// Numbas version: exam_results_page_options
{"name": "Year 11 Algebra quiz - Testing", "metadata": {"description": "<p>This quiz contains questions on algebraic fractions, logarithmic equations, exponential equations, quadratic&nbsp;equations and simultaneous&nbsp;equations.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 1800, "percentPass": "80", "showQuestionGroupNames": true, "showstudentname": true, "question_groups": [{"name": "Year 11MAB", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", "", "", "", "", "", "", "", ""], "questions": [{"name": "Evaluate f(x)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"maxValue": "{a1}*({a3}^{a2})+{b1}*{a3}-{c1}", "variableReplacements": [], "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "strictPrecision": false, "precisionType": "dp", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showFeedbackIcon": true, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "minValue": "{a1}*({a3}^{a2})+{b1}*{a3}-{c1}", "showPrecisionHint": true, "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "mustBeReduced": false, "marks": 1}], "showFeedbackIcon": true, "prompt": "<p>Evaluate \\(f(\\var{a3})\\)</p>\n<p style=\"padding-left: 30px;\">\\(f(\\var{a3})\\) =&nbsp;[[0]]</p>", "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "a2", "b1", "c1", "a3"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Evaluating a function</p>"}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a1": {"definition": "random(2..12#1)", "description": "", "group": "Ungrouped variables", "name": "a1", "templateType": "randrange"}, "a3": {"definition": "random(1..6#1)", "description": "", "group": "Ungrouped variables", "name": "a3", "templateType": "randrange"}, "c1": {"definition": "random(1..15#1)", "description": "", "group": "Ungrouped variables", "name": "c1", "templateType": "randrange"}, "a2": {"definition": "random(2..5#1)", "description": "", "group": "Ungrouped variables", "name": "a2", "templateType": "randrange"}, "b1": {"definition": "random(3..12#1)", "description": "", "group": "Ungrouped variables", "name": "b1", "templateType": "randrange"}}, "advice": "<p>\\(f(x)=\\var{a1}x^{\\var{a2}}+\\var{b1}x-\\var{c1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\var{a3}\\)</p>\n<p style=\"padding-left: 30px;\">\\(f(\\var{a3})=\\var{a1}*(\\var{a3})^{\\var{a2}}+\\var{b1}*(\\var{a3})-\\var{c1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(f(\\var{a3})=\\simplify{{a1}*{a3}^{{a2}}}+\\simplify{{b1}*{a3}}-\\var{c1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(f(\\var{a3})=\\simplify{{a1}*{a3}^{{a2}}+{b1}*{a3}-{c1}}\\)</p>", "statement": "<p>Given the function:</p>\n<p style=\"padding-left: 30px;\">\\(f(x)=\\var{a1}x^{\\var{a2}}+\\var{b1}x-\\var{c1}\\)</p>", "type": "question"}, {"name": "Jill's copy of Jill's copy of Manipulation of formula 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Jill Singleton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2473/"}], "metadata": {"description": "<p>Manipulation of&nbsp;algebraic fractions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["a", "b", "c", "d"], "variables": {"c": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c"}, "b": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b"}, "a": {"description": "", "definition": "random(6..15#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a"}, "d": {"description": "", "definition": "random(8..16#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d"}}, "variable_groups": [], "variablesTest": {"condition": "{a}*{d}>{c}*{b}", "maxRuns": 100}, "rulesets": {}, "parts": [{"type": "gapfill", "prompt": "<p>Express your answer as a fraction:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(V =\\)&nbsp;[[0]]</p>", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"type": "jme", "answer": "((5*{b}-{d})R+8)/(({a}*{d}-{b}*{c})R+7*{a}-3*{c})", "scripts": {}, "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showFeedbackIcon": true, "checkvariablenames": false, "showCorrectAnswer": true, "variableReplacements": [], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": "2"}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": []}], "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "<p>When one fraction equals another fraction we can clear both fractions by cross-multiplying:</p>\n<p style=\"text-align: center;\">\\((\\var{a}V+1)*(\\var{d}R+7)=(\\var{b}R+3)*(\\var{c}V+5)\\)</p>\n<p style=\"text-align: center;\">\\(\\simplify{{a}*{d}}VR+\\simplify{7*{a}}V+\\var{d}R+7=\\simplify{{b}*{c}}VR+\\simplify{5*{b}}R+\\simplify{3*{c}}V+15\\)</p>\n<p>Gathering all the terms involving \\(V\\) to the left hand side and moving all other terms to the right hand side gives</p>\n<p style=\"text-align: center;\">\\(\\simplify{{a}*{d}-{b}*{c}}VR+\\simplify{7*{a}-3*{c}}V=\\simplify{5*{b}-{d}}R+8\\)</p>\n<p>Factoring \\(V\\) out on the left hand side&nbsp;</p>\n<p style=\"text-align: center;\">\\(V(\\simplify{{a}*{d}-{b}*{c}}R+\\simplify{7*{a}-3*{c}})=\\simplify{5*{b}-{d}}R+8\\)</p>\n<p>Thus</p>\n<p style=\"text-align: center;\">\\(V=\\frac{\\simplify{5*{b}-{d}}R+8}{\\simplify{{a}*{d}-{b}*{c}}R+\\simplify{7*{a}-3*{c}}}\\)</p>", "statement": "<p>Rearrange the following expression to make V the subject:</p>\n<p>&nbsp; &nbsp; &nbsp;\\(\\frac{\\var{a}V+1}{\\var{b}R+3}=\\frac{\\var{c}V+5}{\\var{d}R+7}\\)</p>", "type": "question"}, {"name": "Manipulation of formula 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"description": "<p>Manipulation of an exponential function</p>", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "ungrouped_variables": ["k", "c", "m", "d"], "variables": {"c": {"description": "", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "c"}, "m": {"description": "", "definition": "random(1.6..5#0.2)", "templateType": "randrange", "group": "Ungrouped variables", "name": "m"}, "k": {"description": "", "definition": "random(2..12#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "k"}, "d": {"description": "", "definition": "random(1..14#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "d"}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "parts": [{"type": "gapfill", "prompt": "<p>\\(x =\\)&nbsp;[[0]]</p>", "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"type": "jme", "answer": "(ln((1-y/{k})/{c})-{d})/{m}", "scripts": {}, "showpreview": true, "expectedvariablenames": [], "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showFeedbackIcon": true, "checkvariablenames": false, "showCorrectAnswer": true, "variableReplacements": [], "checkingaccuracy": 0.001, "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": "2"}], "marks": 0, "showFeedbackIcon": true, "variableReplacements": []}], "preamble": {"css": "", "js": ""}, "functions": {}, "advice": "<p style=\"text-align: left; padding-left: 60px;\">\\(y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\\)</p>\n<p style=\"text-align: left;\">Working from the outside in,&nbsp;we divide across by \\(\\var{k}\\) &nbsp;&nbsp;</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\frac{y}{\\var{k}}=1-\\var{c}e^{\\var{m}x+\\var{d}}\\)</p>\n<p style=\"text-align: left;\">We can bring the \\(x\\) variable to the left hand side and move the \\(y\\) variable to the right hand side</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\var{c}e^{\\var{m}x+\\var{d}}=1-\\frac{y}{\\var{k}}\\)</p>\n<p style=\"text-align: left;\">Again working from the outside in we divide across by \\(\\var{c}\\)</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(e^{\\var{m}x+\\var{d}}=\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\)</p>\n<p style=\"text-align: left;\">Taking the natural log of both&nbsp;sides eliminates the \\(e\\) from the left hand side.&nbsp;</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\var{m}x+\\var{d}=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)\\)</p>\n<p style=\"text-align: left;\">Subtract \\(\\var{d}\\) from both sides</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\var{m}x=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}\\)</p>\n<p style=\"text-align: left;\">and finally divide by \\(\\var{m}\\) to get</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(x=\\frac{ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}}{\\var{m}}\\)</p>", "statement": "<p>Rearrange the following expression to make \\(x\\) the subject:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\\)</p>", "type": "question"}, {"name": "Solve a logarithmic equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Calculate the value of \\(x\\) that satisfies the equation when&nbsp; \\(y=\\var{d}\\).</p>\n<p>Input your answer correct to three decimal places.</p>\n<p>\\(x = \\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "((10^(d/a))-c)/b", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "variableReplacements": [], "strictPrecision": false, "marks": "2", "minValue": "((10^(d/a))-c)/b", "precisionType": "dp", "showPrecisionHint": false, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "precision": "3"}]}], "statement": "<p>Given the following logarithmic equation:</p>\n<p style=\"padding-left: 30px;\">\\(y=\\var{a}log(\\var{b}x+\\var{c}))\\)</p>\n<p></p>", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "d"], "functions": {}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve a logarithmic equation</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p style=\"text-align: left; padding-left: 30px;\">\\(\\var{a}log(\\var{b}x+\\var{c})=\\var{d}\\)</p>\n<p>Divide across by \\(\\var{a}\\)</p>\n<p style=\"padding-left: 