// Numbas version: finer_feedback_settings
{"name": "Practice Test FY001", "metadata": {"description": "<p>NUMBAS practice exam for FY001 - Your score won't be counted for your mark.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 3600, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Practice", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "Ugur's copy of Factorising Quadratics 2 - Intermediate", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Katie Lester", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/586/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "advice": "<p><strong>Repeated roots questions, using the equation from Part a:</strong></p>\n<p style=\"text-align: center;\">$\\simplify{{f1}x^2+{f2}*x+{f3}}=0$</p>\n<p style=\"text-align: left;\">These are solved in the exact same way as the previous set of quadratics, but the 'roots' (solutions) are&nbsp;the same.</p>\n<p style=\"text-align: left;\">This type of equation is described as having 'repeated roots'.</p>\n<p style=\"text-align: left;\">The graphed functions would only touch the $x$-axis once either at their peak or trough.</p>\n<p style=\"text-align: left;\">Looking at the above equation, you may be able to instantly see that the brackets must both include the quantity $\\var{c}$.</p>\n<p style=\"text-align: left;\">If not, don't worry, just go through the same process of trial and error as taught previously until you reach your solution of</p>\n<p style=\"text-align: left;\">$\\simplify{({a}x+{c})({b}x+{d})}$.</p>\n<p style=\"text-align: left;\">Then, noticing this can be further simplified, write the final simplification as:</p>\n<p style=\"text-align: center;\">$\\simplify{({a}x+{c})^2}$.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"><em></em></p>\n<p style=\"text-align: left;\"><em></em></p>\n<p style=\"text-align: left;\"><em>Note:</em>&nbsp;If the $x^2$ term has a coefficient and simplifies to, for example, $(2x+4)(x+4)$, this is not defined as having repeated roots, since the solutions are:</p>\n<p style=\"text-align: left;\">$2x+4=0$&emsp;&emsp;so&emsp;&emsp;&emsp;$x=\\frac{-4}{2}=-2$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: left;\">$x+4=0$&emsp;&emsp;&nbsp;&nbsp;so&emsp;&emsp;&emsp;$x=-4$</p>\n<p style=\"text-align: left;\">so are in fact different.</p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\"><strong>Zero constants equations, using Part d:</strong></p>\n<p style=\"text-align: center;\">$\\simplify{{f14}x^2+{f24}*x+{f34}}=0$</p>\n<p style=\"text-align: left;\">Some equations will not have a constant at the end, meaning one of the constants is zero.</p>\n<p style=\"text-align: left;\">In this case, if there is still an $x$ term, you know one of the constants must still be non-zero.</p>\n<p style=\"text-align: left;\">You must therefore, take out a factor of a number $\\times x$.</p>\n<p style=\"text-align: left;\">Here, the factor could be $\\simplify{{a4}x}$ or&nbsp;$\\simplify{{b4}x}$.</p>\n<p style=\"text-align: left;\">It doesn't matter which you choose, but in this example we will use&nbsp;$\\simplify{{a4}x}$.</p>\n<p style=\"text-align: left;\">In order to pull out this factor, the rest of the equation must be divided by&nbsp;$\\simplify{{a4}x}$:</p>\n<p style=\"text-align: left;\">$\\frac{\\simplify{{f14}x^2+{f24}*x+{f34}}}{\\simplify{{a4}*x}}=\\simplify{{f14}x/{a4}+{f24}/{a4}}$.</p>\n<p style=\"text-align: left;\">Therefore, the solution is:</p>\n<p style=\"text-align: center;\">$\\simplify{{a4}x*({f14}x/{a4}+{f24}/{a4})}$.