// Numbas version: exam_results_page_options
{"question_groups": [{"name": "", "questions": [{"name": "Express given fractions with different denominators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"q": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5,7,11,13,17)", "description": "", "name": "q"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..29 except a)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2*p2", "description": "", "name": "c"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11 except t)", "description": "", "name": "t2"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..7 except s)", "description": "", "name": "s2"}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "p*q", "description": "", "name": "k"}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5,7,11,13,17)", "description": "", "name": "p2"}, "l": {"templateType": "anything", "group": "Ungrouped variables", "definition": "2*p2*q2", "description": "", "name": "l"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b*q2", "description": "", "name": "n"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "name": "s"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5,7,11,13,17)", "description": "", "name": "q2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..29)", "description": "", "name": "a"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5,7,11,13,17)", "description": "", "name": "p"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11)", "description": "", "name": "t"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*q", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "p2", "c", "b", "s2", "k", "m", "l", "n", "q", "p", "s", "q2", "t", "t2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "$\\dfrac{\\var{a}}{\\var{p}}$
\n If expressed with denominator $\\var{k}$, the numerator is [[0]].
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{m}", "minValue": "{m}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$\\dfrac{\\var{s*t}}{\\var{t*k}}$
\n If expressed with denominator $\\var{k}$, the numerator is [[0]].
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{s}", "minValue": "{s}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$\\dfrac{\\var{b}}{\\var{2*p2}}$
\n If expressed with denominator $\\var{l}$, the numerator is [[0]].
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{n}", "minValue": "{n}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$\\dfrac{\\var{t2*s2}}{\\var{t2*l}}$
\n If expressed with denominator $\\var{l}$, the numerator is [[0]].
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{s2}", "minValue": "{s2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Express each the following fractions as a fraction with the given denominator. Write the numerator in the given box.
", "tags": ["checked2015", "Fractions", "fractions", "SFY0001"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of numerators and denominators of numerical fractions.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) Observe that $\\var{k} \\div \\var{p} =\\var{q}$, so we multiply top and bottom by $\\var{q}$ to get $\\dfrac{\\var{a*q}}{\\var{p*q}}$.
\nThe numerator is then the number on the top, i.e., $\\var{a*q}$.
\nb) Observe that $\\var{t*k} =\\var{t} \\times \\var{k}$, so we divide top and bottom by $\\var{t}$ to get $\\dfrac{\\var{s}}{\\var{k}}$.
\nThe numerator is then the number on the top, i.e., $\\var{s}$.
\nc) Observe that $\\var{l} \\div \\var{2*p2} =\\var{q2}$, so we multiply top and bottom by $\\var{q2}$ to get $\\dfrac{\\var{b*q2}}{\\var{2*p2*q2}}$.
\nThe numerator is then the number on the top, i.e., $\\var{b*q2}$.
\nd) Observe that $\\var{t2*l} =\\var{t2} \\times \\var{l}$, so we divide top and bottom by $\\var{t2}$ to get $\\dfrac{\\var{s2}}{\\var{l}}$.
\nThe numerator is then the number on the top, i.e., $\\var{s2}$.
\n\n"}, {"name": "Fractions: Lowest form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(c[x]*a[x],x,0..3)", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(-9..-5),random(5..9),random(11..19),random(35..61)]", "name": "a", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(chcp(a[x],2,9,random(2..9)),x,0..2)+chcp(a[3],40,80,random(40..80))", "name": "b", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(2..9),random(2..9),random(2..9),random(40..70)]", "name": "c", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(c[x]*b[x],x,0..3)", "name": "f", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "f"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"chcp": {"type": "number", "language": "jme", "definition": "if(gcd(a,d)=1,d,chcp(a,b,c,random(b..c)))", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["d", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "{a[0]}/{b[0]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form without brackets.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 4, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"answer": "{a[1]}/{b[1]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form without brackets.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 3, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"answer": "{a[2]}/{b[2]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form. Do not include brackets in your answer.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 4, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"answer": "{a[3]}/{b[3]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form. Do not include brackets in your answer.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 5, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1.5, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "Given a fraction $\\displaystyle \\frac{a}{b}$ then it is in lowest form if $a$ and $b$ have no common factors.
\n If $c$ was a common factor then we could cancel the $c$ and we have converted the fraction into a fraction with smaller numbers.
\n For example the fraction $\\displaystyle \\frac{18}{24}=\\frac{9 \\times 2}{12 \\times 2} = \\frac{9}{12}$ as we can cancel the common factor $2$.
\n But we are not yet finished as $\\displaystyle \\frac{9}{12}=\\frac{3 \\times 3}{4 \\times 3} = \\frac{3}{4}$ on cancelling the common factor $3$. We cannot go any further as $3$ and $4$ have no common factors (other than $1$, which is never considered as a factor).
\n Of course we could have spotted that $6$ was a common factor as $\\displaystyle \\frac{18}{24}=\\frac{3 \\times 6}{4 \\times 6}=\\frac{3}{4}$ , but it is perfectly OK to do it in stages as we did above. Just make sure that your final fraction does not have common factors.
\n
", "marks": 0, "scripts": {}}], "prompt": "$\\displaystyle \\simplify[noc]{{d[0]}/{f[0]}}\\;=$[[0]],$\\;\\;\\displaystyle \\simplify[noc]{{d[1]}/{f[1]}}\\;=$[[1]],$\\;\\;\\displaystyle \\simplify[noc]{{d[2]}/{f[2]}}\\;=$[[2]],$\\;\\;\\displaystyle \\simplify[noc]{{d[3]}/{f[3]}}\\;=$[[3]]
\n Input as fractions and do not include brackets in your answer.
\n You can click on Show steps for help. You will not lose any marks if you do.
", "stepsPenalty": 0}], "statement": "Reduce the following fractions to their lowest form.
", "tags": ["Fractions", "SFY0001", "cancellation", "cancelling", "cancelling ", "checked2015", "common factor", "denominator", "lowest form", "numerator"], "rulesets": {"noc": ["std", "!simplifyFractions"], "std": ["all", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "11/08/2012:
\n Added tags.
\n Added description.
\n Function chcp(a,b,c,d) gives number coprime to a in the range b..c, d is usually random(b..c) for redundant reasons!
\n Note that the answer is constrained by max length as well as requiring / and no brackets.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Reducing fractions to their lowest form by cancelling common factors in the numerator and denominator. There are 4 questions.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "We have:
\n $\\displaystyle \\simplify[noc]{{d[0]}/{f[0]}}=\\simplify[]{({a[0]}*{c[0]})/({b[0]}*{c[0]})}=\\simplify[all]{{a[0]}/{b[0]}}$. Common factor $\\var{c[0]}$.
\n $\\displaystyle \\simplify[noc]{{d[1]}/{f[1]}}=\\simplify[]{({a[1]}*{c[1]})/({b[1]}*{c[1]})}=\\simplify[all]{{a[1]}/{b[1]}}$. Common factor $\\var{c[1]}$.
\n $\\displaystyle \\simplify[noc]{{d[2]}/{f[2]}}=\\simplify[]{({a[2]}*{c[2]})/({b[2]}*{c[2]})}=\\simplify[all]{{a[2]}/{b[2]}}$. Common factor $\\var{c[2]}$.
\n $\\displaystyle \\simplify[noc]{{d[3]}/{f[3]}}=\\simplify[]{({a[3]}*{c[3]})/({b[3]}*{c[3]})}=\\simplify[all]{{a[3]}/{b[3]}}$. Common factor $\\var{c[3]}$.
"}, {"name": "Precedence of operators - everything except brackets", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"e2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except [d2,b2])", "name": "e2", "description": ""}, "b3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..-2)", "name": "b3", "description": ""}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except [d1,b1])", "name": "e1", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4 except c1)", "name": "f1", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except a1)", "name": "b1", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except a2)", "name": "b2", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except a1)", "name": "d1", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4 except b2)", "name": "c2", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except a2)", "name": "d2", "description": ""}, "c3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "c3", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4 except b1)", "name": "c1", "description": ""}, "a3": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except[-1,0,1])", "name": "a3", "description": ""}, "f2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4 except c2)", "name": "f2", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a2", "description": ""}}, "ungrouped_variables": ["f1", "f2", "e1", "a1", "a3", "a2", "b1", "b2", "b3", "c3", "c2", "c1", "d1", "d2", "e2"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a1*b1^c1+d1*e1^f1}", "prompt": "$\\var{a1} \\times \\var{b1}^\\var{c1} + \\var{d1} \\times \\var{e1} ^ \\var{f1}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a1*b1^c1+d1*e1^f1}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a2*b2^c2-d2*e2^f2}", "prompt": "$\\var{a2} \\times \\var{b2}^\\var{c2} - \\var{d2} \\times \\var{e2} ^ \\var{f2}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a2*b2^c2-d2*e2^f2}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a3*b3^c3}", "prompt": "$\\var{a3} \\times (\\var{b3})^\\var{c3}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a3*b3^c3}"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Evaluate the following expressions:
", "tags": ["BEDMAS", "BIDMAS", "BODMAS", "checked2015", "Precedence of Operators", "SFY0001"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the precedence of operators using BIDMAS applied to integers. These questions only test IDMAS. That is Indices, Division/Multiplication and Addition/Subtraction.
"}, "advice": "First evaluate any powers. Then evaluate the expression remaining using DMAS. Pay particular attention to minus signs in part (c). Thus:
\na)
\n$\\var{a1} \\times \\var{b1}^\\var{c1} + \\var{d1} \\times \\var{e1} ^ \\var{f1}=\\var{a1} \\times \\var{b1^c1}+\\var{d1} \\times \\var{e1^f1}=\\var{a1*b1^c1} +\\var{d1*e1^f1}=\\var{a1*b1^c1+d1*e1^f1}$
\nb)
\n$\\var{a2} \\times \\var{b2}^\\var{c2} - \\var{d2} \\times \\var{e2} ^ \\var{f2}=\\var{a2} \\times \\var{b2^c2}-\\var{d2} \\times \\var{e2^f2}=\\var{a2*b2^c2} -\\var{d2*e2^f2}=\\var{a2*b2^c2-d2*e2^f2}$
\nc)
\n$\\var{a3} \\times (\\var{b3})^\\var{c3}=\\var{a3} \\times (\\var{b3^c3})=\\var{a3*b3^c3}$
"}, {"name": "Precedence of operators - everything except indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "g1", "description": ""}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6)", "name": "e1", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..7)", "name": "d", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8 except [a,b])", "name": "c", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..11 except[a,b,c,d,e])", "name": "f", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except a1)", "name": "b1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..11 except [a1,b1,c1])", "name": "d1", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a1", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except a2)", "name": "b2", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except [a2,b2])", "name": "c2", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..11 except [a2,b2,c2])", "name": "d2", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except [a1,b1])", "name": "c1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a", "description": ""}, "h": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(7..15)", "name": "h", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..11 except a)", "name": "b", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "g", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "a2", "description": ""}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..7 except e1)", "name": "f1", "description": ""}}, "ungrouped_variables": ["a", "f1", "c", "b", "e1", "d", "g", "f", "h", "a1", "a2", "b1", "b2", "c2", "c1", "g1", "d2", "d1"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a*b+a*c}", "prompt": "$\\var{a} \\times (\\var{b}+\\var{c})$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a*b+a*c}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a*c + b*c}", "prompt": "$(\\var{a}+\\var{b}) \\times \\var{c}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a*c + b*c}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{d}", "prompt": "$\\var{(a-b)*d} \\div (\\var{a}-\\var{b})$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{d}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{(a1+b1)*(f1-e1)}", "prompt": "$(\\var{a1} + \\var{b1}) \\times (\\var{f1}-\\var{e1})$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{(a1+b1)*(f1-e1)}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{d2}", "prompt": "$(\\var{d2*(a2+b2)-c2}+\\var{c2}) \\div (\\var{a2} + \\var{b2})$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{d2}"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Evaluate the following expressions:
", "tags": ["BEDMAS", "BIDMAS", "BODMAS", "checked2015", "Precedence of Operators", "SFY0001"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the precedence of operators using BIDMAS. These questions only test BDMAS. That is, they test Brackets, Division/Multiplication and Addition/Subtraction.
"}, "advice": "First work through the expression from left to right, evaluating any expressions inside brackets. Thus:
\na)
\n$\\var{a} \\times (\\var{b}+\\var{c})=\\var{a} \\times \\var{b+c}=\\var{a*b+a*c}$
\nb)
\n$(\\var{a}+\\var{b}) \\times \\var{c}=\\var{a+b} \\times \\var{c}=\\var{a*c + b*c}$
\nc)
\n$\\var{(a-b)*d} \\div (\\var{a}-\\var{b})=\\var{(a-b)*d} \\div (\\var{a-b})=\\var{d}$
\nd)
\n$(\\var{a1} + \\var{b1}) \\times (\\var{f1}-\\var{e1})=\\var{a1+b1} \\times (\\var{f1-e1})=\\var{(a1+b1)*(f1-e1)}$
\ne)
\n$(\\var{d2*(a2+b2)-c2}+\\var{c2}) \\div (\\var{a2} + \\var{b2})=\\var{d2*(a2+b2)} \\div \\var{a2 + b2}=\\var{d2}$
"}, {"name": "Precedence of operators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"g1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "g1"}, "e1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "e1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..7)", "description": "", "name": "d"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..8 except [a,b])", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..11 except[a,b,c,d,e])", "description": "", "name": "f"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9 except a1)", "description": "", "name": "b1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..11 except [a1,b1,c1])", "description": "", "name": "d1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9 except a2)", "description": "", "name": "b2"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9 except [a2,b2])", "description": "", "name": "c2"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9 except [a1,b1])", "description": "", "name": "c1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..15)", "description": "", "name": "h"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11 except a)", "description": "", "name": "b"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "g"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "name": "a2"}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..7)", "description": "", "name": "f1"}}, "ungrouped_variables": ["a", "f1", "c", "b", "e1", "d", "g", "f", "h", "a1", "a2", "b1", "b2", "c2", "c1", "g1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a+b*c}", "prompt": "$\\var{a}+\\var{b} \\times\\var{c}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a+b*c}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a*b+c}", "prompt": "$\\var{a} \\times \\var{b}+\\var{c}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a*b+c}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{h-a2}", "prompt": "$\\var{h}-\\var{a2*b2} \\div \\var{b2}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{h-a2}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a*b + c-f*g}", "prompt": "$\\var{a} \\times \\var{b}+\\var{c*d} \\div \\var{d} - \\var{f} \\times \\var{g}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a*b + c-f*g}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a1-c1*d1-e1*g1}", "prompt": "$\\var{a1*b1} \\div \\var{b1}-\\var{c1} \\times \\var{d1} - \\var{e1*f1} \\div \\var{f1} \\times \\var{g1}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a1-c1*d1-e1*g1}"}], "statement": "Evaluate the following expressions:
", "tags": ["BEDMAS", "BIDMAS", "BODMAS", "checked2015", "Precedence of Operators", "SFY0001"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the precedence of operators using BIDMAS, applied to integers. These questions only test DMAS. That is, only Division/Multiplcation and Addition/Subtraction.
"}, "advice": "First work through the expression from left to right, evaluating any multiplications and divisions as you come to them. You should be left with an expression involving only pluses and minuses. Evaluate this expression, again working from left to right. Thus:
\na)
\n$\\var{a}+\\var{b} \\times \\var{c}=\\var{a}+\\var{b*c}=\\var{a+b*c}$
\nb)
\n$\\var{a} \\times \\var{b}+\\var{c}=\\var{a*b}+\\var{c}=\\var{a*b+c}$
\nc)
\n$\\var{h}-\\var{a2*b2} \\div \\var{b2}=\\var{h}-\\var{a2}=\\var{h-a2}$
\nd)
\n$\\var{a} \\times \\var{b}+\\var{c*d} \\div \\var{d} - \\var{f} \\times \\var{g}=\\var{a*b}+\\var{c}-\\var{f*g}=\\var{a*b + c-f*g}$
\ne)
\n$\\var{a1*b1} \\div \\var{b1}-\\var{c1} \\times \\var{d1} - \\var{e1*f1} \\div \\var{f1} \\times \\var{g1}=\\var{a1}-\\var{c1*d1}-\\var{e1*g1}=\\var{a1-c1*d1-e1*g1}$
\n"}, {"name": "Precedence of operators", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "r*c^d", "name": "b", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5)", "name": "c", "description": ""}, "i1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except [c,f])", "name": "i1", "description": ""}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "t", "description": ""}, "k": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..7)", "name": "k", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "d2", "description": ""}, "l": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6)", "name": "l", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..11)", "name": "n", "description": ""}, "h": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "h", "description": ""}, "f2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..11 except e2)", "name": "f2", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(b/(c^d)) - e1*f^g +t*h*i1^j-t*k", "name": "a", "description": ""}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "q*s", "name": "p", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..11)", "name": "a2", "description": ""}, "q": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "q", "description": ""}, "e2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..11)", "name": "e2", "description": ""}, "e1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8)", "name": "e1", "description": ""}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "d", "description": ""}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..7)", "name": "m", "description": ""}, "j": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "j", "description": ""}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5)", "name": "c2", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..7)", "name": "b2", "description": ""}, "r": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9)", "name": "r", "description": ""}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(3..11)", "name": "s", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except c)", "name": "f", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..4)", "name": "g", "description": ""}, "g2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..3)", "name": "g2", "description": ""}}, "ungrouped_variables": ["f2", "b2", "d2", "g2", "i1", "a2", "c2", "e1", "e2", "a", "c", "b", "d", "g", "f", "h", "k", "j", "m", "l", "n", "q", "p", "s", "r", "t"], "rulesets": {}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{t*(l-m*n+s)}", "prompt": "$[(\\var{a}-\\var{b} \\div \\var{c}^\\var{d})+\\var{e1} \\times \\var{f}^\\var{g}] \\div (\\var{h} \\times \\var{i1}^\\var{j} -\\var{k}) \\times [\\var{l} -\\var{m} \\times \\var{n} + \\var{p} \\div \\var{q}]$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{t*(l-m*n+s)}"}, {"showCorrectAnswer": true, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "minValue": "{a2+b2*(-c2)^d2 +(e2-f2)^g2}", "prompt": "$\\var{a2}+\\var{b2} \\times (\\var{-c2})^\\var{d2} +(\\var{e2}-\\var{f2})^\\var{g2}$
", "showPrecisionHint": false, "scripts": {}, "type": "numberentry", "correctAnswerFraction": false, "variableReplacements": [], "marks": 1, "maxValue": "{a2+b2*(-c2)^d2 +(e2-f2)^g2}"}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Evaluate the following expressions:
", "tags": ["BEDMAS", "BIDMAS", "BODMAS", "checked2015", "Precedence of Operators", "SFY0001"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the precedence of operators using BIDMAS. That is, they test Brackets, Indices, Division/Multiplication and Addition/Subtraction.