30px;\">\\(log(\\var{b}x+\\var{c})=\\var{d}/\\var{a}=\\simplify{{d}/{a}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{b}x+\\var{c}=10^{\\simplify{{d}/{a}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{b}x+\\var{c}=\\simplify{10^{{d}/{a}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{b}x=\\simplify{10^{{d}/{a}}}-\\var{c}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{b}x=\\simplify{10^{{d}/{a}}-{c}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\simplify{(10^{{d}/{a}}-{c})/{b}}\\)</p>", "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..6#1)"}, "c": {"name": "c", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..10#1)"}, "a": {"name": "a", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..8#1)"}, "d": {"name": "d", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..6#1)"}}, "type": "question"}, {"name": "Solve an exponential equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "advice": "<p style=\"text-align: center;\">\\(\\var{k}=\\var{a}e^{\\var{m}x+{\\var{c}}}\\)</p>\n<p style=\"text-align: center;\">\\(\\frac{\\var{k}}{\\var{a}}=e^{\\var{m}x+\\var{c}}\\)</p>\n<p style=\"text-align: center;\">\\(ln\\left(\\frac{\\var{k}}{\\var{a}}\\right)=\\var{m}x+\\var{c}\\)</p>\n<p style=\"text-align: center;\">\\(ln\\left(\\frac{\\var{k}}{\\var{a}}\\right)-\\var{c}=\\var{m}x\\)</p>\n<p style=\"text-align: center;\">\\(\\frac{ln\\left(\\frac{\\var{k}}{\\var{a}}\\right)-\\var{c}}{\\var{m}}=x\\)</p>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve an exponential equation</p>"}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a", "m", "c", "k"], "variable_groups": [], "functions": {}, "statement": "<p>Given the equation \\(f(x)=\\var{a}e^{\\var{m}x+\\var{c}}\\)</p>\n<p>Determine the value for \\(x\\) that satisfies the relation \\(f(x)=\\var{k}\\)</p>", "tags": [], "rulesets": {}, "parts": [{"marks": 0, "prompt": "<p>Input your answer correct to three decimal places.</p>\n<p style=\"padding-left: 30px;\">\\(x = \\) [[0]]</p>", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "type": "gapfill", "gaps": [{"mustBeReducedPC": 0, "precisionPartialCredit": 0, "maxValue": "(ln({k}/{a})-{c})/{m}", "precision": "3", "strictPrecision": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerFraction": false, "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "type": "numberentry", "allowFractions": false, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "showPrecisionHint": false, "marks": "2", "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "minValue": "(ln({k}/{a})-{c})/{m}"}]}], "variables": {"a": {"templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": "", "definition": "random(4..20#1)"}, "c": {"templateType": "randrange", "group": "Ungrouped variables", "name": "c", "description": "", "definition": "random(0.1..3#0.2)"}, "m": {"templateType": "randrange", "group": "Ungrouped variables", "name": "m", "description": "", "definition": "random(0.1..1.5#0.1)"}, "k": {"templateType": "randrange", "group": "Ungrouped variables", "name": "k", "description": "", "definition": "random(100..150#1)"}}, "variablesTest": {"maxRuns": 100, "condition": "c>0 or c<0 "}, "type": "question"}, {"name": "Solving quadratic equations 1(a)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "statement": "<p>There are two values that satisfy the quadratic equation:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x^2+\\simplify{{{a1}*{b1}*{c1}}}=\\simplify{{a1}*{b1}+{a1}{c1}}x\\)</p>", "rulesets": {}, "variable_groups": [], "tags": [], "functions": {}, "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Type in the greater of the two values that satisfies the equation.</p>\n<p>Input your answer correct to three decimal places. &nbsp;\\(x = \\)&nbsp;[[0]]</p>\n<p>Type in the lesser of the two values that satisfies the equation.&nbsp;</p>\n<p>Input your answer correct to three decimal places. &nbsp;\\(x = \\)&nbsp;[[1]]</p>", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "scripts": {}, "variableReplacements": [], "maxValue": "{b1}", "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "minValue": "{b1}", "marks": 1, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "showFeedbackIcon": true, "precision": "3"}, {"notationStyles": ["plain", "en", "si-en"], "precisionPartialCredit": 0, "correctAnswerStyle": "plain", "scripts": {}, "variableReplacements": [], "maxValue": "{c1}", "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "minValue": "{c1}", "marks": 1, "strictPrecision": false, "precisionType": "dp", "showPrecisionHint": false, "showFeedbackIcon": true, "precision": "3"}]}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving quadratic equations using a formula,</p>"}, "variablesTest": {"condition": "", "maxRuns": "1"}, "advice": "<p>The formula for solving a quadratic equation of the form &nbsp;\\(ax^2+bx+c=0\\) &nbsp;is given by</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\\)</p>\n<p>In this example &nbsp;\\(a=\\var{a1},\\,\\,\\,b=\\simplify{+-{a1}*({b1}+{c1})}\\) &nbsp;and &nbsp;\\(c=\\simplify{{a1}*{b1}*{c1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{\\var{b}\\pm \\sqrt{(-\\var{b})^2-4*\\var{a1}*\\var{c}}}{2*\\var{a1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{\\var{b}\\pm \\sqrt{\\simplify{{b}^2-4*{a1}*{c}}}}{\\simplify{2*{a1}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\simplify{{b}+({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}+({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\\) &nbsp; &nbsp; &nbsp; &nbsp;or &nbsp; &nbsp; &nbsp; &nbsp;\\(x=\\simplify{{b}-({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}-({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\\)</p>\n<p style=\"padding-left: 60px;\"></p>\n<p></p>", "ungrouped_variables": ["a1", "b1", "c1", "b", "c"], "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "anything", "definition": "{a1}*({b1}+{c1})", "description": "", "group": "Ungrouped variables"}, "a1": {"name": "a1", "templateType": "randrange", "definition": "random(1..6#1)", "description": "", "group": "Ungrouped variables"}, "c": {"name": "c", "templateType": "anything", "definition": "{a1}*{b1}*{c1}", "description": "", "group": "Ungrouped variables"}, "b1": {"name": "b1", "templateType": "randrange", "definition": "random(11..25#1)", "description": "", "group": "Ungrouped variables"}, "c1": {"name": "c1", "templateType": "randrange", "definition": "random(1..10#1)", "description": "", "group": "Ungrouped variables"}}, "type": "question"}, {"name": "Solving quadratic equations 1(b)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Type in the greater of the two values that satisfies the equation. Input your answer correct to three decimal places.</p>\n<p style=\"text-align: left; padding-left: 30px;\">\\(x\\) =&nbsp;[[0]]</p>\n<p>Type in the lesser of the two values that satisfies the equation.&nbsp;Input your answer correct to three decimal places.</p>\n<p style=\"padding-left: 30px;\">\\(x\\) =&nbsp;[[1]]</p>", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "-{b1}/(2*{a1})+sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "mustBeReduced": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "strictPrecision": false, "marks": 1, "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": false, "minValue": "-{b1}/(2*{a1})+sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "notationStyles": ["plain", "en", "si-en"], "precision": "3"}, {"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "-{b1}/(2*{a1})-sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "mustBeReduced": false, "showFeedbackIcon": true, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "allowFractions": false, "strictPrecision": false, "marks": 1, "variableReplacements": [], "precisionType": "dp", "showPrecisionHint": false, "minValue": "-{b1}/(2*{a1})-sqrt({b1}^2-4*{a1}*{c1})/(2*{a1})", "notationStyles": ["plain", "en", "si-en"], "precision": "3"}]}], "statement": "<p>There are two values that satisfy the quadratic function below when&nbsp; \\(y=\\var{c1}\\):</p>\n<p style=\"padding-left: 30px;\">\\(y=\\var{a1}x^2+\\var{b1}x\\)</p>", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a1", "b1", "c1"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving quadratic equations using a formula,</p>"}, "variablesTest": {"condition": "b1^2>4*a1*c1", "maxRuns": "1"}, "advice": "<p>The formula for solving a quadratic equation of the form &nbsp;\\(ax^2+bx+c=0\\) &nbsp;is given by</p>\n<p>\\(x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\\)</p>\n<p>In this example &nbsp;\\(a=\\var{a1},\\,\\,\\,b=\\var{b1}\\) &nbsp;and &nbsp;\\(c=\\var{c1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(x=\\frac{-\\var{b1}\\pm \\sqrt{\\var{b1}^2-4\\times\\var{a1}\\times\\var{c1}}}{2\\times\\var{a1}}\\)</p>\n<p style=\"padding-left: 60px;\">\\(x=\\frac{-\\var{b1}\\pm \\sqrt{\\simplify{{b1}^2-4*{a1}*{c1}}}}{\\simplify{2*{a1}}}\\)</p>\n<p style=\"padding-left: 60px;\">\\(x=\\simplify{(-{b1}+ ({b1}^2-4*{a1}*{c1})^0.5)/(2*{a1})}\\) &nbsp; or &nbsp; \\(x=\\simplify{(-{b1}- ({b1}^2-4*{a1}*{c1})^0.5)/(2*{a1})}\\)</p>", "preamble": {"css": "", "js": ""}, "variables": {"a1": {"name": "a1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..6#1)"}, "b1": {"name": "b1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(16..25#1)"}, "c1": {"name": "c1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..10#1)"}}, "tags": [], "type": "question"}, {"name": "Solving quadratic equations 1(c)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "rulesets": {}, "functions": {}, "type": "question", "preamble": {"css": "", "js": ""}, "statement": "<p>The following equation can be converted into a quadratic