</p>", "rulesets": {}, "parts": [{"prompt": "<p>$\\simplify{{f1}x^2+{f2}*x+{f3}}$</p>\n<p>$=$[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a}x+{c})({b}x+{d})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1], "musthave": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["(", ")"], "partialCredit": 0}}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\simplify{{f12}x^2+{f22}*x+{f32}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a2}x+{c2})({b2}x+{d2})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$\\simplify{{f13}x^2+{f23}*x+{f33}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a3}x+{c3})({b3}x+{d3})", "marks": "2", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "<p>$\\simplify{{f14}x^2+{f24}*x+{f34}}$</p>\n<p>$=$[[0]]&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>The answer must be factorised.</p>", "showStrings": true, "strings": ["x^"], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": ["x"], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "({a4}x+{c4})({b4}x+{d4})", "marks": "3", "checkvariablenames": true, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "steps": [{"prompt": "<p>Hint:&nbsp;Remember one or more of the constants could be zero.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 0, "scripts": {}, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "<p><strong>Simplify the following quadratics into two linear factors in the form $(ax+c)(bx+d)$.</strong></p>\n<p>Quadratics are defined as having 'repeated roots' if the both roots in the solution are the same. This can be written as $(x+a)(x+a)$, but can be simplified to $(x+a)^2$.</p>", "variable_groups": [{"variables": ["a", "b", "c", "d", "f1", "f2", "f3"], "name": "Part a"}, {"variables": ["a2", "b2", "c2", "d2", "f12", "f22", "f32"], "name": "Part b"}, {"variables": ["a3", "b3", "c3", "d3", "f13", "f23", "f33"], "name": "Part c"}, {"variables": ["a4", "b4", "c4", "d4", "f14", "f24", "f34"], "name": "Part d"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"f1": {"definition": "a*b", "templateType": "anything", "group": "Part a", "name": "f1", "description": ""}, "f2": {"definition": "(b*c)+(a*d)", "templateType": "anything", "group": "Part a", "name": "f2", "description": ""}, "f3": {"definition": "c*d", "templateType": "anything", "group": "Part a", "name": "f3", "description": ""}, "b4": {"definition": "random(1..2 except a4)", "templateType": "anything", "group": "Part d", "name": "b4", "description": ""}, "b2": {"definition": "random(1..3)", "templateType": "anything", "group": "Part b", "name": "b2", "description": ""}, "b3": {"definition": "random(2..3)", "templateType": "anything", "group": "Part c", "name": "b3", "description": ""}, "d4": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part d", "name": "d4", "description": ""}, "d2": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part b", "name": "d2", "description": ""}, "d3": {"definition": "random(-15..15 except 0)", "templateType": "anything", "group": "Part c", "name": "d3", "description": ""}, "f23": {"definition": "(b3*c3)+(a3*d3)", "templateType": "anything", "group": "Part c", "name": "f23", "description": ""}, "f22": {"definition": "(b2*c2)+(a2*d2)", "templateType": "anything", "group": "Part b", "name": "f22", "description": ""}, "f24": {"definition": "(b4*c4)+(a4*d4)", "templateType": "anything", "group": "Part d", "name": "f24", "description": ""}, "a3": {"definition": "random(1..3)", "templateType": "anything", "group": "Part c", "name": "a3", "description": ""}, "a2": {"definition": "random(2..3)", "templateType": "anything", "group": "Part b", "name": "a2", "description": ""}, "a4": {"definition": "random(1..3)", "templateType": "anything", "group": "Part d", "name": "a4", "description": ""}, "c3": {"definition": "random(-10..10)", "templateType": "anything", "group": "Part c", "name": "c3", "description": ""}, "c2": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part b", "name": "c2", "description": ""}, "c4": {"definition": "0", "templateType": "anything", "group": "Part d", "name": "c4", "description": ""}, "a": {"definition": "1", "templateType": "anything", "group": "Part