// Numbas version: exam_results_page_options
{"name": "1.4.2.Exercises", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.4.2.Exercises", "tags": [], "metadata": {"description": "<p>6 Exercises from HELM Book 1.4 Part 2: Multiplication and Division of Algebraic Fractions.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Where possible, simplify each of the following:</p>", "advice": "<p>Remember: to divide by a fraction, multiply by the reciprocal of the second fraction.</p>\n<p>To multiply two fractions: multiply the numerators together and multiply the denominators together. Look for factors common to the numerator and denominator that can be cancelled out. Note that it can sometimes be more efficient to cancel factors before carrying out the multiplications.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1ans": {"name": "q1ans", "group": "q1q2", "definition": "(q1c[0]/q1c[1])*(q1c[2]/q1c[3])", "description": "", "templateType": "anything", "can_override": false}, "q2ans": {"name": "q2ans", "group": "q1q2", "definition": "(q1c[4]/q1c[5])/(q1c[6]/q1c[7])", "description": "", "templateType": "anything", "can_override": false}, "q1n": {"name": "q1n", "group": "q1q2", "definition": "repeat(random(2..15),8)", "description": "<p>the base values for the constants for q1 and q2:</p>\n<p>Q1: n1,d1,n2,d2</p>\n<p>Q2: n1,d1,n2,d2</p>", "templateType": "anything", "can_override": false}, "q1idx": {"name": "q1idx", "group": "q1q2", "definition": "repeat(random(0,1),6)", "description": "<p>decides wiether to take the common factor from the top (0) or the bottom (1), left (0) or right (1), and then whether to attach it to left (0) or right(1).</p>", "templateType": "anything", "can_override": false}, "q1c": {"name": "q1c", "group": "q1q2", "definition": "[if(q1idx[0]=1 and q1idx[2]=0, q1n[0]*q1gcd[0],q1n[0]),\n  if(q1idx[0]=0 and q1idx[2]=0, q1n[1]*q1gcd[0],q1n[1]),\n  if(q1idx[0]=1 and q1idx[2]=1, q1n[2]*q1gcd[0],q1n[2]),\n  if(q1idx[0]=0 and q1idx[2]=1, q1n[3]*q1gcd[0],q1n[3]),\n  if(q1idx[3]=1 and q1idx[5]=0, q1n[4]*q1gcd[1],q1n[4]),\n  if(q1idx[3]=0 and q1idx[5]=0, q1n[5]*q1gcd[1],q1n[5]),\n  if(q1idx[3]=1 and q1idx[5]=1, q1n[6]*q1gcd[1],q1n[6]),\n  if(q1idx[0]=0 and q1idx[5]=1, q1n[7]*q1gcd[1],q1n[7])\n]", "description": "<p>the constants for q1 and q2:</p>\n<p>Q1: n1,d1,n2,d2,gcd</p>\n<p>Q2: n1,d1,n2,d2,gcd</p>", "templateType": "anything", "can_override": false}, "q1gcd": {"name": "q1gcd", "group": "q1q2", "definition": "[q1n[q1idx[1]*2+q1idx[0]],\n q1n[q1idx[3]*2+q1idx[2]]\n]", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "[\"a\",\"b\",\"c\",\"d\",\"f\",\"g\",\"h\",\"k\",\"m\",\"n\",\"p\",\"q\",\"r\",\"s\",\"t\",\"u\",\"v\",\"w\",\"x\",\"y\",\"z\"]", "description": "", "templateType": "anything", "can_override": false}, "q3v": {"name": "q3v", "group": "q3", "definition": "shuffle(alphabet)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "q3a": {"name": "q3a", "group": "q3", "definition": "q3an/q3ad", "description": "", "templateType": "anything", "can_override": false}, "q3n": {"name": "q3n", "group": "q3", "definition": "random(1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "q3d": {"name": "q3d", "group": "q3", "definition": "random(1,2,3)", "description": "", "templateType": "anything", "can_override": false}, "q3ans": {"name": "q3ans", "group": "q3", "definition": "expression((q3a*q3n)/q3d + \"*(\"+q3v[0]+\"+\"+q3v[1]+\")\")", "description": "", "templateType": "anything", "can_override": false}, "q3b": {"name": "q3b", "group": "q3", "definition": "simplify(expression(q3n+\"*(\"+q3v[0]+\"+\"+q3v[1]+\")/\"+q3d),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "q3an,q3ad": {"name": "q3an,q3ad", "group": "Ungrouped