equation:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+\\frac{\\simplify{{a1}*{b1}*{c1}}}{x}=\\simplify{{a1}*({b1}+{c1})}\\)</p>", "showQuestionGroupNames": false, "tags": [], "variable_groups": [], "variablesTest": {"maxRuns": "1", "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving quadratic equations using a formula</p>"}, "advice": "<p>\\(\\var{a1}x+\\frac{\\simplify{{a1}*{b1}*{c1}}}{x}=\\simplify{{a1}*({b1}+{c1})}\\)</p>\n<p>We clear the fraction in the equation by multiplying across by \\(x\\)</p>\n<p>\\(\\var{a1}x^2+\\simplify{{a1}*{b1}*{c1}}=\\simplify{{a1}*({b1}+{c1})}x\\)</p>\n<p>Bringing all the terms to the left hand side and putting them in order of their powers of \\(x\\) gives</p>\n<p>\\(\\var{a1}x^2-\\simplify{{a1}*({b1}+{c1})}x+\\simplify{{a1}*{b1}*{c1}}=0\\)</p>\n<p>The formula for solving a quadratic equation of the form &nbsp;\\(ax^2+bx+c=0\\) &nbsp;is given by</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\\)</p>\n<p>In this example &nbsp;\\(a=\\var{a1},\\,\\,\\,b=\\simplify{+-{a1}*({b1}+{c1})}\\) &nbsp;and &nbsp;\\(c=\\simplify{{a1}*{b1}*{c1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{\\var{b}\\pm \\sqrt{(-\\var{b})^2-4*\\var{a1}*\\var{c}}}{2*\\var{a1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{\\var{b}\\pm \\sqrt{\\simplify{{b}^2-4*{a1}*{c}}}}{\\simplify{2*{a1}}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\simplify{{b}+({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}+({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\\) &nbsp; &nbsp; &nbsp; &nbsp;or &nbsp; &nbsp; &nbsp; &nbsp;\\(x=\\simplify{{b}-({b}^2-4*{a1}*{c})^0.5}/\\simplify{2*{a1}}=\\simplify{({b}-({b}^2-4*{a1}*{c})^0.5)/(2*{a1})}\\)</p>\n<p style=\"padding-left: 60px;\"></p>\n<p></p>", "ungrouped_variables": ["a1", "b1", "c1", "b", "c"], "variables": {"a1": {"templateType": "randrange", "definition": "random(1..6#1)", "description": "", "name": "a1", "group": "Ungrouped variables"}, "c1": {"templateType": "randrange", "definition": "random(1..10#1)", "description": "", "name": "c1", "group": "Ungrouped variables"}, "b": {"templateType": "anything", "definition": "{a1}*({b1}+{c1})", "description": "", "name": "b", "group": "Ungrouped variables"}, "b1": {"templateType": "randrange", "definition": "random(11..25#1)", "description": "", "name": "b1", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "definition": "{a1}*{b1}*{c1}", "description": "", "name": "c", "group": "Ungrouped variables"}}, "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "parts": [{"showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "type": "gapfill", "prompt": "<p>Type in the greater of the two values that satisfies the equation.&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[0]]</p>\n<p>Type in the lesser of the two values that satisfies the equation.&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[1]]</p>", "marks": 0, "gaps": [{"variableReplacements": [], "marks": 1, "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "minValue": "{b1}", "strictPrecision": false, "type": "numberentry", "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{b1}", "precisionPartialCredit": 0, "precision": "3", "showPrecisionHint": true, "precisionType": "dp", "variableReplacementStrategy": "originalfirst"}, {"variableReplacements": [], "marks": 1, "scripts": {}, "allowFractions": false, "showCorrectAnswer": true, "minValue": "{c1}", "strictPrecision": false, "type": "numberentry", "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{c1}", "precisionPartialCredit": 0, "precision": "3", "showPrecisionHint": true, "precisionType": "dp", "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst"}]}, {"name": "Solving two linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "parts": [{"scripts": {}, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "<p>Input the value for \\(x\\)&nbsp;as an exact fraction.</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[0]]</p>\n<p>Input the value for \\(y\\)&nbsp;as an exact fraction.</p>\n<p style=\"padding-left: 30px;\">\\(y = \\)&nbsp;[[1]]</p>", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "({d}*{r1}-{b}*{r2})/({a}*{d}-{b}*{c})", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "allowFractions": true, "minValue": "({d}*{r1}-{b}*{r2})/({a}*{d}-{b}*{c})", "marks": 1, "variableReplacements": [], "showFeedbackIcon": true}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "scripts": {}, "maxValue": "(-{c}*{r1}+{a}*{r2})/({a}*{d}-{b}*{c})", "mustBeReduced": false, "showCorrectAnswer": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "allowFractions": true, "minValue": "(-{c}*{r1}+{a}*{r2})/({a}*{d}-{b}*{c})", "marks": 1, "variableReplacements": [], "showFeedbackIcon": true}]}], "statement": "<p>Solve the following system of simultaneous equations:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a}x+\\var{b}y=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c}x+\\var{d}y=\\var{r2}\\)</p>", "rulesets": {}, "variable_groups": [], "ungrouped_variables": ["a", "b", "c", "d", "r1", "r2"], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solving