"}, "advice": "First work through the expression from left to right, evaluating any expressions inside brackets, being careful to evaluate powers (indices) before MDAS. Thus:
\na)
\n$[(\\var{a}-\\var{b} \\div \\var{c}^\\var{d})+\\var{e1} \\times \\var{f}^\\var{g}] \\div (\\var{h} \\times \\var{i1}^\\var{j} -\\var{k}) \\times [\\var{l} -\\var{m} \\times \\var{n} + \\var{p} \\div \\var{q}]$
\n$=[(\\var{a}-\\var{b} \\div \\var{c^d})+\\var{e1} \\times \\var{f^g}] \\div (\\var{h} \\times \\var{i1^j} -\\var{k}) \\times [\\var{l} -\\var{m} \\times \\var{n} + \\var{p} \\div \\var{q}]$
\n$=[(\\var{a}-\\var{b / (c^d)})+\\var{e1*f^g}] \\div (\\var{h * i1^j} -\\var{k}) \\times [\\var{l} -\\var{m*n} + \\var{p / q}]$
\n$=[\\var{a-b / (c^d)+e1*f^g}] \\div (\\var{h * i1^j -k}) \\times [\\var{l -m*n + p / q}]=\\var{t*(l-m*n+s)}$
\nb)
\n$\\var{a2}+\\var{b2} \\times (\\var{-c2})^\\var{d2} +(\\var{e2}-\\var{f2})^\\var{g2}=\\var{a2}+\\var{b2} \\times (\\var{(-c2)^d2}) +(\\var{e2-f2})^\\var{g2}=\\var{a2}+\\var{b2*((-c2)^d2)} +\\var{(e2-f2)^g2}=\\var{a2+b2*(-c2)^d2 +(e2-f2)^g2}$
"}, {"name": "Reduce fractions to lowest terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "gcd(a,b)", "name": "d", "description": ""}, "j": {"group": "Ungrouped variables", "templateType": "anything", "definition": "f*h", "name": "j", "description": ""}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a/d", "name": "f", "description": ""}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..50 except a2)", "name": "b2", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..50)", "name": "a", "description": ""}, "k": {"group": "Ungrouped variables", "templateType": "anything", "definition": "g*h", "name": "k", "description": ""}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "gcd(a2,b2)", "name": "d2", "description": ""}, "h": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..25)", "name": "h", "description": ""}, "f2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a2/d2", "name": "f2", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..50 except a)", "name": "b", "description": ""}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b/d", "name": "g", "description": ""}, "g2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b2/d2", "name": "g2", "description": ""}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..50)", "name": "a2", "description": ""}, "h2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..25)", "name": "h2", "description": ""}, "k2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "g2*h2", "name": "k2", "description": ""}, "j2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "f2*h2", "name": "j2", "description": ""}}, "ungrouped_variables": ["a", "f2", "b", "d", "g", "f", "h2", "h", "k", "j", "j2", "k2", "a2", "b2", "d2", "g2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{f}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{f}"}, {"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{g}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{g}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "$\\dfrac{\\var{j}}{\\var{k}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{f2}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{f2}"}, {"showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "showCorrectAnswer": true, "minValue": "{g2}", "correctAnswerFraction": false, "marks": 1, "maxValue": "{g2}"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "$\\dfrac{\\var{j2}}{\\var{k2}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Reduce the following fractions to lowest terms. Write the numerator and denominator of the lowest term fraction in the boxes provided.
", "tags": ["checked2015", "Fractions", "Lowest terms", "SFY0001"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.
"}, "advice": "a) We observe that the highest common factor (the largest number dividing both) of $\\var{j}, \\var{k}$ is $\\var{h}$, so we divide top and bottom by $\\var{h}$ to get $\\dfrac{\\var{f}}{\\var{g}}$. Therefore, in lowest terms, the numerator is $\\var{f}$, the denominator is $\\var{g}$.
\n b) We observe that the highest common factor (the largest number dividing both) of $\\var{j2}, \\var{k2}$ is $\\var{h2}$, so we divide top and bottom by $\\var{h2}$ to get $\\dfrac{\\var{f2}}{\\var{g2}}$. Therefore, in lowest terms, the numerator is $\\var{f2}$, the denominator is $\\var{g2}$.
"}, {"name": "Combine fractions and reduce to lowest terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"g1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a1*d1-b1*c1,b1*d1)", "description": "", "name": "g1"}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..13 except r1)", "description": "", "name": "s1"}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..11)", "description": "", "name": "r1"}, "u11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*u1/r1", "description": "", "name": "u11"}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a1*d1+b1*c1,b1*d1)", "description": "", "name": "f1"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1/t1", "description": "", "name": "b1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r1/t1", "description": "", "name": "a1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "v1/w1", "description": "", "name": "d1"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(r1,s1)", "description": "", "name": "t1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "u1/w1", "description": "", "name": "c1"}, "j1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a1*d1,b1*c1)", "description": "", "name": "j1"}, "v1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..13 except [u1,s1,u11])", "description": "", "name": "v1"}, "u1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..11)", "description": "", "name": "u1"}, "h1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a1*c1,b1*d1)", "description": "", "name": "h1"}, "w1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(u1,v1)", "description": "", "name": "w1"}}, "ungrouped_variables": ["f1", "r1", "g1", "s1", "h1", "u1", "j1", "t1", "a1", "v1", "b1", "w1", "u11", "c1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "$\\dfrac{\\var{a1}}{\\var{b1}} + \\dfrac{\\var{c1}}{\\var{d1}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a1*d1+b1*c1)/f1}", "minValue": "{(a1*d1+b1*c1)/f1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{b1*d1/f1}", "minValue": "{b1*d1/f1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\n $\\dfrac{\\var{a1}}{\\var{b1}} - \\dfrac{\\var{c1}}{\\var{d1}}$
In lowest terms, the numerator is [[0]], the denominator is [[1]]
\t\t \n ", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a1*d1-b1*c1)/g1}", "minValue": "{(a1*d1-b1*c1)/g1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{b1*d1/g1}", "minValue": "{b1*d1/g1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$\\dfrac{\\var{a1}}{\\var{b1}} \\times \\dfrac{\\var{c1}}{\\var{d1}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{a1*c1/h1}", "minValue": "{a1*c1/h1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{b1*d1/h1}", "minValue": "{b1*d1/h1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\n $\\dfrac{\\var{a1}}{\\var{b1}} \\div \\dfrac{\\var{c1}}{\\var{d1}}$
In lowest terms, the numerator is [[0]], the denominator is [[1]]
\t\t \n ", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{a1*d1/j1}", "minValue": "{a1*d1/j1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{b1*c1/j1}", "minValue": "{b1*c1/j1}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Evaluate the following as fractions in lowest terms. Write the numerator and denominator of the lowest term fraction in the boxes provided.
", "tags": ["Arithmetic", "arithmetic", "checked2015", "Fractions", "fractions", "Lowest terms", "SFY0001"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of numerators and denominators of numerical fractions, and reduction to lowest terms.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "For addition and subtraction, write fractions so that they have a common denominator and then perform addition or subtraction on the numerators. One method of doing this is 'cross-multiplication'. The rules are :
\n\\[\\simplify{a/b+ c/d=(a*d+b*c)/(b*d)}.\\]
\\[\\simplify{a/b- c/d=(a*d-b*c)/(b*d)}.\\]
\nFor multiplication and division the rules are simpler:
\n\\[\\simplify{(a/b)} \\times \\simplify{(c/d)}=\\simplify{(a*c)/(b*d)}.\\]
\\[\\simplify{(a/b)} / \\simplify{(c/d)}=\\simplify{(a*d)/(b*c)}.\\]
\nHaving applied these rules, it will be necessary to reduce the resulting fractions to lowest terms. We get:
\n\na) $\\dfrac{\\var{a1}}{\\var{b1}} + \\dfrac{\\var{c1}}{\\var{d1}}=\\dfrac{\\var{a1} \\times \\var{d1} + \\var{b1} \\times \\var{c1}}{\\var{b1} \\times \\var{d1}}=\\dfrac{\\var{a1*d1 + b1*c1}}{\\var{b1*d1}}$. In lowest terms this is $\\dfrac{\\var{(a1*d1 + b1*c1)/f1}}{\\var{b1*d1/f1}}$.
\n\nb) $\\dfrac{\\var{a1}}{\\var{b1}} - \\dfrac{\\var{c1}}{\\var{d1}}=\\dfrac{\\var{a1} \\times \\var{d1} - \\var{b1} \\times \\var{c1}}{\\var{b1} \\times \\var{d1}}=\\dfrac{\\var{a1*d1 - b1*c1}}{\\var{b1*d1}}$. In lowest terms this is $\\dfrac{\\var{(a1*d1 - b1*c1)/g1}}{\\var{b1*d1/g1}}$.
\n\nc) $\\dfrac{\\var{a1}}{\\var{b1}} \\times \\dfrac{\\var{c1}}{\\var{d1}}=\\dfrac{\\var{a1} \\times \\var{c1}}{\\var{b1} \\times \\var{d1}}=\\dfrac{\\var{a1*c1}}{\\var{b1*d1}}$. In lowest terms this is $\\dfrac{\\var{a1*c1/h1}}{\\var{b1*d1/h1}}$
\n\nd) $\\dfrac{\\var{a1}}{\\var{b1}} \\div \\dfrac{\\var{c1}}{\\var{d1}}=\\dfrac{\\var{a1}}{\\var{b1}} \\times \\dfrac{\\var{d1}}{\\var{c1}}=\\dfrac{\\var{a1} \\times \\var{d1}}{\\var{b1} \\times \\var{c1}}=\\dfrac{\\var{a1*d1}}{\\var{b1*c1}}$. In lowest terms this is $\\dfrac{\\var{a1*d1/j1}}{\\var{b1*c1/j1}}$
\n\n"}, {"name": "Combining algebraic fractions 0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator"], "metadata": {"description": "Express $\\displaystyle a \\pm \\frac{c}{x + d}$ as an algebraic single fraction.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Add the following together and express as a single algebraic fraction.
\n
", "advice": "We have:
\n \\[\\simplify[std]{{a} + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c}) / (({a2}*x + {d})) = ({a*a2} * x + {a * d + c}) / ( ({a2}*x + {d}))}\\]
", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9 except a2)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a,-a])", "description": "", "templateType": "anything"}, "nb": {"name": "nb", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "d", "nb", "a2", "s1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Express \\[\\simplify[std]{{a} + ({c} / ({a2}x + {d}))}\\] as a single algebraic fraction.
\n Input the fraction here: [[0]].
\n You can click on Show steps for help. You will lose 1 mark if you do so.
\n
", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "The formula for adding these expressions is :
\\[\\simplify[std]{a + {s1} * (c / d) = (ad + {s1} * bc) / d}\\]
\n and for this exercise we have $\\simplify{d={a2}x+{d}}$.
\n
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a*a2}x+{a*d+c})/({a2}x+{d})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-05, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [10, 11], "checkVariableNames": false, "minlength": {"length": 12, "partialCredit": 0, "message": "Input as a single fraction.
"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}]}, {"name": "Combining algebraic fractions 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "name": "s1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "a", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "name": "d", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "b", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "c", "description": ""}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "name": "nb", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a2", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "({a*a2 + c*a1} * x + {a * d + c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a single fraction.
", "showStrings": false, "partialCredit": 0, "strings": [")-", ")+"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\n Note that in your answer you do not need to expand the denominator.
", "marks": 0, "scripts": {}}], "prompt": "Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps if you need help.You will lose one mark if you do so.
\n
", "stepsPenalty": 1}], "statement": "\n Add the following two fractions together and express as a single fraction over a common denominator.
\n
\n ", "tags": ["SFY0001", "algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t5/08/2012:
\n \t\t Added tags.
\n \t\t Added description.
\n \t\t Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t 12/08/2012:
\n \t\t Back to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t Used the except feature of ranges to get non-degenerate examples.
\n \t\t Checked calculation.OK.
\n \t\t Improved display in content areas.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{a}{x + b} \\pm \\frac{c}{x + d}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The formula for {nb} fractions is :
\n \\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
Hence we have:
\\[\\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\]
"}, {"name": "Combining algebraic fractions 3.0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "name": "s1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b2*a2/c)])", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except a)", "name": "c", "description": ""}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "name": "nb", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "name": "a", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b2", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a2", "description": ""}}, "ungrouped_variables": ["a", "c", "d", "s1", "a2", "b1", "b2", "nb"], "functions": {}, "parts": [{"customName": "", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"prompt": "Note that:
\\[\\simplify[std]{a + (c / d) = (a*d + c) / d}\\]
", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "useCustomName": false, "prompt": "\n Express \\[\\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps to get more information. You will lose one mark if you do so.
\n
\n ", "stepsPenalty": 1, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "( {c+b1*a2} * x + {b1 * d + b2 })/ ( ({a2}*x + {d}))", "scripts": {}, "answerSimplification": "std", "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [10, 11], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 1e-05, "variableReplacements": [], "failureRate": 1, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "customName": "", "checkVariableNames": false, "unitTests": [], "mustmatchpattern": {"message": "Enter your answer as a single fraction.", "pattern": "?/(`!$n)", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "showPreview": true, "marks": 2}], "type": "gapfill", "unitTests": [], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Express the following as a single fraction.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions"], "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle b+ \\frac{dx+p}{x + q}$ as an algebraic single fraction.
"}, "advice": "The formula for adding these expressions is:
\n \\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]
\n and for this exercise we have $\\simplify{a={b1}}$, $\\simplify{c={c}x+{b2}}$, $\\simplify{d={a2}x+{d}}$.
\n Hence we have:
\\[\\begin{eqnarray*} \\simplify[std]{{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \\simplify{(({b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({b1*a2}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ( {b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Combining algebraic fractions 3.1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "name": "s1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a2", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b2*a2/c)])", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except a)", "name": "c", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b1", "description": ""}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "name": "nb", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b2", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "a", "description": ""}}, "ungrouped_variables": ["a", "c", "d", "s1", "a2", "b1", "b2", "nb"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "({a*a2}*x^2 + {c+b1*a2+a*d} * x + {b1 * d + b2 })/ ( ({a2}*x + {d}))", "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a single fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["+(", "-(", ")+", ")-"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "Note that:
\\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]
\n
", "marks": 0, "scripts": {}}], "prompt": "Express \\[\\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps to get more information. You will lose one mark if you do so.
\n
", "stepsPenalty": 1}], "statement": "Express the following as a single fraction.
\n
", "tags": ["SFY0001", "algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions"], "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "18/08/2012:
\n Added tags.
\n Added description.
\n Modified copy of Combining algebraic fractions 3.
\n Checked calculations.OK
\n
\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle ax+b+ \\frac{dx+p}{x + q}$ as an algebraic single fraction.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The formula for adding these expressions is:
\n \\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]
\n and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={c}x+{b2}}$, $\\simplify{d={a2}x+{d}}$.
\n Hence we have:
\\[\\begin{eqnarray*} \\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2} * x^2 + {a * d +b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Combining algebraic fractions 3.2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "name": "s1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sgn(c)*random(1..5 except [round(c*d/a2)])", "description": "", "name": "b2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "c"}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "name": "nb"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,round(a*b/a1)])", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "name": "d"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "b1", "b2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a*a2+a1*c}*x^2 + {b*c+a1*b2+b1*a2+a*d} * x + {b1 * d + b2 * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a single fraction.
", "showStrings": false, "partialCredit": 0, "strings": [")-", ")+"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps for more information. You will lose one mark if you do so.
\n
", "steps": [{"type": "information", "prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\n Note that in your answer you do not need to expand the denominator.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n Add the following two fractions together and express as a single fraction over a common denominator.
\n
\n \n ", "tags": ["SFY0001", "algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t5/08/2012:
\n \t\t \t\t Added tags.
\n \t\t \t\t Added description.
\n \t\t \t\t Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t \t\t 12/08/2012:
\n \t\t \t\t Back to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\t Used the except feature of ranges to get non-degenerate examples.
\n \t\t \t\t Checked calculation.OK.
\n \t\t \t\t Improved display in content areas.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{ax+b}{x + c} \\pm \\frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The formula for {nb} fractions is :
\n \\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\n Hence we have:
\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Combining algebraic fractions 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "name": "s1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sgn(c)*random(1..5 except [round(c*d/a2)])", "description": "", "name": "b2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "c"}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "name": "nb"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,round(a*b/a1)])", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "name": "d"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "b1", "b2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a*a2+a1*c}*x^2 + {b*c+a1*b2+b1*a2+a*d} * x + {b1 * d + b2 * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a single fraction.
", "showStrings": false, "partialCredit": 0, "strings": [")-", ")+"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps for more information. You will lose one mark if you do so.