a", "name": "a", "description": ""}, "c": {"definition": "random(-10..10 except 0)", "templateType": "anything", "group": "Part a", "name": "c", "description": ""}, "b": {"definition": "1", "templateType": "anything", "group": "Part a", "name": "b", "description": ""}, "d": {"definition": "c", "templateType": "anything", "group": "Part a", "name": "d", "description": ""}, "f32": {"definition": "c2*d2", "templateType": "anything", "group": "Part b", "name": "f32", "description": ""}, "f33": {"definition": "c3*d3", "templateType": "anything", "group": "Part c", "name": "f33", "description": ""}, "f34": {"definition": "c4*d4", "templateType": "anything", "group": "Part d", "name": "f34", "description": ""}, "f12": {"definition": "a2*b2", "templateType": "anything", "group": "Part b", "name": "f12", "description": ""}, "f13": {"definition": "a3*b3", "templateType": "anything", "group": "Part c", "name": "f13", "description": ""}, "f14": {"definition": "a4*b4", "templateType": "anything", "group": "Part d", "name": "f14", "description": ""}}, "metadata": {"notes": "", "description": "<p>Factorising further basic quadratics into linear expressions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Ugur's copy of Algebra. Algebraic fractions. Addition, subtraction, mult and division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Katy Dobson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/854/"}, {"name": "Lovkush Agarwal", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1358/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "tags": [], "metadata": {"description": "<p>Practice with adding, subtracting and dividing basic algebraic fractions</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Express the following as a single fraction and simplify where possible.</p>\n<p>Remember to use brackets appropriately and to use a * for multiplication for letters.</p>\n<p>-----------------------------------</p>", "advice": "<p>See Lecture 4.1 and Workshop 4.2 for background and examples.</p>\n<p>For background on fractions see 2.3.&nbsp; For background on algebraic manipulations see 2.6 and 3.3.</p>", "rulesets": {}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "values[0]", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "values[4]", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "values[2]", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "values[1]", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "values[3]", "description": "", "templateType": "anything"}, "values": {"name": "values", "group": "Ungrouped variables", "definition": "shuffle([5,7,8,9])+[random(5..9)]", "description": "<p>a/x&nbsp; + 1/bx</p>\n<p>c/x - a/d</p>\n<p>a/x * d/by</p>\n<p>d/x&nbsp; /&nbsp; &nbsp;c/ay</p>\n<p>a/x -&nbsp; 1 / (c-fx)</p>\n<p>c/(d + y) - (b)/(a+ x)</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["values", "a", "b", "c", "d", "f"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>\\[ \\frac{\\var{a}}{x} + \\frac{1}{\\var{b}x} \\]</p>", "answer": "({a*b}+1)/({b}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "$n/($n*x)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>\\[ \\frac{\\var{c}}{x} -&nbsp;\\frac{\\var{a}}{\\var{d}} \\]</p>", "answer": "({c*d}-{a}*x)/({d}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "($n + `+-($n*x))/($n*x)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>\\[ \\frac{\\var{a}}{x} \\times&nbsp;\\frac{\\var{d}}{\\var{b}y} \\]</p>", "answer": "({a*d})/({b}x*y)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "($n)/($n*x*y)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "0.5", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>\\[ \\frac{\\var{d}}{x} \\div&nbsp;\\frac{\\var{c}}{\\var{a}y} \\]</p>", "answer": "({a*d}y)/({c}x)", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "($n*y)/($n*x)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">\\[ \\dfrac{\\var{a}}{x} + \\dfrac{1}{(\\var{c} -\\var{f}x)} \\]</span></p>", "answer": "({a*c} + {1 - a*f}*x)/(x*({c} -{f}x))", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "($n + `+-($n*x))/?