// Numbas version: exam_results_page_options
{"name": "Surds 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a1", "a2", "b1", "b2", "c1", "c2", "c3", "c4", "c5", "c6", "c7", "d1", "d2", "d3", "d4", "d5", "d6", "d7", "e1", "e2", "e3", "e4", "e5", "e6", "e7", "e8", "e9", "e10", "e11", "f1", "f2", "f3", "f4", "f5", "f6", "f7", "f8", "f9", "f10", "f11", "g1", "g2", "g3", "g4", "g12", "g5", "g6", "g7", "g8", "g9", "g10", "g11", "g13"], "name": "Surds 3", "tags": [], "advice": "<p>The best approach here is to first rationalise the denominator. We do this by multiplying both top and bottom by an approriate value.</p>\n<p>For example to rationalise the denominator for an expression like $\\frac{a}{\\sqrt{b}}$, we multiply numerator and denominator by $\\sqrt{b}$ to get $\\frac{a\\sqrt{b}}{b}$</p>\n<p>Similarly to rationalise the denominator for an expression like $\\frac{a+\\sqrt{b}}{c+\\sqrt{d}}$, we multiply numerator and denominator by $c-\\sqrt{d}$ to get $\\frac{(a+\\sqrt{b})(c-\\sqrt{d})}{c^2-d}$</p>\n<p>Dont forget that:</p>\n<p>$\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}$</p>\n<p>$\\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}}$</p>\n<p>$(a+\\sqrt{b})(a-\\sqrt{b})=a^2-b$</p>\n<p>Remember to check that your answer is in its simplest form.</p>", "rulesets": {}, "parts": [{"prompt": "<p><fraction><numerator>1</numerator><denominator>$\\var{a1}-\\sqrt{\\var{a2}}$</denominator></fraction>&nbsp;$=$&nbsp;<fraction><numerator>[[0]]+$\\sqrt{\\var{a2}}$</numerator><denominator>[[1]]</denominator></fraction>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "a1", "minValue": "a1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "a1^2 - a2", "minValue": "a1^2 - a2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><br/><fraction><numerator>1</numerator><denominator>$\\var{b1}-\\sqrt{\\var{b2}}$</denominator></fraction>&nbsp;$=$&nbsp;<fraction><numerator>[[0]]+$\\sqrt{\\var{b2}}$</numerator><denominator>[[1]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "b1", "minValue": "b1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "b1^2 - b2", "minValue": "b1^2 - b2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><fraction><numerator>$\\var{c3}$</numerator><denominator>$\\var{c1}-\\sqrt{\\var{c2}}$</denominator></fraction>$=$<fraction><numerator>[[0]]$+$[[1]]$\\sqrt{\\var{c2}}$</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "c6*c1", "minValue": "c6*c1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "c6", "minValue": "c6", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "c7", "minValue": "c7", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><fraction><numerator>$\\var{d3}$</numerator><denominator>$\\var{d1}-\\sqrt{\\var{d2}}$</denominator></fraction> $=$&nbsp;<fraction><numerator>[[0]]$+$[[1]]$\\sqrt{\\var{d2}}$</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "d6*d1", "minValue": "d6*d1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "d6", "minValue": "d6", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "d7", "minValue": "d7", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><fraction><numerator>$\\var{e1}-\\sqrt{\\var{e3}}$</numerator><denominator>$\\var{e2}-\\sqrt{\\var{e3}}$</denominator></fraction> $=$&nbsp;<fraction><numerator>[[0]]$+$[[1]]$\\sqrt{\\var{e3}}$</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "e9", "minValue": "e9", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "-e10", "minValue": "-e10", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "e11", "minValue": "e11", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><fraction><numerator>$\\var{f1}+\\sqrt{\\var{f3}}$</numerator><denominator>$\\var{f2}-\\sqrt{\\var{f3}}$</denominator></fraction>$ =&nbsp;$&nbsp;<fraction><numerator>[[0]]$+$[[1]]$\\sqrt{\\var{f3}}$</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "f9", "minValue": "f9", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "f10", "minValue": "f10", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "f11", "minValue": "f11", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p><fraction><numerator>$\\var{g1}+\\sqrt{\\var{g13}}$</numerator><denominator>$\\var{g2}+2\\sqrt{\\var{g3}}$</denominator></fraction>$ =$&nbsp;<fraction><numerator>[[0]]$+$[[1]]$\\sqrt{\\var{g3}}$</numerator><denominator>[[2]]</denominator></fraction></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "g9", "minValue": "g9", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g10", "minValue": "g10", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "g11", "minValue": "g11", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "<p>Express each of the following in the form $\\frac{a+b\\sqrt{c}}{d}$ where a, b, c and d are integers,&nbsp;with&nbsp;a, b, and d having no common factors</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "fraction {\n  display: inline-block;\n  vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n  float: left;\n  width: 100%;\n  text-align: center;\n  line-height: 2.5em;\n}\nfraction > numerator {\n  border-bottom: 1px solid;\n  padding-bottom: 5px;\n}\nfraction > denominator {\n  padding-top: 5px;\n}\nfraction input {\n    line-height: 1em;\n}\n\nfraction .part {\n    margin: 0;\n}\n\n.table-responsive, .fractiontable {\n  display:inline-block;\n}\n.fractiontable {\n  padding: 0; \n  border: 0;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n\n.fractiontable tr {\n    height: 3em;\n}", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "variables": {"f1": {"definition": "random([3,5,2,7,11])", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "random([3,5,7,11] except f1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "random([2,3,5] except f1 except f2)", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}, "f4": {"definition": "f1*f2+f3", "templateType": "anything", "group": "Ungrouped variables", "name": "f4", "description": ""}, "f5": {"definition": "f2+f1", "templateType": "anything", "group": "Ungrouped