", "steps": [{"type": "information", "prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.
\n Note that in your answer you do not need to expand the denominator.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n Add the following two fractions together and express as a single fraction over a common denominator.
\n
\n \n \n ", "tags": ["SFY0001", "algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t \t\t5/08/2012:
\n \t\t \t\t \t\t Added tags.
\n \t\t \t\t \t\t Added description.
\n \t\t \t\t \t\t Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t \t\t \t\t 12/08/2012:
\n \t\t \t\t \t\t Back to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\t \t\t Used the except feature of ranges to get non-degenerate examples.
\n \t\t \t\t \t\t Checked calculation.OK.
\n \t\t \t\t \t\t Improved display in content areas.
\n \t\t \t\t \n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{ax+b}{cx + d} \\pm \\frac{rx+s}{px + q}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The formula for {nb} fractions is :
\n \\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b={a1}x+{b}}$, $\\simplify{d={a2}x+{d}}$.
\n Hence we have:
\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Compute more complicated combinations of fractions and reduce to lowest terms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "u2"}, "x21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*v2*x2/(r2*v2-s2*u2)", "description": "", "name": "x21"}, "e2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "x2/z2", "description": "", "name": "e2"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11 except r2)", "description": "", "name": "s2"}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "r2"}, "o2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "o2"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "u2/w2", "description": "", "name": "c2"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "v2/w2", "description": "", "name": "d2"}, "v2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11 except [u2,s2,u21])", "description": "", "name": "v2"}, "j2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a2*d2*f2-b2*c2*f2+b2*d2*e2,b2*d2*f2)", "description": "", "name": "j2"}, "f2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "y2/z2", "description": "", "name": "f2"}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11 except o2)", "description": "", "name": "p2"}, "l2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a2*d2*f2+b2*c2*e2,b2*d2*f2)", "description": "", "name": "l2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r2/t2", "description": "", "name": "a2"}, "h2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "p2/q2", "description": "", "name": "h2"}, "i2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a2*d2*f2+b2*c2*f2+b2*d2*e2,b2*d2*f2)", "description": "", "name": "i2"}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "x2"}, "x22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "-x21", "description": "", "name": "x22"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(x2,y2)", "description": "", "name": "z2"}, "y2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..11 except [u2,s2,v2,x2,x21,x22])", "description": "", "name": "y2"}, "u21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*u2/r2", "description": "", "name": "u21"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2/t2", "description": "", "name": "b2"}, "w2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(u2,v2)", "description": "", "name": "w2"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(r2,s2)", "description": "", "name": "t2"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a2*d2*f2*h2+b2*c2*f2*g2+b2*d2*e2*h2,b2*d2*f2*h2)", "description": "", "name": "m2"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(o2,p2)", "description": "", "name": "q2"}, "g2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "o2/q2", "description": "", "name": "g2"}, "k2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a2*d2*f2-b2*c2*f2-b2*d2*e2,b2*d2*f2)", "description": "", "name": "k2"}}, "ungrouped_variables": ["f2", "h2", "j2", "w2", "b2", "y2", "d2", "q2", "s2", "u2", "k2", "m2", "x21", "o2", "x22", "g2", "i2", "v2", "a2", "x2", "c2", "z2", "e2", "p2", "r2", "t2", "l2", "u21"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"prompt": "$\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}}{\\var{d2}} + \\dfrac{\\var{e2}}{\\var{f2}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a2*d2*f2+b2*c2*f2+b2*d2*e2)/i2}", "minValue": "{(a2*d2*f2+b2*c2*f2+b2*d2*e2)/i2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(b2*d2*f2)/i2}", "minValue": "{(b2*d2*f2)/i2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\n $\\dfrac{\\var{a2}}{\\var{b2}} - \\dfrac{\\var{c2}}{\\var{d2}} + \\dfrac{\\var{e2}}{\\var{f2}}$
In lowest terms, the numerator is [[0]], the denominator is [[1]]
\n \n ", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a2*d2*f2-b2*c2*f2+b2*d2*e2)/j2}", "minValue": "{(a2*d2*f2-b2*c2*f2+b2*d2*e2)/j2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(b2*d2*f2)/j2}", "minValue": "{(b2*d2*f2)/j2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "\n $\\dfrac{\\var{a2}}{\\var{b2}} - \\dfrac{\\var{c2}}{\\var{d2}} - \\dfrac{\\var{e2}}{\\var{f2}}$
In lowest terms, the numerator is [[0]], the denominator is [[1]]
\n \n ", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a2*d2*f2-b2*c2*f2-b2*d2*e2)/k2}", "minValue": "{(a2*d2*f2-b2*c2*f2-b2*d2*e2)/k2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(b2*d2*f2)/k2}", "minValue": "{(b2*d2*f2)/k2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}}{\\var{d2}} \\times \\dfrac{\\var{e2}}{\\var{f2}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a2*d2*f2+b2*c2*e2)/l2}", "minValue": "{(a2*d2*f2+b2*c2*e2)/l2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(b2*d2*f2)/l2}", "minValue": "{(b2*d2*f2)/l2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}, {"prompt": "$\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}}{\\var{d2}} \\times \\dfrac{\\var{g2}}{\\var{h2}}+\\dfrac{\\var{e2}}{\\var{f2}}$
\n In lowest terms, the numerator is [[0]], the denominator is [[1]]
", "scripts": {}, "gaps": [{"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(a2*d2*f2*h2+b2*c2*f2*g2+b2*d2*e2*h2)/m2}", "minValue": "{(a2*d2*f2*h2+b2*c2*f2*g2+b2*d2*e2*h2)/m2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}, {"correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "allowFractions": false, "type": "numberentry", "maxValue": "{(b2*d2*f2*h2)/m2}", "minValue": "{(b2*d2*f2*h2)/m2}", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "statement": "Evaluate the following as fractions in lowest terms. Write the numerator and denominator of the lowest term fraction in the boxes provided.
", "tags": ["checked2015", "Fractions", "fractions", "Lowest terms", "SFY0001"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Harder questions testing addition, subtraction, multiplication and division of numerical fractions and reduction to lowest terms. They also test BIDMAS in the context of fractions.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Perform the various operations required in the order dictated by BIDMAS.
\nFor addition and subtraction, write fractions so that they have a common denominator and then perform addition or subtraction on the numerators. One method of doing this is 'cross-multiplication'. The rules are :
\n\\[\\simplify{a/b+ c/d=(a*d+b*c)/(b*d)}.\\]
\\[\\simplify{a/b- c/d=(a*d-b*c)/(b*d)}.\\]
\nFor multiplication and division the rules are simpler:
\n\\[\\simplify{(a/b)} \\times \\simplify{(c/d)=(a*c)/(b*d)}.\\]
\\[\\simplify{(a/b)} / \\simplify{(c/d)}=\\simplify{(a*d)/(b*c)}.\\]
\nHaving applied these rules, it will be necessary to reduce the resulting fractions to lowest terms. In some of the following there might be slightly simpler approaches possible.
\na) $\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}}{\\var{d2}} + \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2}+ \\var{b2*c2}}{\\var{b2*d2}} + \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2+ b2*c2}}{\\var{b2*d2}} + \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2+ b2*c2} \\times \\var{f2}+\\var{b2*d2} \\times \\var{e2}}{\\var{b2*d2} \\times \\var{f2}}=\\dfrac{\\var{(a2*d2+ b2*c2)*f2}+\\var{b2*d2*e2}}{\\var{b2*d2*f2}}=\\dfrac{\\var{(a2*d2+ b2*c2)*f2+b2*d2*e2}}{\\var{b2*d2*f2}}$.
\nIn lowest terms, this is $\\dfrac{\\var{(a2*d2*f2+b2*c2*f2+b2*d2*e2)/i2}}{\\var{(b2*d2*f2)/i2}}$.
\n\nb) $\\dfrac{\\var{a2}}{\\var{b2}} - \\dfrac{\\var{c2}}{\\var{d2}} + \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2}- \\var{b2*c2}}{\\var{b2*d2}} + \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2- b2*c2}}{\\var{b2*d2}} + \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2- b2*c2} \\times \\var{f2}+\\var{b2*d2} \\times \\var{e2}}{\\var{b2*d2} \\times \\var{f2}}=\\dfrac{\\var{(a2*d2- b2*c2)*f2}+\\var{b2*d2*e2}}{\\var{b2*d2*f2}}=\\dfrac{\\var{(a2*d2- b2*c2)*f2+b2*d2*e2}}{\\var{b2*d2*f2}}$.
\nIn lowest terms, this is $\\dfrac{\\var{(a2*d2*f2-b2*c2*f2+b2*d2*e2)/j2}}{\\var{(b2*d2*f2)/j2}}$.
\n\nc) $\\dfrac{\\var{a2}}{\\var{b2}} - \\dfrac{\\var{c2}}{\\var{d2}} - \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2}- \\var{b2*c2}}{\\var{b2*d2}} - \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2- b2*c2}}{\\var{b2*d2}} - \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2- b2*c2}\\times \\var{f2}-\\var{b2*d2} \\times \\var{e2}}{\\var{b2*d2} \\times \\var{f2}}=\\dfrac{\\var{(a2*d2- b2*c2)*f2}-\\var{b2*d2*e2}}{\\var{b2*d2*f2}}=\\dfrac{\\var{(a2*d2- b2*c2)*f2-b2*d2*e2}}{\\var{b2*d2*f2}}$.
\nIn lowest terms, this is $\\dfrac{\\var{(a2*d2*f2-b2*c2*f2-b2*d2*e2)/k2}}{\\var{(b2*d2*f2)/k2}}$.
\n\nd) $\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}}{\\var{d2}} \\times \\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}\\times \\var{e2}}{\\var{d2} \\times \\var{f2}}=\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2*e2}}{\\var{d2*f2}} =\\dfrac{\\var{a2} \\times \\var{d2*f2}}{\\var{b2} \\times \\var{d2*f2}} + \\dfrac{\\var{b2} \\times \\var{c2*e2}}{\\var{b2}\\times \\var{d2*f2}}=\\dfrac{\\var{a2*d2*f2}+\\var{b2*c2*e2}}{\\var{b2*d2*f2}}=\\dfrac{\\var{a2*d2*f2+b2*c2*e2}}{\\var{b2*d2*f2}} $.
\nIn lowest terms this is $\\dfrac{\\var{(a2*d2*f2+b2*c2*e2)/l2}}{\\var{(b2*d2*f2)/l2}} $
\n\ne) $\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2}}{\\var{d2}} \\times \\dfrac{\\var{g2}}{\\var{h2}}+\\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2}}{\\var{b2}} +\\left[ \\dfrac{\\var{c2}}{\\var{d2}} \\times \\dfrac{\\var{g2}}{\\var{h2}}\\right]+\\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2}}{\\var{b2}} +\\left[ \\dfrac{\\var{c2} \\times \\var{g2} }{\\var{d2} \\times \\var{h2}} \\right]+\\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2*g2} }{\\var{d2*h2}} +\\dfrac{\\var{e2}}{\\var{f2}}$
\n$=\\left[\\dfrac{\\var{a2}}{\\var{b2}} + \\dfrac{\\var{c2*g2} }{\\var{d2*h2}} \\right]+\\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2} \\times \\var{d2*h2}+\\var{b2} \\times \\var{c2*g2}}{\\var{b2} \\times \\var{d2*h2}} +\\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2*h2+b2*c2*g2}}{\\var{b2*d2*h2}} +\\dfrac{\\var{e2}}{\\var{f2}}=\\dfrac{\\var{a2*d2*h2+b2*c2*g2} \\times \\var{f2}+\\var{b2*d2*h2} \\times \\var{e2}}{\\var{b2*d2*h2} \\times \\var{f2}}=\\dfrac{\\var{(a2*d2*h2+b2*c2*g2)*f2+b2*d2*h2*e2}}{\\var{b2*d2*h2*f2}}$.
\nIn lowest terms this is $\\dfrac{\\var{((a2*d2*h2+b2*c2*g2)*f2+b2*d2*h2*e2)/m2}}{\\var{b2*d2*h2*f2/m2}}$.
\n\n"}, {"name": "Index laws - combinations of powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"x": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3,6,9,10,12,13,16,18,20,21)", "description": "", "name": "x"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..12 except m)", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,5,7)", "description": "", "name": "b"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "", "name": "m"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..12 except [m1,n])", "description": "", "name": "n1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(8..20)", "description": "", "name": "a"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2,3,5,7,11,13,17)", "description": "", "name": "t"}, "y": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(4,5,7,8,11,14,15,17,19,23)", "description": "", "name": "y"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9 except m)", "description": "", "name": "m1"}}, "ungrouped_variables": ["a", "b", "m", "n", "m1", "t", "n1", "y", "x"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{m+n}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express the following as a single power of $\\var{a}$:
\n \\[\\var{a}^{\\var{m}} \\times \\var{a}^{\\var{n}}\\]
\n Enter the power [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{m1*n1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express the following as a single power of $\\var{b}$:
\n \\[(\\var{b}^{\\var{m1}})^{\\var{n1}}\\]
\n Enter the power [[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{x*y}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{t}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express the following as a power of a single number. The power should be a prime.
\n \\[\\var{x}^{\\var{t}} \\times \\var{y}^{\\var{t}}\\]
\n Enter the number [[0]] and the power [[1]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "", "tags": ["checked2015", "exponents", "Index Laws", "indices", "powers", "SFY0001"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing rather basic understanding of the index laws.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "(a) Add the powers: $\\var{m} + \\var{n}=\\var{m+n}$ (first index law).
\n (b) Multiply the powers $\\var{m1} \\times \\var{n1}=\\var{m1*n1}$ (second index law).
\n (c) Multiply $\\var{x}$ and $\\var{y}$ to get $\\var{x*y}$, with the power unchanged at $\\var{t}$. [This is the only way for the power to be a prime.]
"}, {"name": "Index Laws 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "t1"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t*f", "description": "", "name": "n"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "f"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "d"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "t"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*f+b*c*f*t-d*t", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "f", "m", "t1", "a1", "t", "n"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a1}/{t1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\sqrt[\\var{t1}]{\\var{2^{a1}}}}$
The power is [[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{m}/{n}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\dfrac{\\sqrt[\\var{t}]{\\var{2^a}}*(\\var{2^b})^{\\var{c}}}{(\\var{2^d})^{1/\\var{f}}}}$
In lowest terms, the power is [[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Express the following as a power of 2, writing the power in the box provided. Write the power as a fraction or as an integer.
", "tags": ["Index Laws", "SFY0001", "checked2015", "exponents", "indices", "powers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the index laws.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n Write each of the numbers as a power of 2, write the n'th root as: to the power 1/n. Then apply the index laws to combine the powers. Thus \\[{\\LARGE\\sqrt[\\var{t1}]{\\var{2^{a1}}}=(2^{\\var{a1}})^{1/ \\var{t1}} =2^{\\var{a1} /\\var{t1}}=2^{\\simplify{{a1}/{t1}}}}\\] and \\[{\\LARGE\\dfrac{\\sqrt[\\var{t}]{\\var{2^a}}*(\\var{2^b})^{\\var{c}}}{(\\var{2^d})^{1/\\var{f}}}=\\dfrac{(2^{\\var{a}})^{1/ \\var{t}} * (2^{\\var{b}})^{\\var{c}}}{(2^{\\var{d}})^{1/\\var{f}}}=\\dfrac{2^{\\var{a}/\\var{t}} * 2^{\\var{b* c}}}{2^{\\var{d}/\\var{f}}}}\\] \n \\[{\\LARGE=2^{\\var{a}/\\var{t}} * 2^{\\var{b* c}}*2^{-\\var{d}/\\var{f}}=2^{\\simplify{{m}/{n}}}}.\\]\n
\n "}, {"name": "Index Laws 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3)", "description": "", "name": "c"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5 except t2)", "description": "", "name": "s2"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a1"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "c2"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except 0)", "description": "", "name": "d2"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t*f", "description": "", "name": "n"}, "f2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5 except [b,0])", "description": "", "name": "f2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "name": "b"}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t2*s2", "description": "", "name": "n2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "name": "a2"}, "h2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except [d2,0])", "description": "", "name": "h2"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "d"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a*f+b*c*f*t-d*t", "description": "", "name": "m"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "f"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "t2"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-4..5 except 0)", "description": "", "name": "b2"}, "p2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b2+c2*d2", "description": "", "name": "p2"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "t1"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t2*b2+t2*c2*d2-s2*f2-s2*g2*h2", "description": "", "name": "m2"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "f2+g2*h2", "description": "", "name": "q2"}, "g2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-3..3 except [c2,0])", "description": "", "name": "g2"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "t"}}, "ungrouped_variables": ["f2", "h2", "t2", "b2", "n", "d2", "q2", "s2", "m2", "a", "g2", "a1", "a2", "c2", "p2", "c", "b", "d", "f", "m", "t1", "t", "n2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a1}/{t1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\sqrt[\\var{t1}]{a^\\var{a1}}}$
The power is [[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{m}/{n}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\dfrac{\\sqrt[\\var{t}]{a^\\var{a}}*(a^\\var{b})^{\\var{c}}}{(a^\\var{d})^{1/\\var{f}}}}$
In lowest terms, the power is [[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{m2}/{n2}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\dfrac{\\sqrt[\\var{s2}]{a^{\\var{b2}}*(a^{\\var{c2}})^{\\var{d2}}}}{\\sqrt[\\var{t2}]{a^{\\var{f2}}*(a^{\\var{g2}})^{\\var{h2}}}}}$
In lowest terms, the power is [[0]]
\n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Express the following as a power of a, writing the power in the box provided. Write the power as a fraction or as an integer.