two simultaneous linear equations</p>"}, "variablesTest": {"condition": "{a}*{d}>{b}*{c}\n", "maxRuns": 100}, "advice": "<p>equation (i) &nbsp; &nbsp;\\(\\var{a}x+\\var{b}y=\\var{r1}\\)</p>\n<p>equation (ii) &nbsp; &nbsp;\\(\\var{c}x+\\var{d}y=\\var{r2}\\)</p>\n<p>If we decide to eliminate the \\(x\\) variables we need to have the same number of \\(x\\) in both equations</p>\n<p>\\(\\var{c}\\)*equation (i) &nbsp; &nbsp; &nbsp;\\(\\simplify{{c}*{a}}x+\\simplify{{c}*{b}}y=\\simplify{{c}*{r1}}\\)</p>\n<p>\\(\\var{a}\\)*equation (ii) &nbsp; &nbsp;&nbsp;\\(\\simplify{{c}*{a}}x+\\simplify{{d}*{a}}y=\\simplify{{a}*{r2}}\\)</p>\n<p>Subtracting gives:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(\\simplify{{c}*{b}-{d}*{a}}y=\\simplify{{c}*{r1}-{a}*{r2}}\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(y=\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})}\\)</p>\n<p>Substituting this solution for \\(y\\) into equation (i) gives</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;\\(\\var{a}x+\\var{b}*(\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})})=\\var{r1}\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(\\var{a}x=\\var{r1}-\\var{b}*(\\simplify{({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})})\\)</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(\\var{a}x=\\simplify{{r1}-{b}*({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a})}\\)</p>\n<p></p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;\\(x=\\simplify{({r1}-{b}*({c}*{r1}-{a}*{r2})/({c}*{b}-{d}*{a}))/{a}}\\)</p>\n<p></p>", "preamble": {"css": "", "js": ""}, "variables": {"b": {"name": "b", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..12#1)"}, "d": {"name": "d", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..12#1)"}, "r1": {"name": "r1", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(10..40#1)"}, "c": {"name": "c", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(2..11#1)"}, "a": {"name": "a", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(1..10#1)"}, "r2": {"name": "r2", "templateType": "randrange", "group": "Ungrouped variables", "description": "", "definition": "random(16..50#1)"}}, "tags": [], "type": "question"}, {"name": "Solve a system of three simultaneous linear equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "tags": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a1", "b1", "c1", "r1", "r2", "r3"], "variables": {"b1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..10#1)", "name": "b1"}, "r1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(20..42#1)", "name": "r1"}, "a1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(2..8#1)", "name": "a1"}, "c1": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(3..12#1)", "name": "c1"}, "r3": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(30..60#1)", "name": "r3"}, "r2": {"group": "Ungrouped variables", "description": "", "templateType": "randrange", "definition": "random(18..50#1)", "name": "r2"}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve a system of three simultaneous linear equations</p>"}, "parts": [{"type": "gapfill", "prompt": "<p>Input the value of \\(x\\) that satisfies the three equations.</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[0]]</p>\n<p>Input the value of \\(y\\) that satisfies the three equations.</p>\n<p style=\"padding-left: 30px;\">\\(y = \\)&nbsp;[[1]]</p>\n<p>Input the value of \\(z\\) that satisfies the three equations.</p>\n<p style=\"padding-left: 30px;\">\\(z = \\)&nbsp;[[2]]</p>", "scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "gaps": [{"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "minValue": "((18-{b1}*{c1})*(3*{r1}-{r3})+({c1}-12)*(6*{r2}-{b1}*{r3}))/((3*{a1}-5)*(18-{b1}*{c1})-(12-5*{b1})*(12-{c1}))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": "3", "showCorrectAnswer": true, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "((18-{b1}*{c1})*(3*{r1}-{r3})+({c1}-12)*(6*{r2}-{b1}*{r3}))/((3*{a1}-5)*(18-{b1}*{c1})-(12-5*{b1})*(12-{c1}))", "mustBeReduced": false, "showPrecisionHint": true, "marks": 1, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "minValue": "((15-2*{c1})*(2*{r1}-{a1}*{r2})+(3*{a1}-8)*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": "3", "showCorrectAnswer": true, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "((15-2*{c1})*(2*{r1}-{a1}*{r2})+(3*{a1}-8)*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "mustBeReduced": false, "showPrecisionHint": true, "marks": 1, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "minValue": "((12-5*{b1})*(2*{r1}-{a1}*{r2})+(4-{b1}*{a1})*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "precision": "3", "showCorrectAnswer": true, "mustBeReducedPC": 