colbourn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2083/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>A simple situational question about a box of chocolates, asking how many of each type there are, what percentage of the box they represent, the probability of picking one and ratios of different types.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p></p>", "advice": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<h4><strong>a)</strong></h4>\n<p>100% represents the whole box of chocolates. As there are 5 different kinds of chocolate in the box and they are all represented equally, to calculate the percentage chocolates which are caramel, divide 100 by 5.</p>\n<p style=\"text-align: center;\">Caramel chocolate = $\\displaystyle\\frac{100}{5}$ = $20$% of the box.</p>\n<p></p>\n<p></p>\n<p><strong>b)</strong>&nbsp;</p>\n<p>The original number of chocolates&nbsp;in the box is stated. We worked out above that each type of chocolate makes up 20% of the box, so we need to work out 20% of {chocs}.</p>\n<p>To do this, either divide {chocs} by 100 and multiply by 20, OR multiply {chocs} by 0.2. The two methods will give the same result.</p>\n<p style=\"text-align: center;\">Method 1: $\\displaystyle\\frac{\\var{chocs}}{100}$ x $20$&nbsp;= $\\var{type}$;</p>\n<p style=\"text-align: center;\"><strong>OR</strong></p>\n<p style=\"text-align: center;\">Method 2: $\\var{chocs}$ x $0.2$ = $\\var{type}$.</p>\n<p></p>\n<p></p>\n<p><strong>c)</strong></p>\n<p>There are now {type} fewer chocolates in the box, but the remaining chocolates&nbsp;now represent 100% of the box. There are now only 4 types of chocolate in it and there is still equal representation inside the box.</p>\n<p>Use the method from part <strong>a)</strong> to find out the equal share of each chocolate type.</p>\n<p style=\"text-align: center;\">Each type = $\\displaystyle\\frac{100}{4}$ = $25$% of the box.</p>\n<p></p>\n<p></p>\n<p><strong>d)</strong>&nbsp;</p>\n<p><em>i)</em></p>\n<p>The first section asks you to compare plain chocolate and dark chocolate. It states that there are {p} plain chocolates and {d} dark chocolates left in the box.</p>\n<p>Insert the numbers of each into the gaps.</p>\n<p style=\"text-align: center;\">Plain $\\var{p}$ : $\\var{d}$ Dark</p>\n<p style=\"text-align: left;\">From this, we should look to see if this answer can be simplified down. To do this, we need to find the greatest common divisor of $\\var{p}$ and&nbsp;$\\var{d}$.&nbsp;</p>\n<p style=\"text-align: left;\">The greatest common divisor is $\\var{gcd}$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd&gt;1\">Using this value to simplify down the ratio by dividing each term by the value, the final answer is</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"gcd&gt;1\">Plain $\\var{ratio_plain}$ : $\\var{ratio_dark}$ Dark.</span></p>\n<p><span data-jme-visible=\"gcd&gt;1\">This states that for every {ratio_plain} plain {if(ratio_plain=1,\"chocolate\",\"chocolates\")}, there {if(ratio_dark=1,\"is\",\"are\")} {ratio_dark} dark {if(ratio_dark=1,\"chocolate\",\"chocolates\")}.</span></p>\n<p><span data-jme-visible=\"gcd=1\">Therefore, it is not possible to simplify further and the final answer is</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"gcd=1\">Plain $\\var{p}$ : $\\var{d}$ Dark.</span></p>\n<p><span data-jme-visible=\"gcd=1\">This states that for every {p} plain {if(p=1,\"chocolate\",\"chocolates\")}, there {if(d=1,\"is\",\"are\")}{d} dark {if(d=1,\"chocolate\",\"chocolates\")}.</span></p>\n<p></p>\n<p><em>ii)</em></p>\n<p>The second section asks you to compare coconut chocolates and the rest of the box. It states that there are {c} coconut chocolates. To calculate the number of chocolates in the rest of the box, add together the stated amounts of plain, dark and nutty chocolates:</p>\n<p style=\"text-align: center;\">$\\var{p}+\\var{d}+\\var{n}$ = $\\var{rob}$.</p>\n<p>Insert these two figures into the gaps.</p>\n<p style=\"text-align: center;\">Coconut $\\var{c}$ : $\\var{rob}$ Other chocolates</p>\n<p style=\"text-align: left;\">From this, we should look to see if this answer can be simplified down. To do this, we need to find the greatest common divisor of $\\var{c}$ and&nbsp;$\\var{rob}$.&nbsp;</p>\n<p style=\"text-align: left;\">The greatest common divisor is $\\var{gcd2}$.</p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd2 &gt; 1\">Using this value to simplify down the ratio by dividing each term by the value, the final answer is</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"gcd2 &gt; 1\">Coconut $\\var{ratio_coconut}$ : $\\var{ratio_rest}$ Other chocolates.</span></p>\n<p><span data-jme-visible=\"gcd2 &gt; 1\">This states that for every {ratio_coconut} coconut {if(ratio_coconut=1,\"chocolate\",\"chocolates\")}, there {if(ratio_rest=1,\"is\",\"are\")} {ratio_rest} other {if(ratio_rest=1,\"chocolate\",\"chocolates\")} in the box.</span></p>\n<p><strong></strong><span data-jme-visible=\"gcd2 = 1\">Therefore, it is not possible to simplify further and the final answer is&nbsp;</span></p>\n<p style=\"text-align: center;\"><span data-jme-visible=\"gcd2 = 1\">Coconut $\\var{c}$ : $\\var{rob}$ Other chocolates.</span></p>\n<p style=\"text-align: left;\"><span data-jme-visible=\"gcd2 = 1\">This states that for every {c} coconut {if(c=1,\"chocolate\",\"chocolates\")}, there {if(rob=1,\"is\",\"are\")} {rob} other {if(rob=1,\"chocolate\",\"chocolates\")} in the box.</span></p>", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"ratio_dark": {"name": "ratio_dark", "group": "Ungrouped variables", "definition": "d/gcd(p,d)", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Number of dark chocolates in ratio of plain to dark.</p>", "templateType": "anything", "can_override": false}, "prob": {"name": "prob", "group": "Ungrouped variables", "definition": "precround({n/{a},2)", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Probability that a nutty chocolate is selected from the box on day 3.</p>", "templateType": "anything", "can_override": false}, "type": {"name": "type", "group": "Ungrouped variables", "definition": "chocs/5", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Number of each type of chocolate in the box initially.</p>", "templateType": "anything", "can_override": false}, "minusc": {"name": "minusc", "group": "Ungrouped variables", "definition": "{chocs-type}", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Number of chocolates in the box minus caramel.</p>", "templateType": "anything", "can_override": false}, "chocs": {"name": "chocs", "group": "Ungrouped variables", "definition": "random(70 .. 95#5)", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Total number of chocolates in the box before eating.</p>", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped 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none;\"></div>\n<p>Number of nutty chocolates on day 3.</p>", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..3)*p", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Number of dark chocolates on day 3.</p>", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Number of plain chocolates on day 3.</p>", "templateType": "anything", "can_override": false}, "ratio_coconut": {"name": "ratio_coconut", "group": "Ungrouped variables", "definition": "c/gcd(c, rob)", "description": "<div id=\"MathJax_Message\" style=\"display: none;\"></div>\n<p>Number of coconut chocolates in ratio of coconut&nbsp;to rest of 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