", "tags": ["Index Laws", "SFY0001", "checked2015", "exponents", "indices", "powers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the index laws.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n Write each of the numbers as a power of a, write the n'th root as: to the power 1/n. Then apply the index laws to combine the powers. Thus \\[{\\LARGE\\sqrt[\\var{t1}]{a^\\var{a1}}=(a^\\var{a1})^{1/\\var{t1}}=a^{\\simplify{{a1}/{t1}}}},\\]\n \\[{\\LARGE\\dfrac{\\sqrt[\\var{t}]{a^\\var{a}}*(a^\\var{b})^{\\var{c}}}{(a^\\var{d})^{1/\\var{f}}}=\\dfrac{(a^{\\var{a}})^{1/ \\var{t}} * (a^{\\var{b}})^{\\var{c}}}{(a^{\\var{d}})^{1/\\var{f}}}=\\dfrac{a^{\\var{a}/\\var{t}} * a^{\\var{b* c}}}{a^{\\var{d}/\\var{f}}}}\\] \n \\[{\\LARGE=a^{\\var{a}/\\var{t}} * a^{\\var{b* c}}*a^{-\\var{d}/\\var{f}}=a^{\\simplify{{m}/{n}}}},\\]\n \\[{\\LARGE\\dfrac{\\sqrt[\\var{s2}]{a^{\\var{b2}}*(a^{\\var{c2}})^{\\var{d2}}}}{\\sqrt[\\var{t2}]{a^{\\var{f2}}*(a^{\\var{g2}})^{\\var{h2}}}}=\\dfrac{(a^{\\var{b2}}*a^{\\var{c2*d2}})^{1/\\var{s2}}}{(a^{\\var{f2}}*a^{\\var{g2*h2}})^{1/\\var{t2}}}}\\]\n \\[{\\LARGE=\\dfrac{(a^{\\var{b2+c2*d2}})^{1/\\var{s2}}}{(a^{\\var{f2+g2*h2}})^{1/\\var{t2}}}=\\dfrac{a^{\\simplify{{p2}/{s2}}}}{a^{\\simplify{{q2}/{t2}}}}=a^{\\simplify{{m2}/{n2}}}}\\]\n
\n "}, {"name": "Index Laws 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "d"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except b)", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except b)", "description": "", "name": "f"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "t1"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "b1"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "t"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "name": "a1"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c*d*h*t+g*t-k*l*h*t-n*h", "description": "", "name": "m3"}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except [c,g,j])", "description": "", "name": "k"}, "l": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5 except d)", "description": "", "name": "l"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b*d*h*t+f*t-j*l*h*t-m*h", "description": "", "name": "m2"}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except [c,g,k,m])", "description": "", "name": "n"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5 except t)", "description": "", "name": "h"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "name": "b"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except [c,f])", "description": "", "name": "g"}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "t*h", "description": "", "name": "n2"}, "j": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except [b,f])", "description": "", "name": "j"}, "m": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4 except [b,f,j])", "description": "", "name": "m"}}, "ungrouped_variables": ["c", "b", "d", "g", "f", "h", "k", "j", "m", "l", "t1", "a1", "t", "m3", "m2", "n2", "n", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a1}/{t1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}, {"answer": "{b1}/{t1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\sqrt[\\var{t1}]{\\var{2^{a1}*3^{b1}}}}$
$\\alpha =$ [[0]], $\\beta =$ [[1]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{m2}/{n2}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}, {"answer": "{m3}/{n2}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\dfrac{(\\var{2^b * 3^c})^{\\var{d}}*(\\var{2^f * 3^g})^{1/\\var{h}}}{(\\var{2^j *3^k})^{\\var{l}}*\\sqrt[\\var{t}]{\\var{2^m *3^n}}}}$
$\\alpha =$ [[0]], $\\beta =$ [[1]]
\n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Express the following in the form $2^{\\alpha} 3^{\\beta}$, writing the $\\alpha$ and $\\beta$ in the boxes provided. Write the powers as fractions or as integers.
", "tags": ["Index Laws", "SFY0001", "checked2015", "exponents", "indices", "powers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the index laws.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n Write each of the numbers as a power of 2 multiplied by a power of 3, write the n'th root as: to the power 1/n. Then apply the index laws to combine the powers. Thus\n \\[{\\LARGE\\sqrt[\\var{t1}]{\\var{2^{a1}*3^{b1}}}=(2^{\\var{a1}}*3^{\\var{b1}})^{1/\\var{t1}}=(2^{\\var{a1}})^{1/\\var{t1}}*(3^{\\var{b1}})^{1/\\var{t1}}=2^{\\simplify{{a1}/{t1}}} 3^{\\simplify{{b1}/{t1}}}},\\]\n \\[{\\LARGE\\dfrac{(\\var{2^b * 3^c})^{\\var{d}}*(\\var{2^f * 3^g})^{1/\\var{h}}}{(\\var{2^j *3^k})^{\\var{l}}*\\sqrt[\\var{t}]{\\var{2^m *3^n}}}=\\dfrac{(2^{\\var{b}} * 3^{\\var{c}})^{\\var{d}}*(2^{\\var{f}} * 3^{\\var{g}})^{1/\\var{h}}}{(2^{\\var{j}} *3^{\\var{k}})^{\\var{l}}*(2^{\\var{m}} *3^{\\var{n}})^{1/\\var{t}}}}\\]\n \\[{\\LARGE =\\dfrac{(2^{\\var{b*d}} * 3^{\\var{c*d}})*(2^{\\var{f}/\\var{h}} * 3^{\\var{g}/\\var{h}})}{(2^{\\var{j*l}} *3^{\\var{k*l}})*(2^{\\var{m}/\\var{t}} *3^{\\var{n}/\\var{t}})} =\\dfrac{2^{\\var{b*d}} * 3^{\\var{c*d}}*2^{\\var{f}/\\var{h}} * 3^{\\var{g}/\\var{h}}}{2^{\\var{j*l}} *3^{\\var{k*l}}*2^{\\var{m}/\\var{t}} *3^{\\var{n}/\\var{t}}}}\\]\n \\[{\\LARGE =2^{\\var{b*d}} *2^{\\var{f}/\\var{h}}*2^{\\var{-j*l}}*2^{-\\var{m}/\\var{t}} * 3^{\\var{c*d}} * 3^{\\var{g}/\\var{h}} *3^{-\\var{k*l}}*3^{-\\var{n}/\\var{t}} =2^{\\simplify{{m2}/{n2}}} 3^{\\simplify{{m3}/{n2}}}}\\]\n
\n "}, {"name": "Index Laws 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "d"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4 except b)", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4 except b)", "description": "", "name": "f"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..5)", "description": "", "name": "t1"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "b1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "name": "a1"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c*d*h+g-k*l*h", "description": "", "name": "m3"}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4 except [c,g,j])", "description": "", "name": "k"}, "l": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..4 except d)", "description": "", "name": "l"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b*d*h+f-j*l*h", "description": "", "name": "m2"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..3)", "description": "", "name": "h"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "b"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4 except [c,f])", "description": "", "name": "g"}, "j": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4 except [b,f])", "description": "", "name": "j"}}, "ungrouped_variables": ["c", "b", "d", "g", "f", "h", "k", "j", "l", "t1", "a1", "b1", "m3", "m2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{a1}/{t1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}, {"answer": "{b1}/{t1}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE\\sqrt[\\var{t1}]{x^{\\var{a1}}*y^{\\var{b1}}}}$
$\\alpha =$ [[0]], $\\beta =$ [[1]]
\n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "{m2}/{h}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}, {"answer": "{m3}/{h}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n ${\\LARGE \\dfrac{(x^{\\var{b}} * y^{\\var{c}})^{\\var{d}}*(x^{\\var{f}} * y^{\\var{g}})^{1/\\var{h}}}{(x^{\\var{j}} *y^{\\var{k}})^{\\var{l}}}}$
$\\alpha =$ [[0]], $\\beta =$ [[1]]
\n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Express the following in the form $x^{\\alpha} y^{\\beta}$, writing the $\\alpha$ and $\\beta$ in the boxes provided. Write the powers as fractions or as integers.
", "tags": ["Index Laws", "SFY0001", "checked2015", "exponents", "indices", "powers"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Questions testing understanding of the index laws.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n Write the n'th root as: to the power 1/n. Then apply the index laws to combine the powers.\n \n Write each of the numbers as a power of $x$ multiplied by a power of $y$, write the n'th root as: to the power 1/n. Then apply the index laws to combine the powers. Thus\n \\[{\\LARGE\\sqrt[\\var{t1}]{x^{\\var{a1}}*y^{\\var{b1}}}=(x^{\\var{a1}}*y^{\\var{b1}})^{1/\\var{t1}}=(x^{\\var{a1}})^{1/\\var{t1}}*(y^{\\var{b1}})^{1/\\var{t1}}=x^{\\simplify{{a1}/{t1}}} y^{\\simplify{{b1}/{t1}}}},\\]\n \\[{\\LARGE\\dfrac{(x^{\\var{b}} * y^{\\var{c}})^{\\var{d}}*(x^{\\var{f}} * y^{\\var{g}})^{1/\\var{h}}}{(x^{\\var{j}} *y^{\\var{k}})^{\\var{l}}} =\\dfrac{(x^{\\var{b*d}} * y^{\\var{c*d}})*(x^{\\var{f}/\\var{h}} * y^{\\var{g}/\\var{h}})}{(x^{\\var{j*l}} *y^{\\var{k*l}})} }\\]\n \\[{\\LARGE =x^{\\var{b*d}} *x^{\\var{f}/\\var{h}}*x^{\\var{-j*l}} * y^{\\var{c*d}} * y^{\\var{g}/\\var{h}} *y^{-\\var{k*l}} =x^{\\simplify{{m2}/{h}}} y^{\\simplify{{m3}/{h}}}}\\]\n
\n "}, {"name": "Solve a linear equation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9)", "name": "d", "description": ""}, "an1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "d-b", "name": "an1", "description": ""}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except a)", "name": "c", "description": ""}, "an2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "a-c", "name": "an2", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "name": "b", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "name": "a", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "an1", "an2"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{an1}/{an2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer, not as a decimal.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\n \\[\\simplify{{a} * x + {b} = {c} * x + {d}}\\]
\n $x=\\;$ [[0]]
\n \n ", "marks": 0}], "statement": "\n Solve the following equation for $x$.
\n Input your answer as a fraction or an integer as appropriate and not as a decimal.
\n \n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "Solving equations", "subject of an equation"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle ax+b = cx+d$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "If $\\simplify{{a}*x + {b}}=\\simplify{{c}*x + {d}}$, then we first subtract $\\var{c}x$ from each side.
\n We obtain $\\simplify{{a}x - {c}x+ {b}}= \\var{d}$.
\n Next we subtract $\\var{b}$ from each side (or add $\\var{-b}$ if you prefer - it means the same thing), to obtain $\\var{a}x - \\var{c}x = \\simplify{{d}-{b}}$.
\n In other words $\\simplify{{a}-{c}}x= \\simplify{{d}-{b}}$.
\n Now divide both sides by $\\simplify{{a-c}}$ to obtain $x=\\dfrac{\\simplify{{d-b}}}{\\simplify{{a-c}}}$ $=\\Large{\\simplify{{an1}/{an2}}}$.
"}, {"name": "Solve an equation in algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"q": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p*c/a", "description": "", "name": "q"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "description": "", "name": "d"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5 except [p,abs(b)])", "description": "", "name": "a"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "m*a/g", "description": "", "name": "c"}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except r)", "description": "", "name": "t"}, "r": {"group": "Ungrouped variables", "templateType": "anything", "definition": "(p*d+c*s-b*q)/a", "description": "", "name": "r"}, "an1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b*t-s*d", "description": "", "name": "an1"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "description": "", "name": "s"}, "an2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "p*d+s*c-a*t-b*q", "description": "", "name": "an2"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "description": "", "name": "b"}, "g": {"group": "Ungrouped variables", "templateType": "anything", "definition": "gcd(a,p)", "description": "", "name": "g"}, "p": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "p"}, "m": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..3)", "description": "", "name": "m"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "m", "q", "p", "s", "r", "t", "an2", "an1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "{an1}/{an2}", "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer, not as a decimal.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\n \\[\\simplify{({p}*x+{s}) / ({a} * x + {b}) = ({q}*x+{t}) / ({c} * x + {d})}\\]
\n $x=\\;$ [[0]]
\n If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\n \n \n ", "steps": [{"type": "information", "prompt": "\n Cross-multiply to get:
\\[\\simplify{({p}*x+{s})*({c} * x + {d})=({q}*x+{t})*({a} * x + {b})}\\]
Multiplying out to get \\[\\simplify{{p*c}x^2 +{p*d+c*s}x+{s*d}={q*a}x^2 +{q*b+t*a}x+{t*b}}.\\] Subtract the $x^2$ term from each side to leave a linear equation:
\n Solve this equation for $x$.
\n \n ", "showCorrectAnswer": true, "marks": 0, "scripts": {}}], "showCorrectAnswer": true, "stepsPenalty": 1}], "statement": "\n Solve the following equation for $x$.
\n Input your answer as a fraction or an integer as appropriate and not as a decimal.
\n \n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "subject of an equation"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle \\frac{px+s}{ax+b} = \\frac{qx+t}{cx+d}$ with $pc=qa$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Cross-multiply to get:
\\[\\simplify{({p}*x+{s})*({c} * x + {d})=({q}*x+{t})*({a} * x + {b})}\\]
Multiplying out we get \\[\\simplify{{p*c}x^2 +{p*d+c*s}x+{s*d}={q*a}x^2 +{q*b+t*a}x+{t*b}}\\] Subtracting ${\\var{a*q}}x^2$ from each side we are left with \\[\\simplify{{p*d+c*s}x+{s*d}={q*b+t*a}x+{t*b}}\\] which we solve as a linear equation: \\[\\simplify{{p*d+c*s-q*b-t*a}x={t*b-s*d}}\\] and so \\[\\simplify{x={an1}/{an2}}.\\]
"}, {"name": "Solve an equation with reciprocals", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"an1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b*t-s*d", "description": "", "name": "an1"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,t])", "description": "", "name": "d"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except [s,abs(d),a*t/s])", "description": "", "name": "c"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8)", "description": "", "name": "s"}, "an2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*c-a*t", "description": "", "name": "an2"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..9 except [s,abs(b)])", "description": "", "name": "a"}, "t": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..8 except s)", "description": "", "name": "t"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except [0,s])", "description": "", "name": "b"}}, "ungrouped_variables": ["a", "c", "b", "d", "s", "t", "an2", "an1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "{an1}/{an2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction or an integer, not as a decimal.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "\n Rearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now solve this linear equation.
\n \n ", "marks": 0, "scripts": {}}], "prompt": "\n \\[\\simplify{{s} / ({a} * x + {b}) = {t} / ({c} * x + {d})}\\]
\n $x=\\;$ [[0]]
\n If you want help in solving the equation, click on Show steps. If you do so then you will lose 1 mark.
\n \n \n ", "stepsPenalty": 1}], "statement": "\n Solve the following equation for $x$.
\n Input your answer as a fraction or an integer as appropriate and not as a decimal.
\n ", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "changing the subject of an equation", "checked2015", "rearranging equations", "SFY0001", "solving", "solving equations", "subject of an equation"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t\t\t\t\t \t\t \t\t\t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$: $\\displaystyle \\frac{s}{ax+b} = \\frac{t}{cx+d}$
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Rearrange the equation by cross-multiplying to get:
\\[\\simplify{{s}*({c} * x + {d}) = {t} *({a} * x + {b})}\\]
Multiply out to get \\[\\simplify{{s*c}*x+{s*d}={t*a}*x+{t*b}}.\\] Now this is a linear equation which is solved in the following steps: \\[\\simplify{{s*c-t*a}*x={t*b-s*d}}\\] and then \\[\\simplify{x={t*b-s*d}/{s*c-t*a}}.\\]
"}, {"name": "Solving equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s1", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(a2=2,random(3,5,7,9),a2=3,random(2,4,5,7),a2=4,random(3,5,7,9),a2=5,random(3,4,6,7,9),a2=6,random(4,5,7,8,9),a2=7,random(3,4,5,6,8,9),a2=8,random(3,5,6,7,9),a2=9,random(2,4,5,7,8),9)", "name": "a1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sa*random(2..9)", "name": "a", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sc*random(1..9)", "name": "c", "description": ""}, "sc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sc", "description": ""}, "sc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sc1", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*b2=a1*b,b2+1,b2)", "name": "b1", "description": ""}, "that": {"templateType": "anything", "group": "Ungrouped variables", "definition": "lcm(abs(b),abs(b1))/abs(b1)", "name": "that", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(a)", "name": "a2", "description": ""}, "sb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sb", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..9)", "name": "b2", "description": ""}, "this": {"templateType": "anything", "group": "Ungrouped variables", "definition": "lcm(abs(b),abs(b1))/abs(b)", "name": "this", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sc1*random(1..9)", "name": "c1", "description": ""}, "aort": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b*b1>0,'take away the equation','add the equation')", "name": "aort", "description": ""}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b*b1>0,-1,1)", "name": "s6", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sb*random(1..9)", "name": "b", "description": ""}, "sa": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "sa", "description": ""}, "fromorto": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b*b1>0,'from','to')", "name": "fromorto", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "that", "this", "sc1", "s1", "s6", "a1", "aort", "a2", "b1", "b2", "sc", "sb", "sa", "fromorto", "c1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "showQuestionGroupNames": false, "variable_groups": [], "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "{c*b1-b*c1}/{b1*a-a1*b}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}, {"answer": "{c*a1-a*c1}/{b*a1-a*b1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input as a fraction or an integer not as a decimal
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1} \\end{eqnarray} \\]
\n $x=\\phantom{{}}$[[0]], $y=\\phantom{{}}$[[1]]
\n Input your answers as fractions or integers, not as decimals.