0, "allowFractions": false, "strictPrecision": false, "showFeedbackIcon": true, "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "((12-5*{b1})*(2*{r1}-{a1}*{r2})+(4-{b1}*{a1})*(5*{r2}-2*{r3}))/((4-{a1}*{b1})*(15-2*{c1})-(5*{b1}-12)*(8-3*{a1}))", "mustBeReduced": false, "showPrecisionHint": true, "marks": 1, "precisionType": "dp", "correctAnswerStyle": "plain"}], "marks": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "functions": {}, "advice": "<p style=\"padding-left: 30px; text-align: left;\">(i) &nbsp; &nbsp;\\(\\var{a1}x+2y+4z=\\var{r1}\\)</p>\n<p style=\"padding-left: 30px;\">(ii) &nbsp; \\(2x+\\var{b1}y+3z=\\var{r2}\\)</p>\n<p style=\"padding-left: 30px;\">(iii) &nbsp;\\(5x+6y+\\var{c1}z=\\var{r3}\\)</p>\n<p>First reduce the three equations in three unknowns to a two equations in two unknowns problem by eliminating one of the variables.</p>\n<p>We can eliminate \\(x\\) using equations (i) and (ii)</p>\n<p style=\"padding-left: 30px;\">2*(i) &nbsp; &nbsp; \\(\\simplify{2*{a1}}x+4y+8z=\\simplify{2*{r1}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}\\)*(ii) &nbsp; &nbsp;\\(\\simplify{2*{a1}}x+\\simplify{{a1}*{b1}}y+\\simplify{3*{a1}}z=\\simplify{{a1}*{r2}}\\)</p>\n<p>Subtracting gives us a new equation</p>\n<p style=\"padding-left: 30px;\">(iv) &nbsp; &nbsp;\\(\\simplify{(4-{a1}{b1})y+(8-3*{a1})z}=\\simplify{2*{r1}-{a1}*{r2}}\\)</p>\n<p>We can also eliminate \\(x\\) using equations (ii) and (iii)</p>\n<p style=\"padding-left: 30px;\">5*(ii) &nbsp; &nbsp;\\(10x +\\simplify{5*{b1}}y+15z=\\simplify{5*{r2}}\\)</p>\n<p style=\"padding-left: 30px;\">2*(iii) &nbsp; \\(10x+12y+\\simplify{2*{c1}}z=\\simplify{2*{r3}}\\)</p>\n<p>Subtracting gives us another new equation</p>\n<p style=\"padding-left: 30px;\">(v) &nbsp; &nbsp; \\(\\simplify{(5*{b1}-12)y+(15-2*{c1})z}=\\simplify{5*{r2}-2*{r3}}\\)</p>\n<p>We could then eliminate the \\(y\\) from these two new equations</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{5*{b1}-12}\\)*(iv) &nbsp; &nbsp;\\(\\simplify{(5*{b1}-12)*(4-{a1}{b1})y+(5*{b1}-12)*(8-3*{a1})z}=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{4-{a1}{b1}}\\)*(v) &nbsp; &nbsp;\\(\\simplify{(4-{a1}{b1})*(5*{b1}-12)y+(4-{a1}{b1})*(15-2*{c1})z}=\\simplify{(4-{a1}{b1})*(5*{r2}-2*{r3})}\\)</p>\n<p>Subtracting gives us</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}z=\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}\\)</p>\n<p>Thus</p>\n<p style=\"padding-left: 30px;\">\\(z=\\frac{\\simplify{(5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}{b1})*(5*{r2}-2*{r3})}}{\\simplify{(5*{b1}-12)*(8-3*{a1})-(4-{a1}{b1})*(15-2*{c1})}}=\\simplify{decimal{((5*{b1}-12)*(2*{r1}-{a1}*{r2})-(4-{a1}*{b1})*(5*{r2}-2*{r3}))/( &nbsp;(5*{b1}-12)*(8-3*{a1})-(4-{a1}*{b1})*(15-2*{c1}))}}\\)</p>\n<p>We can now back substitute this value for \\(z\\) into equation (iv) to find the correct value for \\(y\\) and then&nbsp;back substitute both these values into equation (i) to calculate \\(x\\).&nbsp;</p>\n<p style=\"padding-left: 30px;\"></p>", "statement": "<p>Solve the following system of&nbsp;three simultaneous linear equations:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+2y+4z=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(2x+\\var{b1}y+3z=\\var{r2}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(5x+6y+\\var{c1}z=\\var{r3}\\)</p>", "type": "question"}, {"name": "Solving a Linear and a Non-linear system of equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}], "metadata": {"description": "<p>Solving a Linear and a Non-linear system of equations</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "statement": "<p>Given two equations:</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+\\var{b1}y=\\var{r1}\\)</p>\n<p style=\"padding-left: 60px;\">and</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}y^2=\\var{r2}\\)</p>\n<p>There are two solutions for \\(x\\) that satisfy both of these equations and&nbsp;for each \\(x\\) value there exists a corresponding \\(y\\) value that forms&nbsp;a solution pair.</p>", "preamble": {"js": "", "css": ""}, "functions": {}, "variable_groups": [], "variables": {"a1": {"name": "a1", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "r1": {"name": "r1", "definition": "random(10..20#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "c1": {"name": "c1", "definition": "random(1..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "r2": {"name": "r2", "definition": "random(20..50#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "b1": {"name": "b1", "definition": "random(2..10#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}, "d1": {"name": "d1", "definition": "random(2..5#1)", "templateType": "randrange", "group": "Ungrouped variables", "description": ""}}, "variablesTest": {"condition": "(2*{a1}*{d1}*{r1})^2>4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})", "maxRuns": 100}, "advice": "<p>To solve a system that involves&nbsp;a linear equation and a non-linear equation we must use the substitution method.