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">\\[ \\frac{\\var{c}}{(\\var{d} + y)} - \\frac{\\var{b}}{(\\var{a} + x)}\\]</span></p>", "answer": "({a*c -b*d} + {c}x - {b}y)/(({d} + y)*({a} + x))", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "mustmatchpattern": {"pattern": "(`+-($n) + `+-($n*x) +  `+-($n*y))/(?)", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}, {"name": "y", "value": ""}]}, {"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": false, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Have you checked that you have entered answers appropriately: used brackets and * where necessary.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": false, "choices": ["Yes", "No"], "matrix": [0, 0], "distractors": ["", ""]}], "type": "question"}, {"name": "Ugur's copy of Indices - Multiplication/Division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "variables": {"d": {"name": "d", "description": "", "group": "Ungrouped variables", "definition": "random(-9..-1)", "templateType": "anything"}, "h": {"name": "h", "description": "", "group": "Ungrouped variables", "definition": "random(2..5 except g)", "templateType": "anything"}, "j": {"name": "j", "description": "", "group": "Ungrouped variables", "definition": "random(2..3 except f)", "templateType": "anything"}, 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"variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "{a}^{f}*k^({b}*{f})", "adaptiveMarkingPenalty": 0, "prompt": "<p>$(\\var{a}k^\\var{b})^\\var{f}$</p>", "type": "jme", "customMarkingAlgorithm": "", "customName": "", "failureRate": 1, "useCustomName": false, "checkVariableNames": false, "answerSimplification": "all", "vsetRangePoints": 5, "checkingAccuracy": 0.001}, {"checkingType": "absdiff", "steps": [{"unitTests": [], "showCorrectAnswer": true, "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "prompt": "<p>Use the following law:</p>\n<p>$x^a \\times x^b = x^{a+b}$</p>\n<p></p>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "type": "information"}], "valuegenerators": [{"name": "y", "value": ""}], "notallowed": {"strings": ["*"], "showStrings": 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"showCorrectAnswer": true, "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "prompt": "<p>Use the following law to answer this question:</p>\n<p>$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$</p>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "type": "information"}], "valuegenerators": [{"name": "h", "value": ""}], "stepsPenalty": 0, "scripts": {}, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showPreview": true, "marks": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "{a}/{b}*h^({c}-{d})", "adaptiveMarkingPenalty": 0, "prompt": "<p><fraction><numerator>$\\var{a}h^\\var{c}$</numerator><denominator>$\\var{b}h^\\var{d}$</denominator></fraction></p>\n<p></p>", "type": "jme", "customMarkingAlgorithm": "", "customName": "", "failureRate": 1, "useCustomName": false, "checkVariableNames": false, "answerSimplification": "all", "vsetRangePoints": 5, "checkingAccuracy": 0.001}, {"checkingType": "absdiff", "steps": [{"unitTests": [], "showCorrectAnswer": true, "adaptiveMarkingPenalty": 0, "extendBaseMarkingAlgorithm": true, "prompt": "<p>This question differs from part f due to the brackets. Using principles of BODMAS, the brackets need to be expanded first.&nbsp;</p>\n<p>$(4d)^\\var{g}$ expands to $4^\\var{g}d^\\var{g}$ and&nbsp;$(2d)^\\var{h}$ expands to $2^\\var{h}d^\\var{h}$</p>\n<p>Now you are left with a simple division question as follows:</p>\n<p>$\\frac{4^{\\var{g}}d^{\\var{g}}}{2^{\\var{h}}d^{\\var{h}}}$</p>\n<p></p>\n<p>Use the principle:</p>\n<p>$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$ to answer the question.