", "marks": 0}], "statement": "Solve the following simultaneous equations for $x$ and $y$. Input your answers as fractions or integers, not as decimals.
", "tags": ["checked2015", "equations", "linear", "mas1601", "MAS1601", "pair of linear equations", "simultaneous", "simultaneous linear equations", "solve linear equations", "solving equations", "Solving equations"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\n Added more tags.
\n Added description.
\n Checked calculation. OK.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Solve for $x$ and $y$: \\[ \\begin{eqnarray} a_1x+b_1y&=&c_1\\\\ a_2x+b_2y&=&c_2 \\end{eqnarray} \\]
"}, "advice": "\\[ \\begin{eqnarray} \\simplify[std]{{a}x+{b}y}&=&\\var{c}&\\mbox{ ........(1)}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1}&\\mbox{ ........(2)} \\end{eqnarray} \\]
To get a solution for $x$ multiply equation (1) by {this} and equation (2) by {that}
\n This gives:
\\[ \\begin{eqnarray} \\simplify[std]{{a*this}x+{b*this}y}&=&\\var{this*c}&\\mbox{ ........(3)}\\\\ \\simplify[std]{{a1*that}x+{b1*that}y}&=&\\var{that*c1}&\\mbox{ ........(4)} \\end{eqnarray} \\]
Now {aort} (4) {fromorto} equation (3) to get
\\[\\simplify[std]{({a*this}+{s6*a1*that})x={this*c}+{s6*that*c1}}\\]
And so we get the solution for $x$:
\\[x = \\simplify{{c*b1-b*c1}/{b1*a-a1*b}}\\]
Substituting this value into any of the equations (1) and (2) gives:
\\[y = \\simplify{{c*a1-a*c1}/{b*a1-a*b1}}\\]
You can check that these solutions are correct by seeing if they satisfy both equations (1) and (2) by substituting these values into the equations.
"}, {"name": "Complete the square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)-a^2", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1.0..4.5#0.5)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "s1", "b"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "(x+{a})^2+{b}", "musthave": {"message": "please input in the form $(x+a)^2+b$
", "showStrings": false, "partialCredit": 0, "strings": ["(", ")", "^"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input your answer in the form $(x+a)^2+b$.
", "showStrings": false, "partialCredit": 0, "strings": ["x^2", "x*x", "x x", "x(", "x*("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$\\simplify{x^2+{2*a}x+ {a^2+b}} = \\phantom{{}}$ [[0]].
", "steps": [{"type": "information", "prompt": "Given the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$:
\n 1. Complete the square for $\\simplify{x^2+{2*a}x}$, by comparing $\\simplify{x^2+{2*a}x}$ to $(x+B)^2=x^2+2Bx+B^2$. This will give us the value of $B$.
\n 2. Using your value of $B$, write $\\simplify{x^2+{2*a}x}$ as $(x+B)^2 - B^2$.
\n 3. Add $\\var{a^2+b}$ to both sides.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Put the following quadratic expression in the form $(x+B)^2+C$ for suitable numbers $B$ and $C$.
\n Note that you have to input your answer in the form $(x+B)^2+C$ and the numbers $B,\\;C$ must be input exactly (they can be entered as integers, as fractions or as exact decimals). If $C=0$, then it may be omitted in the answer.
", "tags": ["algebra", "algebraic manipulation", "checked2015", "complete the square", "completing the square", "quadratics", "SFY0001", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $B$ and $C$ such that $x^2+bx+c = (x+B)^2+C$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given the quadratic $q(x)=\\simplify{x^2+{2*a}x+ {a^2+b}}$ we complete the square by:
\n 1. Considering first $\\simplify{x^2+{2*a}x}$ and comparing to $(x+B)^2=x^2+2Bx+B^2$: both start with $x^2$ and we set $2B=\\var{2*a}$, so $B=\\var{a}$. Thus $\\simplify{(x+{a})^2=x^2+{2*a}x+{a}^2}$ and $\\simplify{x^2+{2*a}x=(x+{a})^2-{a}^2}$.
\n 2. It follows that $\\simplify{x^2+{2*a}x+{a^2+b}=(x+{a})^2-{a}^2}+\\simplify{{a^2+b}=(x+{a})^2+{b}}$.
\n 3. Hence we get \\[q(x) = \\simplify[all]{ (x+{a})^2+{b}}\\]
"}, {"name": "Equation of a line parallel to a given line, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-1,1)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(d1..11)", "name": "b", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+Random(1..4)*s1", "name": "c", "description": ""}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(b-d,c-a)", "name": "n1", "description": ""}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d+1", "name": "d1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)*random(1..4)", "name": "a", "description": ""}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b*c=a*d,1,gcd(n1,b*c-a*d))", "name": "n2", "description": ""}, "k1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b*c-a*d-b*h+d*h)/(c-a)", "name": "k1", "description": ""}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "h", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "name": "d", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b*c-a*d)/(c-a)", "name": "g", "description": ""}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except k1)", "name": "k", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b-d)/(a-c)", "name": "f", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h", "s1", "k1", "n1", "n2", "k", "d1"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({b-d}/{a-c})x+{b*h-d*h+c*k-a*k}/{c-a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "answersimplification": "std", "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "\n The equation of the line is of the form $y=mx+c$.
\n The gradient $m$ will be the same as the gradient of the line $\\displaystyle \\simplify{{(b-d)/n2}x+{(c-a)/n2}y={(b*c-a*d)/n2}}$, so start by calculating the gradient of the second line. Having calculated $m$, calculate the constant term $c$ by noting that $y=\\var{k}$ when $x=\\var{h}$.
\n ", "marks": 0, "scripts": {}}], "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n Find the equation of the straight line which:
\n \n - Is parallel to $\\displaystyle \\simplify{{(b-d)/n2}x+{(c-a)/n2}y={(b*c-a*d)/n2}}$.
\n
\n
\n \n - Passes through the point $(\\var{h},\\var{k})$.
\n
\n
\n Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\n Input $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\n If you input $m$ as a fraction, put brackets ( ) around the fraction. For example, if your answer for $m$ is $\\dfrac{-2}{3}$ and your answer for $c$ is $\\dfrac{7}{5}$, you should write $(-2/3)x+7/5$.
\n Click on Show steps if you need help, you will lose 1 mark if you do so.
\n \n ", "tags": ["MAS1602", "SFY0001", "checked2015", "equation of a straight line", "gradient of a line", "parallel line"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the equation of the straight line parallel to the given line that passes through the given point $(a,b)$.
"}, "advice": "\n The equation of the line is of the form $y=mx+c$.
\n The gradient $m$ will be the same as the gradient of the line $\\displaystyle \\simplify{{(b-d)/n2}x+{(c-a)/n2}y={(b*c-a*d)/n2}}$, which is $\\displaystyle m= \\simplify{{b-d}/{a-c}}$. We can calculate the constant term $c$ by noting that $y=\\var{k}$ when $x=\\var{h}$.
\n Using this we get:
\\[ \\begin{eqnarray} \\var{k}&=&\\simplify[std]{({b-d}/{a-c}){h}+c} \\Rightarrow\\\\ c&=&\\simplify[std]{{k}-({b-d}/{a-c}){h}={(b*h-d*h+c*k-a*k)}/{(c-a)}} \\end{eqnarray} \\]
\n Hence the equation of the line is
\\[y = \\simplify[std]{({b-d}/{a-c})x+{b*h-d*h+c*k-a*k}/{c-a}}\\]
\n \n "}, {"name": "Equation of a straight line", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1=d,b1+random(1..3),b1)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+Random(1..4)*s1", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b-d)/(a-c)", "description": "", "name": "f"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "d"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b*c-a*d)/(c-a)", "description": "", "name": "g"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)*random(1..4)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({b-d}/{a-c})x+{b*c-a*d}/{c-a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "steps": [{"type": "information", "prompt": "The equation of the line is of the form $y=mx+c$.
\n You are given the gradient $m$ and you can calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Find the equation of the straight line which:
\n \n - Has gradient $\\displaystyle \\simplify{{b-d}/{a-c}}$.
\n
\n
\n \n - Passes through the point $(\\var{a},\\var{b})$.
\n
\n
\n Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\n Input $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\n Click on Show steps if you need help, you will lose 1 mark if you do so.
", "tags": ["checked2015", "diagram", "equation of a straight line", "gradient of a line", "mas1601", "MAS1601", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\n Added tags.
\n Added description.
\n Checked calculation.OK.
\n Improved display in content areas. Corrected some minor typos.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the equation of a straight line which has a given gradient $m$ and passes through the given point $(a,b)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The equation of the line is of the form $y=mx+c$.
\n You are given the gradient $\\displaystyle m= \\simplify{{b-d}/{a-c}}$ and we can calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
\n Using this we get:
\\[ \\begin{eqnarray} \\var{b}&=&\\simplify[std]{({b-d}/{a-c}){a}+c} \\Rightarrow\\\\ c&=&\\simplify[std]{{b}-({b-d}/{a-c}){a}={(b*c-a*d)}/{(c-a)}} \\end{eqnarray} \\]
\n Hence the equation of the line is
\\[y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}\\]
"}, {"name": "Equation of a straight line, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1=d,b1+random(1..3),b1)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+Random(1..4)*s1", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b-d)/(a-c)", "description": "", "name": "f"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "d"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b*c-a*d)/(c-a)", "description": "", "name": "g"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)*random(1..4)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "s1", "b1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({b-d}/{a-c})x+{b*c-a*d}/{c-a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "steps": [{"type": "information", "prompt": "\n The equation of the line is of the form $y=mx+c$.
\n Calculate the gradient $m$ between the given points and then calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
\n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n Find the equation of the straight line which passes through the points $(\\var{a},\\var{b})$ and $(\\var{c},\\var{d})$:
\n \n
\n Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\n Input $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\n If you input $m$ as a fraction, put brackets ( ) around the fraction. For example, if your answer for $m$ is $\\dfrac{-2}{3}$ and your answer for $c$ is $\\dfrac{7}{5}$, you should write $(-2/3)x+7/5$.
\n Click on Show steps if you need help, you will lose 1 mark if you do so.
\n ", "tags": ["MAS1602", "SFY0001", "Steps", "checked2015", "equation of a straight line", "gradient of a line"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the equation of the straight line which passes through the points $(a,b)$ and $(c,d)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n The equation of the line is of the form $y=mx+c$.
\n Calculate the gradient $m=\\dfrac{\\var{d}-(\\var{b})}{\\var{c}-(\\var{a})}=\\dfrac{\\var{d-b}}{\\var{c-a}}$ between the given points and then calculate the constant term $c$ by noting that $y=\\var{b}$ when $x=\\var{a}$.
\n Using this we get:
\\[ \\begin{eqnarray} \\var{b}&=&\\simplify[std]{({b-d}/{a-c}){a}+c} \\Rightarrow\\\\ c&=&\\simplify[std]{{b}-({b-d}/{a-c}){a}={(b*c-a*d)}/{(c-a)}} \\end{eqnarray} \\]
\n Hence the equation of the line is
\\[y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}\\]
\n "}, {"name": "Equation of a straight line 0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(a,b)", "description": "", "name": "c1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(gcd(c1,c2)=1,c2,c2+1)", "description": "", "name": "c"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-11..11 except 0)", "description": "", "name": "c2"}}, "ungrouped_variables": ["a", "c2", "c1", "b", "c"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "({-a}/{b})x+{c}/{b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "Express the equation of the given line in the form $y=mx+c$.
\n \\[\\simplify[std]{{a}x+{b}y={c}}\\]
\n Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\n Input $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\n If you input $m$ as a fraction, put brackets ( ) around the fraction. For example, if your answer for $m$ is $-\\dfrac{2}{3}$ and your answer for $c$ is $\\dfrac{7}{5}$, you should write $(-2/3)x+7/5$.
", "tags": ["SFY0001", "checked2015", "equation of a straight line", "standard form"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express the equation of the given line in the form $y=mx+c$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n \n "}, {"name": "Equation of line perpendicular to given line, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(d1..11)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+Random(1..4)*s1", "description": "", "name": "c"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b-d)/(a-c)", "description": "", "name": "f"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "d+1", "description": "", "name": "d1"}, "n1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "gcd(b-d,c-a)", "description": "", "name": "n1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)*random(1..4)", "description": "", "name": "a"}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "k"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "h"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9)", "description": "", "name": "d"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(b*c-a*d)/(c-a)", "description": "", "name": "g"}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b*c=a*d,1,gcd(n1,b*c-a*d))", "description": "", "name": "n2"}}, "ungrouped_variables": ["a", "c", "b", "d", "g", "f", "h", "s1", "n1", "n2", "k", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "({a-c}/{d-b})x+{c*h-a*h+d*k-b*k}/{d-b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$y=\\;\\phantom{{}}$[[0]]
", "steps": [{"type": "information", "prompt": "\n The equation of the line is of the form $y=mx+c$.
\n The gradient $m$ will be the $\\dfrac{-1}{n}$ where $n$ is the gradient of the line $\\displaystyle \\simplify{{(b-d)/n2}x+{(c-a)/n2}y={(b*c-a*d)/n2}}$, so start by calculating the gradient of the second line. Having calculated $n$, calculate $m=\\dfrac{-1}{n}$ and finally calculate the constant term $c$ by noting that $y=\\var{k}$ when $x=\\var{h}$.
\n ", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n Find the equation of the straight line which:
\n \n - Is perpendicular to $\\displaystyle \\simplify{{(b-d)/n2}x+{(c-a)/n2}y={(b*c-a*d)/n2}}$.
\n
\n
\n \n - Passes through the point $(\\var{h},\\var{k})$.
\n
\n
\n Input your answer in the form $mx+c$ for suitable values of $m$ and $c$.
\n Input $m$ and $c$ as fractions or integers as appropriate and not as decimals.
\n If you input $m$ as a fraction, put brackets ( ) around the fraction. For example, if your answer for $m$ is $\\dfrac{-2}{3}$ and your answer for $c$ is $\\dfrac{7}{5}$, you should write $(-2/3)x+7/5$.
\n Click on Show steps if you need help, you will lose 1 mark if you do so.
\n \n ", "tags": ["MAS1602", "SFY0001", "checked2015", "equation of a straight line", "gradient of a line", "perpendicular line"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the equation of the straight line perpendicular to the given line that passes through the given point $(a,b)$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "\n The equation of the line is of the form $y=mx+c$.
\n The gradient $m$ will be the $\\dfrac{-1}{n}$ where $n$ is the gradient of the line $\\displaystyle \\simplify{{(b-d)/n2}x+{(c-a)/n2}y={(b*c-a*d)/n2}}$, which is $\\displaystyle n= \\simplify{{b-d}/{a-c}}$. Having calculated $n$, calculate $\\displaystyle m=\\dfrac{-1}{n} = \\simplify{{a-c}/{d-b}}$. We can calculate the constant term $c$ by noting that $y=\\var{k}$ when $x=\\var{h}$.
\n Using this we get:
\\[ \\begin{eqnarray} \\var{k}&=&\\simplify[std]{({a-c}/{d-b}){h}+c} \\Rightarrow\\\\ c&=&\\simplify[std]{{k}-({a-c}/{d-b}){h}={c*h-a*h+d*k-b*k}/{d-b}} \\end{eqnarray} \\]
\n Hence the equation of the line is
\\[y = \\simplify[std]{({a-c}/{d-b})x+{c*h-a*h+d*k-b*k}/{d-b}}\\]
\n "}, {"name": "Centre and radius of a circle", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "b"}, "r": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "description": "", "name": "r"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-5..5)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "r", "b"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "{a}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{b}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{r}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "The centre is ([[0]],[[1]]). The radius is [[2]].
", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n Find the centre and radius of the circle with equation
\n \\[\\simplify[std]{x^2+y^2 -{2*a}x -{2*b}y+{a^2+b^2-r^2}}=0.\\]
\n Input the coordinates of the centre and the radius as integers.
\n \n \n ", "tags": ["SFY0001", "centre", "checked2015", "equation of a circle", "radius", "standard form"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the centre and radius of a circle when given an equation in standard form.
"}, "advice": "\n Complete the square for each of $x$ and $y$: $~~~\\simplify[std]{x^2-{2*a}x =(x-{a})^2-{a^2}}$ and $\\simplify[std]{y^2 -{2*b}y=(y-{b})^2-{b^2}}$.
\n Therefore $\\simplify[std]{x^2+y^2 -{2*a}x -{2*b}y+{a^2+b^2-r^2}=(x-{a})^2-{a^2}+(y-{b})^2-{b^2}+{a^2+b^2-r^2}}$
\n and the equation of the circle becomes $\\simplify[std]{(x-{a})^2+(y-{b})^2={r^2}}$.
\n It follows that the centre is $(\\var{a}, \\var{b})$ and the radius is $\\var{r}$.
\n "}, {"name": "Complete the square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)-a^2*k", "description": "", "name": "b"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1.0..4.5#0.5)", "description": "", "name": "a"}, "k": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "name": "k"}}, "ungrouped_variables": ["a", "s1", "b", "k"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"answer": "{k}(x+{a})^2+{b}", "musthave": {"message": "please input in the form $a((x+B)^2+C)$
", "showStrings": false, "partialCredit": 0, "strings": ["(", ")", "^"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.0001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "Input your answer in the form $a((x+B)^2+C)$.
", "showStrings": false, "partialCredit": 0, "strings": ["x^2", "x*x", "x x", "x(", "x*("]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "$\\simplify{{k}x^2+{2*a*k}x+ {a^2*k+b}} = \\phantom{{}}$ [[0]].
", "steps": [{"type": "information", "prompt": "Given the quadratic $q(x)=\\simplify{{k}x^2+{2*a*k}x+ {a^2*k+b}}$ we complete the square by:
\n 1. Writing $\\simplify{{k}x^2+{2*a*k}x}$ as $\\var{k}(\\simplify{x^2+{2*a}x})$ and complete the square for $\\simplify{x^2+{2*a}x}$.