variables", "definition": "random([2,1],[3,1],[1,2],[1,3],[2,3])", "description": "", "templateType": "anything", "can_override": false}, "q4a": {"name": "q4a", "group": "q4", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "q4idx": {"name": "q4idx", "group": "q4", "definition": "random(0..3)", "description": "", "templateType": "anything", "can_override": false}, "q4expr": {"name": "q4expr", "group": "q4", "definition": "[ [\"\\\\frac\\{\"+q4v[0]+\"\\}\\{\"+q4v[1]+\"\\}\",\n    \" \\\\times \",\n    \" \\\\frac\\{\"+q4v[0]+\"+\"+q4a+\"\\}\\{\"+q4v[1]+\"+\"+q4a+\"\\}\"\n  ],\n  [\"\\\\frac\\{\"+q4a+\"\\}\\{\"+q4v[1]+\"\\}\",\n    \" \\\\times \",\n    simplify(expression(\"(\"+q4v[0]+\"^2+\"+q4a+\"*\"+q4v[0]+\")/(\"+q4v[1]+\"+\"+q4a+\")\"),\"all\")\n  ],\n  [\"\\\\frac\\{\\\\pi d^2\\}\\{\"+(q4a+1)+\"\\}\",\n    \" \\\\times \",\n    \" \\\\frac\\{Q\\}\\{\\\\pi d^2\\}\"\n  ],\n  [\"\\\\frac\\{Q\\}\\{\\\\frac\\{\\\\pi d^2\\}\\{\"+(q4a+1)+\"\\}\\}\",\n    \"\",\n    \"\"\n  ]\n  ][q4idx]", "description": "", "templateType": "anything", "can_override": false}, "q4v": {"name": "q4v", "group": "q4", "definition": "shuffle(alphabet)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "q4ans": {"name": "q4ans", "group": "q4", "definition": "expression([ \n  \"(\"+q4v[0]+\"*\"+\"(\"+q4v[0]+\"+\"+q4a+\"))/(\"+q4v[1]+\"*\"+\"(\"+q4v[1]+\"+\"+q4a+\"))\",\n  \"(\"+q4v[0]+\"*\"+\"(\"+q4v[0]+\"+\"+q4a+\"))/(\"+q4v[1]+\"*\"+\"(\"+q4v[1]+\"+\"+q4a+\"))\",\n  \"Q/(\"+(q4a+1)+\")\",\n  \"(\"+(q4a+1)+\"*Q)/(\\pi*d^2)\"\n  ][q4idx])", "description": "", "templateType": "anything", "can_override": false}, "q5v": {"name": "q5v", "group": "q5", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q5n": {"name": "q5n", "group": "q5", "definition": "[random(1..10),random(2..5)]", "description": "", "templateType": "anything", "can_override": false}, "q5expr": {"name": "q5expr", "group": "q5", "definition": "expression(\n  \"(\"+(q5n[0]*q5n[1])+\"/\"+(q5n[0]*q5n[1]+1)+\")/(\"+q5v+\"+\"+q5n[0]+\")\"\n  )", "description": "", "templateType": "anything", "can_override": false}, "q6v": {"name": "q6v", "group": "q6", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q6gcd": {"name": "q6gcd", "group": "q6", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "q6idx": {"name": "q6idx", "group": "q6", "definition": "random(0,1)", "description": "<p>0 means that there is no cancelling</p>\n<p>1 means that one x-term is a multiple of the other.</p>", "templateType": "anything", "can_override": false}, "q6ctmp": {"name": "q6ctmp", "group": "q6", "definition": "[random(1..9),// numerator constant\n  random(1..9),random(-9..9 except 0), // denominator 1 constants\n    random(1..9),random(-9..9 except 0) // denominator 2 constants\n]", "description": "", "templateType": "anything", "can_override": false}, "q6c": {"name": "q6c", "group": "q6", "definition": "[ q6ctmp[0],\n  q6ctmp[1]/gcd(q6ctmp[1],q6ctmp[2]),// make sure that the denominator \n  q6ctmp[2]/gcd(q6ctmp[1],q6ctmp[2]),// values are coprime\n  q6ctmp[3]/gcd(q6ctmp[3],q6ctmp[4]),\n  q6ctmp[4]/gcd(q6ctmp[3],q6ctmp[4])\n]", "description": "", "templateType": "anything", "can_override": false}, "q6expr": {"name": "q6expr", "group": "q6", "definition": "[[simplify(expression(q6c[0]+\"/(\"+q6c[1]+\"*\"+q6v+\"+\"+q6c[2]+\")\"),\"all\"),\n  \"\\\\div\",\n  simplify(expression(q6v+\"/(\"+q6c[3]+\"*\"+q6v+\"+\"+q6c[4]+\")\"),\"all\")\n  ]\n  ,\n  [simplify(expression(q6c[0]+\"/(\"+q6c[1]+\"*\"+q6v+\"+\"+q6c[2]+\")\"),\"all\"),\n    \"\\\\div\",\n    simplify(expression(q6v+\"/(\"+q6c[1]*q6gcd+\"*\"+q6v+\"+\"+q6c[2]*q6gcd+\")\"),\"all\")\n  ]\n  ][q6idx]", "description": "", "templateType": "anything", "can_override": false}, "q6num": {"name": "q6num", "group": "q6", "definition": "[\nsimplify(expression(\n    \"(\"+q6c[0]+\"*(\"+q6c[3]+\"*\"+q6v+\"+\"+q6c[4]+\"))\"\n    ),\"all\")  \n,\nexpression(string(q6c[0]*q6gcd))\n][q6idx]", "description": "", "templateType": "anything", "can_override": false}, "q6den": {"name": "q6den", "group": "q6", "definition": "[\nsimplify(expression(\n    \"(\"+q6v+\"*(\"+q6c[1]+\"*\"+q6v+\"+\"+q6c[2]+\"))\"\n    ),\"all\")  \n,\nexpression(q6v)\n][q6idx]", "description": "", "templateType": "anything", "can_override": false}, "q5num": {"name": "q5num", "group": "q5", "definition": "q5n[0]*q5n[1]", "description": "", "templateType": "anything", "can_override": false}, "q5den": {"name": "q5den", "group": "q5", "definition": "expression((q5n[0]*q5n[1]+1)+\"*(\"+q5v+\"+\"+q5n[0]+\")\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q3an,q3ad", "alphabet"], "variable_groups": [{"name": "q1q2", "variables": ["q1n", "q1idx", "q1gcd", "q1c", "q1ans", "q2ans"]}, {"name": "q3", "variables": ["q3v", "q3a", "q3n", "q3d", "q3b", "q3ans"]}, {"name": "q4", "variables": ["q4a", "q4idx", "q4v", "q4expr", "q4ans"]}, {"name": "q5", "variables": ["q5v", "q5n", "q5expr", "q5num", "q5den"]}, {"name": "q6", "variables": ["q6v", "q6gcd", "q6idx", "q6ctmp", "q6c", "q6expr", "q6num", "q6den"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": true, "customName": "Question 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle{\\frac{\\var{q1c[0]}}{\\var{q1c[1]}}\\times \\frac{\\var{q1c[2]}}{\\var{q1c[3]}}}$.</p>", "minValue": "{q1ans}", "maxValue": "{q1ans}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Question 2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle{\\frac{\\var{q1c[4]}}{\\var{q1c[5]}}\\div \\frac{\\var{q1c[6]}}{\\var{q1c[7]}}}$.</p>", "minValue": "{q2ans}", "maxValue": "{q2ans}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": true, "customName": "Question 3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Simplify if possible, but leave your answer in factorised form if there is a common factor.<br/>$\\displaystyle{\\var{expression(string(q3a))}\\times (\\var{q3b})}$.</p>", "answer": "{q3ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$n;a/$n;b*($v+$v) `| $n;a*($v+$v)/$n;b \n`| $n;a;b:1*($v+$v) `| ($v+$v);a:1;b:1\n`| ($v+$v)/$n;b;a:1\n`where gcd(a,b)=1", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "Question 4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle{\\var{latex(q4expr[0])} \\var{latex(q4expr[1])} \\var{latex(q4expr[2])}}\\qquad$ <span data-jme-visible=\"q4idx=3 or q4idx=2\">(to enter $\\pi$, type <code>pi</code>)</span></p>", "answer": "{q4ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": true, "allowUnknownFunctions": false, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "gapfill", "useCustomName": true, "customName": "Question 5", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle{\\var{q5expr}}$</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; border-bottom: 1pt solid;\">[[0]]</td>\n</tr>\n<tr>\n<td>[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "numerator", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{q5num}", "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": 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"variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>$\\displaystyle{\\var{q6expr[0]} \\var{latex(q6expr[1])} \\var{q6expr[2]}}$</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center; border-bottom: 1pt solid;\">[[0]]</td>\n</tr>\n<tr>\n<td>[[1]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "numerator", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{q6num}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 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