</p>\n<p></p>", "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "customName": "", "useCustomName": false, "showFeedbackIcon": true, "marks": 0, "type": "information"}], "valuegenerators": [{"name": "d", "value": ""}], "notallowed": {"strings": ["/"], "showStrings": false, "partialCredit": 0, "message": ""}, "stepsPenalty": 0, "scripts": {}, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "showPreview": true, "marks": 1, "variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "answer": "(4^{g})/(2^{h})*d^{g-h}", "adaptiveMarkingPenalty": 0, "prompt": "<p><fraction><numerator>$(4d)^\\var{g}$</numerator><denominator>$(2d)^\\var{h}$</denominator></fraction></p>\n<p></p>", "type": "jme", "customMarkingAlgorithm": "", "customName": "", "failureRate": 1, "useCustomName": false, "checkVariableNames": false, "answerSimplification": "all", "vsetRangePoints": 5, "checkingAccuracy": 0.001}], "statement": "<p>Simplify each of the following expressions, giving your answer in its simplest form.</p>\n<p>Click 'Show steps' for guidance on which index law is applicable.</p>", "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j"], "advice": "<p>Recall the laws of indices to help solve the problems:</p>\n<p>$x^a \\times x^b = x^{a+b}$</p>\n<p>$x^a \\div x^b = x^{a-b}$</p>\n<p>$x^{-a} = \\frac{1}{x^a}$</p>\n<p>$(x^a)^b = x^{ab}$</p>\n<p>$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$</p>\n<p>$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$</p>\n<p>$x^0 = 1$</p>\n<p></p>\n<p><strong>Worked Solutions:</strong></p>\n<p>Part a) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $x^{(\\var{a}+\\var{b})}=\\simplify{x^{({a}+{b})}}$</p>\n<p>Part b) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $p^{(\\var{c}+\\var{d})}=\\simplify{p^{({c}+{d})}}$</p>\n<p>Part c) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\var{a}^\\var{f}\\times{k^{(\\var{b}\\times\\var{f})}}=\\simplify{{a}^{f}*k^{({b}*{f})}}$</p>\n<p>Part d) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $y^{((\\var{a}+\\var{b})/(\\var{a}\\times\\var{b}))}=y^{\\frac{\\simplify{{a}+{b}}}{\\simplify{{a}*{b}}}}$</p>\n<p>Part e) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $c^{(\\var{a}-\\var{b})}=c^\\simplify{({a}-{b})}$</p>\n<p>Part f) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$\\frac{\\var{a}}{\\var{b}}h^{\\var{c}-\\var{d}}=\\frac{\\var{a}}{\\var{b}}{\\simplify{h^{{c}-{d}}}}$</p>\n<p>Part g) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\frac{4^\\var{g}}{2^\\var{h}}\\times{d^{\\var{g}-\\var{h}}}=\\simplify{(4^{g})/(2^{h})*d^{g-h}}$</p>\n<p>Part h) &nbsp; 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The roots are real and distinct</p>\n<p>2. $\\Delta=0$. The roots are real and equal. Their value is $\\displaystyle \\frac{-b}{2a}$</p>\n<p>3. $\\Delta \\lt 0$. There are no real roots. 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However, entering one correct and the other incorrect gives feedback stating that both are incorrect.</p>"}, "advice": "\n\t  <p><strong>Direct Factorisation</strong></p>\n\t  <p>If you can spot a direct factorisation then this is the quickest way to do this question.</p>\n\t  <p>For this example we have the factorisation</p>\n\t  <p>\\[\\simplify{{a*b} * x ^ 2 + ( {-b*c-a * d}) * x + {c * d} = ({a} * x + { -c}) * ({b} * x + { -d})}\\]</p>\n\t  <p>Hence we find the roots:<br/>\\[\\begin{eqnarray} x&amp;=&amp; \\simplify{{n1-n4}/{2*a*b}}\\\\ x&amp;=&amp; \\simplify{{n1+n4}/{2*a*b}} \\end{eqnarray} \\]</p>\n\t  <p><strong>Other Methods.</strong></p>\n\t  <p>There are several methods of finding the roots &ndash; here are the main methods.