\n 2. Multiplying the completed square for $\\simplify{x^2+{2*a}x}$ by $\\var{k}$.
\n 3. Adding $\\var{a^2*k+b}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Put the following quadratic expression in the form $a(x+B)^2+C$ for suitable numbers $a$, $B$ and $C$.
\n Note that you have to input your answer in the form $a(x+B)^2+C$ and the numbers $a, \\;B,\\;C$ must be input exactly (as integers, fractions or exact decimals).
", "tags": ["algebra", "algebraic manipulation", "checked2015", "complete the square", "completing the square", "quadratics", "SFY0001", "Steps", "steps"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $a$, $B$ and $C$ such that $ax^2+bx+c = a(x+B)^2+C$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given the quadratic $q(x)=\\simplify{{k}x^2+{2*a*k}x+ {a^2*k+b}}$ we complete the square by:
\n 1. Considering first $\\simplify{{k}x^2+{2*a*k}x}$, taking out the factor $\\var{k}$ so that we get $\\var{k}(\\simplify{x^2+{2*a}x})$, and complete the square for $\\simplify{x^2+{2*a}x}$.
\n 2. Comparing $\\simplify{x^2+{2*a}x}$ to $(x+B)^2=x^2+2Bx+B^2$: both start with $x^2$ and we set $2B=\\var{2*a}$, so $B=\\var{a}$. Thus $\\simplify{(x+{a})^2=x^2+{2*a}x+{a}^2}$ and $\\simplify{x^2+{2*a}x=(x+{a})^2-{a}^2}$.
\n 3. It follows that $\\simplify{{k}x^2+{2*a*k}x ={k}((x+{a})^2-{a}^2)={k}(x+{a})^2-{a^2 *k}}$ .
\n 4. Therefore $\\simplify{{k}x^2+{2*a*k}x+ {a^2*k+b}={k}(x+{a})^2 +{b}} $.
\n "}, {"name": "Intersection of a straight line and a circle", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)", "description": "", "name": "s1"}, "q": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(aInput all numbers as fractions or integers as appropriate and not as decimals.", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{n}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{p}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "{q}", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "([[0]],[[1]]) and ([[2]],[[3]])
", "showCorrectAnswer": true, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n Find the points where:
\n \n - the straight line $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$
\n
\n
\n \n - and the circle $\\simplify[std]{x^2+y^2 -{2*k}x -{2*l}y+{2*k*a+2*l*b-a^2-b^2}}=0$
\n
\n
\n meet.\n
\n \n Enter the coordinates for the two points, starting with the point with the smaller $x$-coordinate.
\n Note that, for example, $-3$ is smaller than $-1$. Thus if the points of intersection were $(-1, 4)$ and $(-3,6)$ you would enter $(-3,6)$ for the first point and $(-1, 4)$ for the second.
\n Input the coordinates as integers.
\n \n \n ", "tags": ["SFY0001", "checked2015", "equation of a circle", "equation of a straight line", "quadratic equation"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Find the points of intersection of a straight line and a circle.
"}, "advice": "\n Substitute for $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$ in the equation of the circle.
\n We get $\\simplify[std]{x^2+(({b-d}/{a-c})x+{b*c-a*d}/{c-a})^2 -{2*k}x -{2*l}(({b-d}/{a-c})x+{b*c-a*d}/{c-a})+{2*k*a+2*l*b-a^2-b^2}}=0$.
\n Collecting terms we get $\\simplify[std]{{1+ (b-d)^2/(a-c)^2} x^2+{(-2)*(b-d)(b*c-a*d)/(a-c)^2-2*k -(2*l)*(b-d)/(a-c)}x+{(b*c-a*d)^2/(c-a)^2 -2*l*(b*c-a*d)/(c-a)+2*k*a+2*l*b-a^2-b^2}}=0$.
\n Solving this quadratic equation for $x$ gives solutions $x=\\var{a}$ and $x=\\var{c}$.
\n Substitute these values of $x$ into the equation $\\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$ to obtain the $y$-coordinates: $y=\\var{b}$ (when $x=\\var{a}$) and $y=\\var{d}$ (when $x=\\var{c}$).
\n Enter the coordinate pairs in the order $(\\var{m},\\var{n})$, $(\\var{p}, \\var{q})$.
\n "}, {"name": "Intersection of two circles in two points", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-1,1)", "description": "", "name": "s1"}, "q": {"group": "Ungrouped variables", "templateType": "anything", "definition": "if(a([[0]],[[1]]) and ([[2]],[[3]])", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{m}", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "showPreview": true, "unitTests": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "marks": 1, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}, {"answer": "{n}", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "showPreview": true, "unitTests": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "marks": 1, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}, {"answer": "{p}", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "showPreview": true, "unitTests": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "marks": 1, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}, {"answer": "{q}", "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "answerSimplification": "std", "showPreview": true, "unitTests": [], "notallowed": {"message": "Input all numbers as fractions or integers as appropriate and not as decimals.
", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "marks": 1, "scripts": {}, "vsetRangePoints": 5, "expectedVariableNames": [], "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "failureRate": 1, "showFeedbackIcon": true}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "\n Find the points where the circles:
\n \n - $\\simplify[std]{x^2+y^2 -{2*k2}x -{2*l2}y+{2*k2*a+2*l2*b-a^2-b^2}}=0$\n
\n
\n
\n \n - and $\\simplify[std]{x^2+y^2 -{2*k}x -{2*l}y+{2*k*a+2*l*b-a^2-b^2}}=0$
\n
\n
\n meet.\n
\n \n Enter the coordinates for the two points, starting with the point with the smaller $x$-coordinate.
\n Note that, for example, $-3$ is smaller than $-1$. Thus if the points of intersection were $(-1, 4)$ and $(-3,6)$ you would enter $(-3,6)$ for the first point and $(-1, 4)$ for the second.
\n Input the coordinates as integers.
\n \n \n ", "tags": ["checked2015", "equation of a circle", "equation of a straight line", "Equation of a straight line", "intersection of two circles", "quadratic equation"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find the points of intersection of two circles.
"}, "advice": "\n Subtract the equation for the first circle from the equation for the second. You get $\\simplify[std]{{2*k2-2*k}x+ {2*l2-2*l}y+{2*k*a+2*l*b-2*k2*a-2*l2*b}}=0$.
\n This can be rearranged as: $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$
\n Substitute for $y = \\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$ in the equation of the circle.
\n We get $\\simplify[std]{x^2+(({b-d}/{a-c})x+{b*c-a*d}/{c-a})^2 -{2*k}x -{2*l}(({b-d}/{a-c})x+{b*c-a*d}/{c-a})+{2*k*a+2*l*b-a^2-b^2}}=0$.
\n Collecting terms we get $\\simplify[std]{{1+ (b-d)^2/(a-c)^2} x^2+{(-2)*(b-d)(b*c-a*d)/(a-c)^2-2*k -(2*l)*(b-d)/(a-c)}x+{(b*c-a*d)^2/(c-a)^2 -2*l*(b*c-a*d)/(c-a)+2*k*a+2*l*b-a^2-b^2}}=0$.
\n Solving this quadratic equation for $x$ gives solutions $x=\\var{a}$ and $x=\\var{c}$.
\n Substitute these values of $x$ into the equation $\\simplify[std]{({b-d}/{a-c})x+{b*c-a*d}/{c-a}}$ to obtain the $y$-coordinates: $y=\\var{b}$ (when $x=\\var{a}$) and $y=\\var{d}$ (when $x=\\var{c}$).
\n Enter the coordinate pairs in the order $(\\var{m},\\var{n})$, $(\\var{p}, \\var{q})$.
\n "}, {"name": "Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-bb0", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..20)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(x4))", "description": "", "name": "c31"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(sqrt(x2))", "description": "", "name": "c2"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB3)", "description": "", "name": "t3"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c4..c5 except 0)", "description": "", "name": "c3"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "description": "", "name": "q3"}, "x4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(a3^2-b3^2)", "description": "", "name": "x4"}, "aa0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p0),4)", "description": "", "name": "aa0"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-cc0", "description": "", "name": "bb1"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-bb0-cc0", "description": "", "name": "aa1"}, "x5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3^2+b3^2", "description": "", "name": "x5"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r0),4)", "description": "", "name": "cc0"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(c01,c02)", "description": "", "name": "c1"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "bb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-CC3", "description": "", "name": "bb4"}, "check1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA0-BB0-CC0", "description": "", "name": "check1"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "temp1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a0*t0/s0", "description": "", "name": "temp1"}, "c01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(x1))", "description": "", "name": "c01"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc2)", "description": "", "name": "u2"}, "cc3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r3),4)", "description": "", "name": "cc3"}, "c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(c31,c32)", "description": "", "name": "c4"}, "bb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q3),4)", "description": "", "name": "bb3"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "description": "", "name": "p3"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa2)", "description": "", "name": "s2"}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC5)", "description": "", "name": "u5"}, "c02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(min(a0,b0)*0.05)", "description": "", "name": "c02"}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": "", "name": "q0"}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC3)", "description": "", "name": "u3"}, "cc5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(CC4,3)", "description": "", "name": "cc5"}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q0),4)", "description": "", "name": "bb0"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c1..c2 except 0)", "description": "", "name": "c0"}, "t5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB5)", "description": "", "name": "t5"}, "a0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a0"}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a0^2+b0^2", "description": "", "name": "x2"}, "s0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa0)", "description": "", "name": "s0"}, "cc2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(cc1,3)", "description": "", "name": "cc2"}, "c5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(sqrt(x5))", "description": "", "name": "c5"}, "b0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "b0"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb2)", "description": "", "name": "t2"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb0)", "description": "", "name": "t0"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": "", "name": "r3"}, "x1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(a0^2-b0^2)", "description": "", "name": "x1"}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc0)", "description": "", "name": "u0"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..20)", "description": "", "name": "a3"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA3)", "description": "", "name": "s3"}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(AA4,3)", "description": "", "name": "aa5"}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(min(a3,b3)*0.05)", "description": "", "name": "c32"}, "temp2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b0-temp1", "description": "", "name": "temp2"}}, "ungrouped_variables": ["c4", "s3", "cc0", "temp2", "temp1", "b0", "cc3", "b3", "u2", "q0", "q3", "c0", "cc5", "s2", "s0", "cc1", "u0", "u3", "cc2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "x2", "c31", "c32", "a0", "a3", "bb0", "s5", "c3", "c2", "c1", "x1", "c02", "x4", "x5", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "c5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "check1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{a0}", "maxValue": "{a0}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$A=\\var{AA0}$, $b=\\var{b0}$, $c=\\var{c0}$
\n Side length $a=$ [[0]]
", "steps": [{"type": "information", "prompt": "Use the Cosine Rule to find $a$: $a^2=b^2+c^2-2bc \\cos A$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$B=\\var{BB3}$, $a=\\var{a3}$, $c=\\var{c3}$
\n Side length $b=$ [[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "Suppose that $\\Delta ABC$ is a triangle with all interior angles $< \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following two sides and an angle, determine the third side length. Write down the side length as a whole number.
\n \n \n ![](\"resources/images/Triangle_1.PNG\"/)
", "tags": ["checked2015", "cosine rule", "Cosine Rule", "SFY0001", "Solving triangles", "Triangle", "Two sides and an angle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\tI want acute triangles with side lengths $a,b,c$. I need $|a^2-b^2|<c^2<a^2+b^2$ along with corresponding conditions on $a,b$. In fact the conditions $a^2-b^2<c^2<a^2+b^2$ and $b^2-a^2<c^2<a^2+b^2$ imply also the corresponding conditions on $a,b$. Thus the design of the question involves choosing $a,b$ and then choosing $c$ to meet the required condition. The integer $c$ is chosen randomly between the ceiling of $\\sqrt{|a^2-b^2|}$ and the floor of $\\sqrt{a^2+b^2}$. The first is no greater than the second because $\\max\\{a,b\\}$ lies between them; if $a=b$, then $\\sqrt{a^2+b^2} > 1$. The range of values for $a$ and $b$ may be changed according to taste without invalidating the question, but questions arise about accuracy. My calculations suggest that values of $a,b,c$ between 5 and 100 are safe, but I have been more conservative than that.
\n \t\t The second part tests the ability to apply the same principles as the first part but with a different orientation to the triangle: the first part seeks $b,C,c$ whereas the second seeks $b,A,a$.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Two questions testing the application of the Cosine Rule when given two sides and an angle. In these questions, the triangle is always acute and both of the given side lengths are adjacent to the given angle.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) Use the Cosine Rule to find $a$: $a^2=b^2+c^2-2bc \\cos A$.
\n \\[a^2=\\var{b0}^2+\\var{c0}^2-2 \\times \\var{b0}\\times\\var{c0} \\times \\cos (\\var{aa0})=\\var{b0^2}+\\var{c0^2}-\\var{2*b0*c0} \\times \\var{cos (aa0)}\\]
\n \\[=\\var{b0^2+c0^2-2*b0*c0* cos (aa0)}.\\]
\n Hence $a=\\sqrt{\\var{b0^2+c0^2-2*b0*c0* cos (aa0)}}=\\var{sqrt(b0^2+c0^2-2*b0*c0* cos (aa0))}$. To the nearest integer, this is $\\var{a0}$.
\n b) Use the Cosine Rule to find $b$: $b^2=a^2+c^2-2ac \\cos B$.
\n \\[b^2=\\var{a3}^2+\\var{c3}^2-2 \\times \\var{a3}\\times\\var{c3} \\times \\cos (\\var{bb3})=\\var{a3^2}+\\var{c3^2}-\\var{2*a3*c3} \\times \\var{cos (bb3)}\\]
\n \\[=\\var{a3^2+c3^2-2*a3*c3* cos (bb3)}.\\]
\n Hence $b=\\sqrt{\\var{a3^2+c3^2-2*a3*c3* cos (bb3)}}=\\var{sqrt(a3^2+c3^2-2*a3*c3* cos (bb3))}$. To the nearest integer, this is $\\var{b3}$.
"}, {"name": "Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle C calculated from A,B\n pi-aa0-bb0\n ", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(13..29)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a3^2+b3^2))+1", "description": "", "name": "c31"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB3)", "description": "", "name": "t3"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "check": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA0-BB0-CC0", "description": "", "name": "check"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c31..c32)", "description": "", "name": "c3"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "description": "", "name": "q3"}, "aa0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //The angle A\n precround(arccos(p0),4)\n ", "description": "", "name": "aa0"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle A calculated from B,C\n pi-bb0-cc0\n ", "description": "", "name": "aa1"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //The angle C\n precround(arccos(r0),4)\n ", "description": "", "name": "cc0"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "bb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-CC3", "description": "", "name": "bb4"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa2)", "description": "", "name": "s2"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle B calculated from A,C\n pi-aa0-cc0\n ", "description": "", "name": "bb1"}, "c01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a0^2+b0^2))+1", "description": "", "name": "c01"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc2)", "description": "", "name": "u2"}, "cc3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r3),4)", "description": "", "name": "cc3"}, "bb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q3),4)", "description": "", "name": "bb3"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "description": "", "name": "p3"}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA3)", "description": "", "name": "s3"}, "c02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a0+0.9*b0,b0+0.9*a0))", "description": "", "name": "c02"}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": "", "name": "q0"}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC3)", "description": "", "name": "u3"}, "cc5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(CC4,3)", "description": "", "name": "cc5"}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //The angle B\n precround(arccos(q0),4)\n ", "description": "", "name": "bb0"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c01..c02)", "description": "", "name": "c0"}, "t5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB5)", "description": "", "name": "t5"}, "a0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a0"}, "s0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa0)", "description": "", "name": "s0"}, "cc2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(cc1,3)", "description": "", "name": "cc2"}, "b0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(14..30)", "description": "", "name": "b0"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb2)", "description": "", "name": "t2"}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc0)", "description": "", "name": "u0"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb0)", "description": "", "name": "t0"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": "", "name": "r3"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a3"}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC5)", "description": "", "name": "u5"}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(AA4,3)", "description": "", "name": "aa5"}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a3+0.9*b3,b3+0.9*a3))", "description": "", "name": "c32"}}, "ungrouped_variables": ["s3", "cc0", "b0", "cc3", "b3", "cc2", "check", "q0", "q3", "cc5", "s2", "s0", "cc1", "u0", "u3", "u2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "c31", "c32", "a0", "a3", "s5", "c3", "c0", "c02", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "bb0"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{c0}", "maxValue": "{c0}", "marks": 2, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$a=\\var{a0}$, $b=\\var{b0}$, $C=\\var{CC0}$
\n Side length $c=$ [[0]]
", "steps": [{"type": "information", "prompt": "Use the Cosine Rule to find $c$: $c^2=a^2+b^2-2ab \\cos C$. Take care over the fact that $\\cos(\\var{cc0})$ is negative.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n Suppose that $\\Delta ABC$ is a triangle with $C> \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.
\n \n \n ![](\"resources/images/Triangle_2.PNG\"/)
\n ", "tags": ["checked2015", "cosine rule", "Cosine Rule", "SFY0001", "Solving triangles", "Triangle", "Two sides and an angle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\tI want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2<c^2<(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "A question testing the application of the Cosine Rule when given two sides and an angle. In this question, the triangle is always obtuse and both of the given side lengths are adjacent to the given angle (which is the obtuse angle).
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Use the Cosine Rule to find $c$: $c^2=a^2+b^2-2ab \\cos C$.