</p>\n<p style=\"padding-left: 30px;\">\\(\\var{a1}x+\\var{b1}y=\\var{r1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}y^2=\\var{r2}\\)</p>\n<p>The first equation is a linear equaton, we use this to write one variable in terms of the other.&nbsp;</p>\n<p>For example, we could make y the subject of this equation&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(y=\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\)</p>\n<p>We can then insert this for every \\(y\\) in the non-linear equation to get</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}*\\left(\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\right)^2=\\var{r2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}x^2+\\var{d1}*\\frac{(\\var{r1}-\\var{a1}x)^2}{\\var{b1}^2}=\\var{r2}\\)</p>\n<p>Multiplying across by \\(\\simplify{{b1}^2}\\) gives</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}*{b1}^2}x^2+\\var{d1}(\\var{r1}-\\var{a1}x)^2-\\simplify{{r2}*{b1}^2}=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}*{b1}^2}x^2+\\var{d1}\\left(\\simplify{{r1}^2}-\\simplify{2*{r1}*{a1}}x+\\simplify{{a1}^2}x^2\\right)-\\simplify{{r2}*{b1}^2}=0\\)</p>\n<p>Gathering the like terms together gives</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{({c1}*{b1}^2+{d1}*{a1}^2)x^2-(2*{a1}*{d1}*{r1})x+({d1}*{r1}^2-{r2}*{b1}^2)}=0\\)</p>\n<p>This is a quadratic equation.</p>\n<p>Recall the formula for solving a quadratic equation of the form &nbsp;\\(ax^2+bx+c=0\\) &nbsp;is given by</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{-b\\pm \\sqrt{b^2-4ac}}{2a}\\)</p>\n<p style=\"padding-left: 30px;\">In this example \\(a =&nbsp;\\simplify{({c1}*{b1}^2+{d1}*{a1}^2)}, b = \\simplify{-(2*{a1}*{d1}*{r1})}\\) and \\(c = \\simplify{{d1}*{r1}^2-{r2}*{b1}^2}\\)</p>\n<p>Once we have each \\(x\\) value we insert it into &nbsp; \\(y=\\frac{\\var{r1}-\\var{a1}x}{\\var{b1}}\\) &nbsp; to find the corresponding \\(y\\) value.</p>", "parts": [{"showCorrectAnswer": true, "prompt": "<p>Input the larger of the two \\(x\\) values. &nbsp; &nbsp; &nbsp;\\(x = \\)&nbsp;[[0]]</p>\n<p>Input the \\(y\\) value that&nbsp;corresponds to the previous answer. &nbsp; &nbsp; \\(y = \\)&nbsp;[[1]]</p>\n<p></p>\n<p>Input the lesser of the two \\(x\\) values that satisfies both equations. &nbsp; &nbsp; \\(x = \\)&nbsp;[[2]]</p>\n<p>Input the \\(y\\) value that&nbsp;corresponds to the previous answer. &nbsp; &nbsp; \\(y = \\)&nbsp;[[3]]</p>", "variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "marks": 0, "gaps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{a1}*{d1}*{r1}+sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{a1}*{d1}*{r1}+sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{r1}*{b1}*{c1}-sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{r1}*{b1}*{c1}-sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{a1}*{d1}*{r1}-sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{a1}*{d1}*{r1}-sqrt{(2*{a1}*{d1}*{r1})^2-4*({b1}^2*{c1}+{d1}*{a1}^2)*({d1}*{r1}^2-{b1}^2*{r2})})/(2*({b1}^2*{c1}+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}, {"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "maxValue": "(2*{r1}*{b1}*{c1}+sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "showPrecisionHint": true, "strictPrecision": false, "type": "numberentry", "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "allowFractions": false, "minValue": "(2*{r1}*{b1}*{c1}+sqrt{(2*{r1}*{b1}*{c1})^2-4*({c1}*{b1}^2+{d1}*{a1}^2)*({c1}*{r1}^2-{r2}*{a1}^2)})/(2*({c1}*{b1}^2+{d1}*{a1}^2))", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "marks": 1, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "precisionType": "dp", "precisionPartialCredit": 0}], "type": "gapfill", "variableReplacements": []}], "tags": [], "ungrouped_variables": ["a1", "b1", "r1", "c1", "d1", "r2"], "type": "question"}]}], "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "warnifunattempted", "message": "<p>You should try every question!</p>"}, "preventleave": true, "startpassword": "3.1415"}, "timing": {"allowPause": true, "timeout": {"action": "warn", "message": "<p>Times up!</p>"}, "timedwarning": {"action": "warn", "message": "<p>You are going to time out soon...</p>"}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "<p>This is a trial quiz to see how this program works.</p>", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": [{"message": "<p>Great work!</p>", "threshold": "90"}, {"message": "<p>More practise needed!</p>", "threshold": "40"}]}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Jill Singleton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2473/"}], "extensions": [], "custom_part_types": [], "resources": []}