</p>\n\t  <p><strong>Finding the roots of a quadratic using the standard formula.</strong></p>\n\t  <p>We can use the following formula for finding the roots of a general quadratic equation $ax^2+bx+c=0$</p>\n\t  <p>The two roots are</p>\n\t  <p>\\[ x = \\frac{-b +\\sqrt{b^2-4ac}}{2a}\\mbox{ and } x = \\frac{-b -\\sqrt{b^2-4ac}}{2a}\\]<br/>there are three main types of solutions depending upon the value of the discriminant $\\Delta=b^2-4ac$</p>\n\t  <p>1. $\\Delta \\gt 0$. The roots are real and distinct</p>\n\t  <p>2. $\\Delta=0$. The roots are real and equal. Their common value is $\\displaystyle -\\frac{b}{2a}$</p>\n\t  <p>3. $\\Delta \\lt 0$. There are no real roots. The root are complex and form a complex conjugate pair.</p>\n\t  <p>For this question the discriminant of $\\simplify{{a*b}x^2+{-b*c-a*d}x+{c*d}}$ is $\\Delta = \\simplify[std]{{-n1}^2-4*{a*b*c*d}}=\\var{disc}$</p>\n\t  <p>{rdis}.</p>\n\t  <p>So the {rep} roots are:</p>\n\t  <p>\\[\\begin{eqnarray} x = \\frac{\\var{n1} - \\sqrt{\\var{disc}}}{\\var{n3}} &amp;=&amp; \\frac{\\var{n1} - \\var{n4} }{\\var{n3}} &amp;=&amp; \\simplify{{n1 - n4}/ {n3}}\\\\ x = \\frac{\\var{n1} + \\sqrt{\\var{disc}}}{\\var{n3}} &amp;=&amp; \\frac{\\var{n1} + \\var{n4} }{\\var{n3}} &amp;=&amp; \\simplify{{n1 + n4}/ {n3}} \\end{eqnarray}\\]</p>\n\t  <p><strong>Completing the square.</strong></p>\n\t  <p>First we complete the square for the quadratic expression $\\simplify{{a*b}x^2+{-n1}x+{c*d}}$<br/>\\[\\begin{eqnarray} \\simplify{{a*b}x^2+{-n1}x+{c*d}}&amp;=&amp;\\var{n5}\\left(\\simplify{x^2+({-n1}/{a*b})x+ {c*d}/{a*b}}\\right)\\\\ &amp;=&amp;\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2+ \\simplify{{c*d}/{a*b}-({-n1}/({2*a*b}))^2}\\right)\\\\ &amp;=&amp;\\var{n5}\\left(\\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2 -\\simplify{ {n2^2}/{4*(a*b)^2}}\\right) \\end{eqnarray} \\]<br/>So to solve $\\simplify{{a*b}x^2+{-n1}x+{c*d}}=0$ we have to solve:<br/>\\[\\begin{eqnarray} \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&amp;\\phantom{{}}&amp; -\\simplify{ {n2^2}/{4*(a*b)^2}}=0\\Rightarrow\\\\ \\left(\\simplify{x+({-n1}/{2*a*b})}\\right)^2&amp;\\phantom{{}}&amp;=\\simplify{ {n2^2}/{4*(a*b)^2}=({abs(n2)}/{2*a*b})^2} \\end{eqnarray}\\]<br/>So we get the two {rep} solutions:<br/>\\[\\begin{eqnarray} \\simplify{x+({-n1}/{2*a*b})}&amp;=&amp;\\simplify{-{abs(n2)}/{2*a*b}} \\Rightarrow &amp;x&amp; = \\simplify{({-abs(n2)+n1}/{2*a*b})}\\\\ \\simplify{x+({-n1}/{2*a*b})}&amp;=&amp;\\simplify{({abs(n2)}/{2*a*b})} \\Rightarrow &amp;x&amp; = \\simplify{({n1+abs(n2)}/{2*a*b})} \\end{eqnarray}\\]</p>\n\t  \n\t    \n\t    \n\t  \n\t  \n\t  \n\t  \n\t  \n\t  \n\t    \n\t  \n\t"}]}], "allowPrinting": true, "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": false, "showfrontpage": true, "showresultspage": "review", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "<p>YOU HAVE RUN OUT OF TIME</p>"}, "timedwarning": {"action": "warn", "message": "<p>LAST 5 MINUTES</p>"}}, "feedback": {"showactualmark": false, "showtotalmark": true, "showanswerstate": false, "allowrevealanswer": false, "advicethreshold": 0, "intro": "<p>This is Test for you to practice your knowledge of the material, and also usage of NUMBAS system in exam conditions. It won't be counted towards your final grade.&nbsp;</p>", "reviewshowscore": true, "reviewshowfeedback": false, "reviewshowexpectedanswer": true, "reviewshowadvice": false, "feedbackmessages": [], "enterreviewmodeimmediately": false, "showexpectedanswerswhen": "never", "showpartfeedbackmessageswhen": "never", "showactualmarkwhen": "never", "showtotalmarkwhen": "always", "showanswerstatewhen": "inreview", "showadvicewhen": "never"}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "type": "exam", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/14200/"}], "extensions": [], "custom_part_types": [], "resources": []}