variables", "name": "f5", "description": ""}, "f6": {"definition": "f2^2-f3", "templateType": "anything", "group": "Ungrouped variables", "name": "f6", "description": ""}, "f7": {"definition": "gcd(f4,f5)", "templateType": "anything", "group": "Ungrouped variables", "name": "f7", "description": ""}, "f8": {"definition": "gcd(f7,f6)", "templateType": "anything", "group": "Ungrouped variables", "name": "f8", "description": ""}, "f9": {"definition": "f4/f8", "templateType": "anything", "group": "Ungrouped variables", "name": "f9", "description": ""}, "c3": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "b1": {"definition": "random([3,5,7,11] except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random([2,3,5] except b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "e4": {"definition": "e1*e2-e3", "templateType": "anything", "group": "Ungrouped variables", "name": "e4", "description": ""}, "d6": {"definition": "d3/d5", "templateType": "anything", "group": "Ungrouped variables", "name": "d6", "description": ""}, "d7": {"definition": "d4/d5", "templateType": "anything", "group": "Ungrouped variables", "name": "d7", "description": ""}, "d4": {"definition": "d1^2 - d2", "templateType": "anything", "group": "Ungrouped variables", "name": "d4", "description": ""}, "d5": {"definition": "gcd(d3,d4)", "templateType": "anything", "group": "Ungrouped variables", "name": "d5", "description": ""}, "d2": {"definition": "random([2,3,5] except d1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d3": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d3", "description": ""}, "d1": {"definition": "random([3,5,7,11] except a1 except b1 except c1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "e11": {"definition": "e6/e8", "templateType": "anything", "group": "Ungrouped variables", "name": "e11", "description": ""}, "e1": {"definition": "random([3,5,2,7,11])", "templateType": "anything", "group": "Ungrouped variables", "name": "e1", "description": ""}, "e3": {"definition": "random([2,3,5] except e1 except e2)", "templateType": "anything", "group": "Ungrouped variables", "name": "e3", "description": ""}, "e2": {"definition": "random([3,5,7,11] except e1)", "templateType": "anything", "group": "Ungrouped variables", "name": "e2", "description": ""}, "g7": {"definition": "gcd(g4,g5)", "templateType": "anything", "group": "Ungrouped variables", "name": "g7", "description": ""}, "g6": {"definition": "g2^2-4g3", "templateType": "anything", "group": "Ungrouped variables", "name": "g6", "description": ""}, "g5": {"definition": "((g12)^0.5)*g2-2g1", "templateType": "anything", "group": "Ungrouped variables", "name": "g5", "description": ""}, "g4": {"definition": "g1*g2-2g3*(g12)^0.5", "templateType": "anything", "group": "Ungrouped variables", "name": "g4", "description": ""}, "g3": {"definition": "random([2,3,5] except g1 except g2)", "templateType": "anything", "group": "Ungrouped variables", "name": "g3", "description": ""}, "g2": {"definition": "random([3,5,7,11] except g1)", "templateType": "anything", "group": "Ungrouped variables", "name": "g2", "description": ""}, "g1": {"definition": "random([3,5,2,7,11])", "templateType": "anything", "group": "Ungrouped variables", "name": "g1", "description": ""}, "g9": {"definition": "g4/g8", "templateType": "anything", "group": "Ungrouped variables", "name": "g9", "description": ""}, "g8": {"definition": "gcd(g7,g6)", "templateType": "anything", "group": "Ungrouped variables", "name": "g8", "description": ""}, "a1": {"definition": "random([3,5,7,11])", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a2": {"definition": "random([2,3,5] except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "e9": {"definition": "e4/e8", "templateType": "anything", "group": "Ungrouped variables", "name": "e9", "description": ""}, "e8": {"definition": "gcd(e7,e6)", "templateType": "anything", "group": "Ungrouped variables", "name": "e8", "description": ""}, "e5": {"definition": "e2-e1", "templateType": "anything", "group": "Ungrouped variables", "name": "e5", "description": ""}, "c2": {"definition": "random([2,3,5] except c1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "random([3,5,7,11] except a1 except b1)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "e6": {"definition": "e2^2 - e3", "templateType": "anything", "group": "Ungrouped variables", "name": "e6", "description": ""}, "c7": {"definition": "c4/c5", "templateType": "anything", "group": "Ungrouped variables", "name": "c7", "description": ""}, "c6": {"definition": "c3/c5", "templateType": "anything", "group": "Ungrouped variables", "name": "c6", "description": ""}, "c5": {"definition": "gcd(c3,c4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c5", "description": ""}, "c4": {"definition": "c1^2 - c2", "templateType": "anything", "group": "Ungrouped variables", "name": "c4", "description": ""}, "e7": {"definition": "gcd(e4,e5)", "templateType": "anything", "group": "Ungrouped variables", "name": "e7", "description": ""}, "e10": {"definition": "e5/e8", "templateType": "anything", "group": "Ungrouped variables", "name": "e10", "description": ""}, "f10": {"definition": "f5/f8", "templateType": "anything", "group": "Ungrouped variables", "name": "f10", "description": ""}, "f11": {"definition": "f6/f8", "templateType": "anything", "group": "Ungrouped variables", "name": "f11", "description": ""}, "g13": {"definition": "g3*g12", "templateType": "anything", "group": "Ungrouped variables", "name": "g13", "description": ""}, "g12": {"definition": "random(4,9,16,25)", "templateType": "anything", "group": "Ungrouped variables", "name": "g12", "description": ""}, "g11": {"definition": "g6/g8", "templateType": "anything", "group": "Ungrouped variables", "name": "g11", "description": ""}, "g10": {"definition": "g5/g8", "templateType": "anything", "group": "Ungrouped variables", "name": "g10", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}