\n \\[c^2=\\var{a0}^2+\\var{b0}^2-2 \\times \\var{a0}\\times\\var{b0} \\times \\cos (\\var{cc0})=\\var{a0^2}+\\var{b0^2}-\\var{2*a0*b0} \\times (\\var{cos (cc0)})\\]
\n \\[=\\var{a0^2+b0^2-2*a0*b0* cos (cc0)}.\\]
\n Hence $a=\\sqrt{\\var{a0^2+b0^2-2*a0*b0* cos (cc0)}}=\\var{sqrt(a0^2+b0^2-2*a0*b0* cos (cc0))}$. To the nearest integer, this is $\\var{c0}$.
"}, {"name": "Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-bb0", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..20)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(x4))", "description": "", "name": "c31"}, "area": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(b0*c0*s0/2,3)", "description": "", "name": "area"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(sqrt(x2))", "description": "", "name": "c2"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB3)", "description": "", "name": "t3"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c4..c5 except 0)", "description": "", "name": "c3"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": "", "name": "r3"}, "x4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(a3^2-b3^2)", "description": "", "name": "x4"}, "aa0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p0),4)", "description": "", "name": "aa0"}, "bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-cc0", "description": "", "name": "bb1"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-bb0-cc0", "description": "", "name": "aa1"}, "x5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3^2+b3^2", "description": "", "name": "x5"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r0),4)", "description": "", "name": "cc0"}, "c1": {"templateType": 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[{"stepsPenalty": 1, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{aa0}-0.0001", "maxValue": "{aa0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb0}-0.0001", "maxValue": "{bb0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{cc0}-0.0001", "maxValue": "{cc0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "$a=\\var{a0}$, $b=\\var{b0}$, $c=\\var{c0}$
\n Angle $A=$ [[0]]
\n Angle $B=$ [[1]]
\n Angle $C=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Then use $\\cos^{-1}$ to find $A$. Apply similar rules to find $B$ and $C$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"stepsPenalty": 1, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{Area}-0.001", "maxValue": "{Area}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "The area is [[0]]
", "steps": [{"type": "information", "prompt": "The area uses any of the formulae $\\dfrac{1}{2}ac \\sin B$, $\\dfrac{1}{2}bc \\sin A$ or $\\dfrac{1}{2}ab \\sin C$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "Suppose that $\\Delta ABC$ is a triangle with all interior angles $< \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following three side lengths, determine the three angles using the Cosine Rule. Write down the angles (in radians) as decimals to 4dp. [Before submitting answers, you can check that the sum of the three angles is $\\pi$.]
\n \n Also calculate the area of the triangle, giving your answer as a decimal to 3dp.
\n \n \n
", "tags": ["Area of a triangle", "checked2015", "cosine rule", "Cosine Rule", "SFY0001", "Solving triangles", "Three side lengths", "Triangle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\tI want acute triangles with side lengths $a,b,c$. I need $|a^2-b^2|<c^2<a^2+b^2$ along with corresponding conditions on $a,b$. In fact the conditions $a^2-b^2<c^2<a^2+b^2$ and $b^2-a^2<c^2<a^2+b^2$ imply also the corresponding conditions on $a,b$. Thus the design of the question involves choosing $a,b$ and then choosing $c$ to meet the required condition. The integer $c$ is chosen randomly between the ceiling of $\\sqrt{|a^2-b^2|}$ and the floor of $\\sqrt{a^2+b^2}$. The first is no greater than the second because $\\max\\{a,b\\}$ lies between them; if $a=b$, then $\\sqrt{a^2+b^2} > 1$. The range of values for $a$ and $b$ may be changed according to taste without invalidating the question, but questions arise about accuracy. My calculations suggest that values of $a,b,c$ between 5 and 100 are safe, but I have been more conservative than that.
\n \t\t \t\t The second part tests the ability to apply the same principles as the first part but with a different orientation to the triangle: the first part seeks $b,C,c$ whereas the second seeks $b,A,a$.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "A question testing the application of the Cosine Rule when given three side lengths. In this question, the triangle is always acute. A secondary application is finding the area of a triangle.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "(a) Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Therefore
\n \\[\\cos A =\\dfrac{\\var{b0}^2+\\var{c0}^2-\\var{a0}^2}{2 \\times \\var{b0} \\times \\var{c0}}=\\dfrac{\\var{b0^2+c0^2-a0^2}}{\\var{2 *b0*c0}}\\]
\n \\[=\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)}\\]
\n and so $A=\\cos^{-1}(\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)})=\\var{aa0}$.
\n \n Similarly $\\cos B =\\dfrac{a^2+c^2-b^2}{2ac}$ and $\\cos C =\\dfrac{a^2+b^2-c^2}{2ab}$. So
\n \\[\\cos B =\\dfrac{\\var{a0}^2+\\var{c0}^2-\\var{b0}^2}{2 \\times \\var{a0} \\times \\var{c0}}=\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)}\\]
\n and so $B=\\cos^{-1}(\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)})=\\var{bb0}$.
\n \\[\\cos C =\\dfrac{\\var{a0}^2+\\var{b0}^2-\\var{c0}^2}{2 \\times \\var{a0} \\times \\var{b0}}=\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)}\\]
\n and so $C=\\cos^{-1}(\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)})=\\var{cc0}$.
\n (b) We can use any of the formulae $\\dfrac{1}{2}ac \\sin B$, $\\dfrac{1}{2}bc \\sin A$ or $\\dfrac{1}{2}ab \\sin C$ for the area. For example
\n \\[\\dfrac{1}{2}bc \\sin A = \\dfrac{1}{2} \\times \\var{b0} \\times \\var{c0} \\times \\sin \\var{aa0}\\]
\n \\[=\\dfrac{1}{2} \\times \\var{b0 * c0} \\times \\var{sin(aa0)}=\\var{Area}\\]
"}, {"name": "Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle C calculated from A,B\n pi-aa0-bb0\n \n ", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(13..29)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", 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"name": "u5"}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(AA4,3)", "description": "", "name": "aa5"}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a3+0.9*b3,b3+0.9*a3))", "description": "", "name": "c32"}}, "ungrouped_variables": ["s3", "cc0", "b0", "cc3", "b3", "cc2", "check", "q0", "q3", "area", "cc5", "s2", "s0", "cc1", "u0", "u3", "u2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "c31", "c32", "a0", "a3", "s5", "c3", "c0", "c02", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "bb0"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{aa0}-0.0001", "maxValue": "{aa0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb0}-0.0001", "maxValue": "{bb0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{cc0}-0.0001", "maxValue": "{cc0}+0.0001", "precision": 4, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "$a=\\var{a0}$, $b=\\var{b0}$, $c=\\var{c0}$
\n Angle $A=$ [[0]]
\n Angle $B=$ [[1]]
\n Angle $C=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Then use $\\cos^{-1}$ to find $A$. Apply similar rules to find $B$ and $C$. Take care over $C$: $\\cos C <0$, which means that $C$ lies between $\\dfrac{\\pi}{2}$ and $\\pi$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{Area}-0.001", "maxValue": "{Area}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "The area is [[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "Suppose that $\\Delta ABC$ is a triangle with $C> \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following three side lengths, determine the three angles using the Cosine Rule. Write down the angles (in radians) as decimals to 4dp. [Before submitting answers, you can check that the sum of the three angles is $\\pi$.]
\n \n Also calculate the area of the triangle, giving your answer as a decimal to 3dp.
\n \n
", "tags": ["Area of a triangle", "checked2015", "cosine rule", "Cosine Rule", "SFY0001", "Solving triangles", "Three side lengths", "Triangle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\tI want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2<c^2<(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$.
\n \t\t \t\t \n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "A question testing the application of the Cosine Rule when given three side lengths. In this question, the triangle is always obtuse. A secondary application is finding the area of a triangle.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "(a) Use the Cosine Rule to find $\\cos A$: $\\cos A =\\dfrac{b^2+c^2-a^2}{2bc}$. Therefore
\n \\[\\cos A =\\dfrac{\\var{b0}^2+\\var{c0}^2-\\var{a0}^2}{2 \\times \\var{b0} \\times \\var{c0}}=\\dfrac{\\var{b0^2+c0^2-a0^2}}{\\var{2 *b0*c0}}\\]
\n \\[=\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)}\\]
\n and so $A=\\cos^{-1}(\\var{(b0^2+c0^2-a0^2)/(2 *b0*c0)})=\\var{aa0}$.
\n \n Similarly $\\cos B =\\dfrac{a^2+c^2-b^2}{2ac}$ and $\\cos C =\\dfrac{a^2+b^2-c^2}{2ab}$. So
\n \\[\\cos B =\\dfrac{\\var{a0}^2+\\var{c0}^2-\\var{b0}^2}{2 \\times \\var{a0} \\times \\var{c0}}=\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)}\\]
\n and so $B=\\cos^{-1}(\\var{(a0^2+c0^2-b0^2)/(2 *a0*c0)})=\\var{bb0}$.
\n \\[\\cos C =\\dfrac{\\var{a0}^2+\\var{b0}^2-\\var{c0}^2}{2 \\times \\var{a0} \\times \\var{b0}}=\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)}\\]
\n and so $C=\\cos^{-1}(\\var{(a0^2+b0^2-c0^2)/(2 *a0*b0)})=\\var{cc0}$.
\n (b) We can use any of the formulae $\\dfrac{1}{2}ac \\sin B$, $\\dfrac{1}{2}bc \\sin A$ or $\\dfrac{1}{2}ab \\sin C$ for the area. For example
\n \\[\\dfrac{1}{2}bc \\sin A = \\dfrac{1}{2} \\times \\var{b0} \\times \\var{c0} \\times \\sin \\var{aa0}\\]
\n \\[=\\dfrac{1}{2} \\times \\var{b0 * c0} \\times \\var{sin(aa0)}=\\var{Area}\\]
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\n Side length $b=$ [[0]]
\n Angle $C=$ [[1]]
\n Side length $c=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 1, "showPrecisionHint": false}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{AA5}-0.001", "maxValue": "{AA5}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{a3}", "maxValue": "{a3}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n $B=\\var{BB3}$, $C=\\var{CC3}$, $c=\\var{c3}$
\n Side length $b=$ [[0]]
\n Angle $A=$ [[1]]
\n Side length $a=$ [[2]]
\n \n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "\n Suppose that $\\Delta ABC$ is a triangle with all interior angles $< \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.
\n \n \n
\n \n \n ", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "Solving triangles", "Triangle", "Two angles and a side"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "I want acute triangles with side lengths $a,b,c$. I need $|a^2-b^2|<c^2<a^2+b^2$ along with corresponding conditions on $a,b$. In fact the conditions $a^2-b^2<c^2<a^2+b^2$ and $b^2-a^2<c^2<a^2+b^2$ imply also the corresponding conditions on $a,b$. Thus the design of the question involves choosing $a,b$ and then choosing $c$ to meet the required condition. The integer $c$ is chosen randomly between the ceiling of $\\sqrt{|a^2-b^2|}$ and the floor of $\\sqrt{a^2+b^2}$. The first is no greater than the second because $\\max\\{a,b\\}$ lies between them; if $a=b$, then $\\sqrt{a^2+b^2} > 1$. The range of values for $a$ and $b$ may be changed according to taste without invalidating the question, but questions arise about accuracy. My calculations suggest that values of $a,b,c$ between 5 and 100 are safe, but I have been more conservative than that.
\n The second part tests the ability to apply the same principles as the first part but with a different orientation to the triangle: the first part seeks $b,C,c$ whereas the second seeks $b,A,a$.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Two questions testing the application of the Sine Rule when given two angles and a side. In this question, the triangle is always acute.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t0}}{\\var{s0}}=\\var{a0*t0/s0}$. The closest integer is then $\\var{b0}$.
\n Since $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC1}$. To 3dp, this gives $\\var{CC2}$.
\n We use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u2}}{\\var{s0}}=\\var{a0*u2/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC1}$ would give a slightly different long decimal value of $c$, but the integer value would be the same.
\n b) We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t3}}{\\var{u3}}=\\var{c3*t3/u3}$. The closest integer is then $\\var{b3}$.
\n Since $A+B+C=\\pi$, we calculate $A=\\pi-B-C=\\var{AA4}$. To 3dp, this gives $\\var{AA5}$.
\n We use the Sine Rule to find $a$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $a=\\dfrac{c \\sin A}{\\sin C}=\\dfrac{\\var{c3}* \\var{s5}}{\\var{u3}}=\\var{c3*s5/u3}$. The closest integer is then $\\var{a3}$. Note that this solution uses the 3dp value of $A$; the answer using $\\var{AA4}$ would give a slightly different long decimal value of $a$, but the integer value would be the same.
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"definition": "//The angle A\n precround(arccos(p0),4)", "description": "", "name": "aa0"}, "aa1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "//Angle A calculated from B,C\n pi-bb0-cc0", "description": "", "name": "aa1"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "//The angle C\n precround(arccos(r0),4)", "description": "", "name": "cc0"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "bb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-CC3", "description": "", "name": "bb4"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "s2": {"templateType": 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"anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA3)", "description": "", "name": "s3"}, "c02": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a0+0.9*b0,b0+0.9*a0))", "description": "", "name": "c02"}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": "", "name": "q0"}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC3)", "description": "", "name": "u3"}, "cc5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(CC4,3)", "description": "", "name": "cc5"}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "//The angle B\n precround(arccos(q0),4)", "description": "", "name": "bb0"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c01..c02)", "description": "", "name": "c0"}, "t5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB5)", "description": "", "name": "t5"}, "a0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a0"}, "s0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa0)", "description": "", "name": "s0"}, "cc2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(cc1,3)", "description": "", "name": "cc2"}, "b0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(14..30)", "description": "", "name": "b0"}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb2)", "description": "", "name": "t2"}, "u0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc0)", "description": "", "name": "u0"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb0)", "description": "", "name": "t0"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": "", "name": "r3"}, "aa4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-BB3-CC3", "description": "", "name": "aa4"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..25)", "description": "", "name": "a3"}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC5)", "description": "", "name": "u5"}, "aa5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(AA4,3)", "description": "", "name": "aa5"}, "c32": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(max(a3+0.9*b3,b3+0.9*a3))", "description": "", "name": "c32"}}, "ungrouped_variables": ["s3", "cc0", "b0", "cc3", "b3", "cc2", "check", "q0", "q3", "cc5", "s2", "s0", "cc1", "u0", "u3", "u2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "c31", "c32", "a0", "a3", "s5", "c3", "c0", "c02", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "bb0"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b0}", "maxValue": "{b0}", "marks": 1, "showPrecisionHint": false}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{CC2}-0.001", "maxValue": "{CC2}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{c0}", "maxValue": "{c0}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$A=\\var{AA0}$, $B=\\var{BB0}$, $a=\\var{a0}$
\n Side length $b=$ [[0]]
\n Angle $C=$ [[1]]
\n Side length $c=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 1, "showPrecisionHint": false}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{AA5}-0.001", "maxValue": "{AA5}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{a3}", "maxValue": "{a3}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "\n $B=\\var{BB3}$, $C=\\var{CC3}$, $c=\\var{c3}$
\n Side length $b=$ [[0]]
\n Angle $A=$ [[1]]
\n Side length $a=$ [[2]]
\n \n \n \n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Suppose that $\\Delta ABC$ is a triangle with $C> \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.
\n \n \n
", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "Solving triangles", "Triangle", "Two angles and a side"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "I want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2<c^2<(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$.
\n ", "licence": "Creative Commons Attribution 4.0 International", "description": "Two questions testing the application of the Sine Rule when given two angles and a side. In this question the triangle is obtuse. In one question, the two given angles are both acute. In the second, one of the angles is obtuse.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t0}}{\\var{s0}}=\\var{a0*t0/s0}$. The closest integer is then $\\var{b0}$.
\n Since $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC1}$. To 3dp, this gives $\\var{CC2}$.
\n We use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u2}}{\\var{s0}}=\\var{a0*u2/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC1}$ would give a slightly different long decimal value of $c$, but the integer value would be the same.
\n b) We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t3}}{\\var{u3}}=\\var{c3*t3/u3}$. The closest integer is then $\\var{b3}$.
\n Since $A+B+C=\\pi$, we calculate $A=\\pi-B-C=\\var{AA4}$. To 3dp, this gives $\\var{AA5}$.
\n We use the Sine Rule to find $a$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $a=\\dfrac{c \\sin A}{\\sin C}=\\dfrac{\\var{c3}* \\var{s5}}{\\var{u3}}=\\var{c3*s5/u3}$. The closest integer is then $\\var{a3}$. Note that this solution uses the 3dp value of $A$; the answer using $\\var{AA4}$ would give a slightly different long decimal value of $a$, but the integer value would be the same.
"}, {"name": "Apply the sine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"cc4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3", "description": "", "name": "cc4"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-bb0", "description": "", "name": "cc1"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..20)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(AA5)", "description": "", "name": "s5"}, "c31": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(x4))", "description": "", "name": "c31"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3*sin(cc3)/c3", "description": "", "name": "s4"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(sqrt(x2))", "description": "", "name": "c2"}, "check2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-BB3-CC3", "description": "", "name": "check2"}, "bb41": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-cc3-aa31", "description": "", "name": "bb41"}, "aa2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(aa1,3)", "description": "", "name": "aa2"}, "c3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c4..c5 except 0)", "description": "", "name": "c3"}, "r3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "description": 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"group": "Ungrouped variables", "definition": "sin(cc21)", "description": "", "name": "u21"}, "cc11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-aa0-bb01", "description": "", "name": "cc11"}, "cc0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r0),4)", "description": "", "name": "cc0"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(c01,c02)", "description": "", "name": "c1"}, "bb5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(BB4,3)", "description": "", "name": "bb5"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "bb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA3-CC3", "description": "", "name": "bb4"}, "bb01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arcsin(t1),3)", "description": "", "name": "bb01"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "aa3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(p3),4)", "description": "", "name": "aa3"}, "check1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "pi-AA0-BB0-CC0", "description": "", "name": "check1"}, "temp1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a0*t0/s0", "description": "", "name": "temp1"}, "c01": {"templateType": "anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(x1))", "description": "", "name": "c01"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc2)", "description": "", "name": "u2"}, "cc3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(r3),4)", "description": "", "name": "cc3"}, "c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "max(c31,c32)", "description": "", "name": "c4"}, "bb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q3),4)", "description": "", "name": "bb3"}, "p3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "description": "", "name": "p3"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(aa2)", "description": "", "name": "s2"}, "u5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC5)", "description": "", "name": "u5"}, "t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b0*sin(aa0)/a0", "description": "", "name": "t1"}, "bb51": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb41,3)", "description": "", "name": "bb51"}, "q0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "description": "", "name": "q0"}, "u3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(CC3)", "description": "", "name": "u3"}, "cc5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(CC4,3)", "description": "", "name": "cc5"}, "bb0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(q0),4)", "description": "", "name": "bb0"}, "c0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(c1..c2 except 0)", "description": "", "name": "c0"}, "t5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(BB5)", "description": "", "name": "t5"}, "a0": {"templateType": "anything", "group": 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"t0", "t1", "u5", "c02", "c5", "u21", "cc4", "cc11", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "check1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb01}-0.001", "maxValue": "{bb01}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{CC21}-0.001", "maxValue": "{CC21}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{c0}", "maxValue": "{c0}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$A=\\var{AA0}$, $a=\\var{a0}$, $b=\\var{b0}$
\n Angle $B=$ [[0]]
\n Angle $C=$ [[1]]
\n Side length $c=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Sine Rule to find $\\sin B$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$, and then find $B$. Remember that $A+B+C=\\pi$. Use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{aa31}-0.001", "maxValue": "{aa31}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "{bb51}-0.001", "maxValue": "{bb51}+0.001", "precision": 3, "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 1}, {"type": "numberentry", "correctAnswerFraction": false, "showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "integerAnswer": true, "integerPartialCredit": 0, "minValue": "{b3}", "maxValue": "{b3}", "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$a=\\var{a3}$, $C=\\var{CC3}$, $c=\\var{c3}$
\n Angle $A=$ [[0]]
\n Angle $B=$ [[1]]
\n Side length $b=$ [[2]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "Suppose that $\\Delta ABC$ is a triangle with all interior angles $< \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following angle and two side lengths, use the Sine Rule to determine the other side length and two angles. Write down the side length as a whole number and the angles (in radians) as decimals to 3dp.
\n \n \n
", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "Solving triangles", "Triangle", "Two sides and an angle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\tI want acute triangles with side lengths $a,b,c$. I need $|a^2-b^2|<c^2<a^2+b^2$ along with corresponding conditions on $a,b$. In fact the conditions $a^2-b^2<c^2<a^2+b^2$ and $b^2-a^2<c^2<a^2+b^2$ imply also the corresponding conditions on $a,b$. Thus the design of the question involves choosing $a,b$ and then choosing $c$ to meet the required condition. The integer $c$ is chosen randomly between the ceiling of $\\sqrt{|a^2-b^2|}$ and the floor of $\\sqrt{a^2+b^2}$. The first is no greater than the second because $\\max\\{a,b\\}$ lies between them; if $a=b$, then $\\sqrt{a^2+b^2} > 1$. The range of values for $a$ and $b$ may be changed according to taste without invalidating the question, but questions arise about accuracy. My calculations suggest that values of $a,b,c$ between 5 and 100 are safe, but I have been more conservative than that.
\n \t\t The second part tests the ability to apply the same principles as the first part but with a different orientation to the triangle: the first part seeks $b,C,c$ whereas the second seeks $b,A,a$.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Two questions testing the application of the Sine Rule when given two sides and an angle. In this question, the triangle is always acute and one of the given side lengths is opposite the given angle.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) We use the Sine Rule to find $B$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $\\sin B=\\dfrac{b \\sin A}{a}=\\dfrac{\\var{b0}* \\var{s0}}{\\var{a0}}=\\var{b0*s0/a0}$. To find $B$ we need to calculate $\\sin^{-1} (\\var{b0*s0/a0})$, calculating the angle between $0$ and $\\dfrac{\\pi}{2}$, so $B=\\var{bb01}$ (to 3 decimal places).
\n Since $A+B+C=\\pi$, we calculate $C=\\pi-A-B=\\var{CC11}$. To 3dp, this gives $\\var{CC21}$.
\n We use the Sine Rule to find $c$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $c=\\dfrac{a \\sin C}{\\sin A}=\\dfrac{\\var{a0}* \\var{u21}}{\\var{s0}}=\\var{a0*u21/s0}$. The closest integer is then $\\var{c0}$. Note that this solution uses the 3dp value of $C$; the answer using $\\var{CC11}$ would give a slightly different long decimal value of $c$, but the integer value would be the same.
\n b) We use the Sine Rule to find $A$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $\\sin A=\\dfrac{a \\sin C}{c}=\\dfrac{\\var{a3}* \\var{u3}}{\\var{c3}}=\\var{a3*u3/c3}$. To find $A$ we need to calculate $\\sin^{-1} (\\var{a3*u3/c3})$, calculating the angle between $0$ and $\\dfrac{\\pi}{2}$, so $A=\\var{aa31}$ (to 3 decimal places).
\n Since $A+B+C=\\pi$, we calculate $B=\\pi-A-C=\\var{bb41}$. To 3dp, this gives $\\var{bb51}$.
\n We use the Sine Rule to find $b$: $\\dfrac{b}{\\sin B}=\\dfrac{c}{\\sin C}$. Thus $b=\\dfrac{c \\sin B}{\\sin C}=\\dfrac{\\var{c3}* \\var{t51}}{\\var{u3}}=\\var{c3*t51/u3}$. The closest integer is then $\\var{b3}$. Note that this solution uses the 3dp value of $B$; the answer using $\\var{bb41}$ would give a slightly different long decimal value of $b$, but the integer value would be the same.
"}, {"name": "Apply the sine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"bb1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle B calculated from A,C\n pi-aa0-cc0\n ", "description": "", "name": "bb1"}, "cc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\n //Angle C calculated from A,B\n pi-aa0-bb0\n ", "description": "", "name": "cc1"}, "u2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(cc2)", "description": "", "name": "u2"}, "t0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sin(bb0)", "description": "", "name": "t0"}, "bb21": 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"anything", "group": "Ungrouped variables", "definition": "ceil(sqrt(a0^2+b0^2))+1", "description": "", "name": "c01"}, "p0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "description": "", "name": "p0"}, "r0": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "description": "", "name": "r0"}, "bb2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(bb1,3)", "description": "", "name": "bb2"}}, "ungrouped_variables": ["bb11", "cc1", "cc0", "b0", "cc2", "check", "q0", "s2", "s0", "u1", "u0", "u2", "aa1", "aa0", "aa2", "cc01", "bb21", "a0", "c0", "p0", "r0", "t2", "t0", "t21", "c01", "c02", "bb2", "bb1", "bb0"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 1, "scripts": {}, "gaps": 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"marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "$A=\\var{AA0}$, $a=\\var{a0}$, $c=\\var{c0}$
\n Angle $C=$ [[0]]
\n Angle $B=$ [[1]]
\n Side length $b=$ [[2]]
", "steps": [{"type": "information", "prompt": "Use the Sine Rule to find $\\sin C$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$, and then find $C$. Note that $C$ lies between $\\dfrac{\\pi}{2}$ and $\\pi$, whereas $\\sin^{-1}$ returns a value between $0$ and $\\dfrac{\\pi}{2}$; this means that $C$ will be $\\pi -$ (value calculated using $\\sin^{-1}$). Remember that $A+B+C=\\pi$. Use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$.
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n Suppose that $\\Delta ABC$ is a triangle with $C> \\dfrac{\\pi}{2}$ (so it is an obtuse triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).
\n Given the following two angles and a side length, determine the other two side lengths and the angle. Write down the side lengths as whole numbers and the angle (in radians) as a decimal to 3dp.
\n \n \n
\n ", "tags": ["checked2015", "SFY0001", "sine rule", "Sine Rule", "Solving triangles", "Triangle", "Two sides and an angle"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\tI want an obtuse triangle with side lengths $a,b,c$. I need $a^2+b^2<c^2<(a+b)^2$. I start with $c_1=ceil(\\sqrt{a^2+b^2})+1$, $c_2=\\max\\{b+0.9 a, a + 0.9 b\\}$ to establish a range of values for $c$ so that the triangle is neither too flat nor too close to a right-angled triangle. The upper limit ensures that $-\\cos C \\leq 0.9$ and so $\\sin C \\geq 0.435$. Specifying that $a \\leq 11b, b \\leq 11a$ ensures that $\\sin A, \\sin B$ are not too small and thereby ensures that percentage errors are below 0.5%. This last figure points to $a,b \\leq 100$ and there are benefits in $a,b \\geq 10$.
\n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "A question testing the application of the Sine Rule when given two sides and an angle. In this question the triangle is obtuse and the first angle to be found is obtuse.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "We use the Sine Rule to find $C$: $\\dfrac{a}{\\sin A}=\\dfrac{c}{\\sin C}$. Thus $\\sin C=\\dfrac{c \\sin A}{a}=\\dfrac{\\var{c0}* \\var{s0}}{\\var{a0}}=\\var{c0*s0/a0}$. To find $C$ we need to calculate $\\sin^{-1} (\\var{c0*s0/a0})$, calculating the angle between $\\dfrac{\\pi}{2}$ and $\\pi$, so $C=\\var{cc01}$ (to 3 decimal places). [Using a calculator, $\\sin^{-1} (\\var{c0*s0/a0})$ returns $\\var{arcsin(u1)}$, which lies between $0$ and $\\dfrac{\\pi}{2}$, so the value we need is $\\pi-\\var{arcsin(u1)}=\\var{pi-arcsin(u1)}$.]
\n Since $A+B+C=\\pi$, we calculate $B=\\pi-A-C=\\var{bb11}$. To 3dp, this gives $\\var{bb21}$.
\n We use the Sine Rule to find $b$: $\\dfrac{a}{\\sin A}=\\dfrac{b}{\\sin B}$. Thus $b=\\dfrac{a \\sin B}{\\sin A}=\\dfrac{\\var{a0}* \\var{t21}}{\\var{s0}}=\\var{a0*t21/s0}$. The closest integer is then $\\var{b0}$. Note that this solution uses the 3dp value of $B$; the answer using $\\var{bb11}$ would give a slightly different long decimal value of $b$, but the integer value would be the same.
\n "}, {"name": "Trigonometric functions of a triangle", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"prompt": "Use $\\sin^{-1}$.
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\n $\\theta=$ [[0]]
", "stepsPenalty": 1, "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "{t1}-0.001", "maxValue": "{t1}+0.001", "precision": 3, "unitTests": [], "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 2, "mustBeReducedPC": 0}], "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"prompt": "Use $\\cos^{-1}$.
", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "prompt": "Given $x=\\var{x2}, r=\\var{r2}$, find $\\theta$.
\n $\\theta=$ [[0]]
", "stepsPenalty": 1, "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "{t2}-0.001", "maxValue": "{t2}+0.001", "precision": 3, "unitTests": [], "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 2, "mustBeReducedPC": 0}], "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"prompt": "Use $\\tan^{-1}$.
", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "customMarkingAlgorithm": "", "type": "information", "extendBaseMarkingAlgorithm": true, "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "prompt": "Given $x=\\var{x3}, y=\\var{y3}$, find $\\theta$.
\n $\\theta=$ [[0]]
", "stepsPenalty": 1, "sortAnswers": false, "scripts": {}, "gaps": [{"precisionPartialCredit": 0, "mustBeReduced": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "{t3}-0.001", "maxValue": "{t3}+0.001", "precision": 3, "unitTests": [], "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 2, "mustBeReducedPC": 0}], "type": "gapfill", "unitTests": [], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variables": {"t1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arcsin(y1/r1),3)", "name": "t1", "description": ""}, "r1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(y1..20)", "name": "r1", "description": ""}, "t3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arctan(y3/x3),3)", "name": "t3", "description": ""}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(4..10 except y1)", "name": "x2", "description": ""}, "r2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(x2..20)", "name": "r2", "description": ""}, "t2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arccos(x2/r2),3)", "name": "t2", "description": ""}, "y1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(4..10)", "name": "y1", "description": ""}, "y3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..20)", "name": "y3", "description": ""}, "x3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(3..20)", "name": "x3", "description": ""}}, "ungrouped_variables": ["r1", "r2", "t2", "t3", "t1", "y1", "x2", "x3", "y3"], "functions": {}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "The following parts refer to a right-angled triangle with hypotenuse length denoted by $r$ and horizontal and vertical side lengths denoted by $x$ and $y$. The angle $\\theta$ is as indicated in the diagram below. Each part gives two side lengths and you are asked to deduce the size of the angle $\\theta$ using appropriate inverse trigonometrical functions. Express your answers in radians, written as decimals to 3dp.
\n ![](\"resources/images/Triangle_3.PNG\"/)
", "tags": ["arccos", "arcsin", "arctan", "checked2015", "Inverse trigonometrical functions", "Right-angled triangle", "triangle", "Triangle"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Questions on right-angled triangles asking for the calculation of angles using inverse-trigonometrical functions.
"}, "advice": "(a) $\\sin \\theta =\\dfrac{\\var{y1}}{\\var{r1}}$ so $\\theta= \\sin^{-1} \\left( \\dfrac{\\var{y1}}{\\var{r1}}\\right) = \\var{t1}$.
\n (b) $\\cos \\theta =\\dfrac{\\var{x2}}{\\var{r2}}$ so $\\theta= \\cos^{-1} \\left( \\dfrac{\\var{x2}}{\\var{r2}}\\right) = \\var{t2}$.
\n (c) $\\tan \\theta =\\dfrac{\\var{y3}}{\\var{x3}}$ so $\\theta= \\tan^{-1} \\left( \\dfrac{\\var{y3}}{\\var{x3}}\\right) = \\var{t3}$.
\n "}, {"name": "Combining algebraic fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "name": "s1"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sgn(c)*random(1..5 except [round(c*d/a2)])", "description": "", "name": "b2"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a2"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,-a])", "description": "", "name": "c"}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "name": "nb"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except [0,round(a*b/a1)])", "description": "", "name": "b1"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "description": "", "name": "d"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "description": "", "name": "a1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "b1", "b2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "marks": 0, "scripts": {}, "gaps": [{"answer": "({a*a2+a1*c}*x^2 + {b*c+a1*b2+b1*a2+a*d} * x + {b1 * d + b2 * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "musthave": {"message": "Input as a single fraction with the numerator as a quadratic and all terms expanded in the numerator.
", "showStrings": false, "partialCredit": 0, "strings": ["^"]}, "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "showCorrectAnswer": true, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "std", "variableReplacements": [], "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "Express \\[\\simplify{({a}x+{b1}) / ({a1}x + {b}) + ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\nNote: you do not need to expand the denominator, but you must enter the numerator as a polynomial in $x$.
\nInput the fraction here: [[0]]
\nClick on Show steps for more information. You will lose one mark if you do so.
", "steps": [{"prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / (b*d)}\\]
\nand for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nNote that in your answer you do not need to expand the denominator.
", "scripts": {}, "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0}], "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "stepsPenalty": 1}], "statement": "Add the following two fractions together and express as a single fraction over a common denominator.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator", "MAS1601", "mas1601"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t \t\t \t\t5/08/2012:
\n \t\t \t\t \t\t Added tags.
\n \t\t \t\t \t\t Added description.
\n \t\t \t\t \t\t Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t \t\t \t\t 12/08/2012:
\n \t\t \t\t \t\t Back to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\t \t\t Used the except feature of ranges to get non-degenerate examples.
\n \t\t \t\t \t\t Checked calculation.OK.
\n \t\t \t\t \t\t Improved display in content areas.
\n \t\t \t\t \n \t\t \n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{ax+b}{x + c} \\pm \\frac{dx+p}{x + q}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "let(\n qa,a*a2+a1*c,\n qb,b*c+a1*b2+b1*a2+a*d,\n qc,b1*d+b2*b,\n roots,[-b/a1,-d/a2],\n \n not (((-qb+sqrt(qb*qb+4*qa*qc))/(2*qa) in roots) or ((-qb-sqrt(qb*qb+4*qa*qc))/(2*qa) in roots))\n)", "maxRuns": "300"}, "advice": "The formula for {nb} fractions is :
\n\\[\\simplify[std]{a / b + {s1} * (c / d) = (a*d + {s1} * b*c) / b*d}\\]
\nand for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={abs(c)}x+{abs(b2)}}$, $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\nHence we have:
\\[\\begin{eqnarray*}\\simplify{({a}x+{b1}) / ({a1}*x + {b}) + ({c}x+{b2}) / ({a2}*x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+({a1*c}x^2+{b*c+a1*b2}x+{b*b2})) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2 + c*a1} * x^2 + {a * d +a1*b2+b1*a2+ c * b}x+{b1*d+b*b2}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Combining algebraic fractions 0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator"], "metadata": {"description": "Express $\\displaystyle a \\pm \\frac{c}{x + d}$ as an algebraic single fraction.
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Add the following together and express as a single algebraic fraction.
\n
", "advice": "We have:
\n \\[\\simplify[std]{{a} + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c}) / (({a2}*x + {d})) = ({a*a2} * x + {a * d + c}) / ( ({a2}*x + {d}))}\\]
", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"s1": {"name": "s1", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-9..9 except a2)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a,-a])", "description": "", "templateType": "anything"}, "nb": {"name": "nb", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "d", "nb", "a2", "s1"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "Express \\[\\simplify[std]{{a} + ({c} / ({a2}x + {d}))}\\] as a single algebraic fraction.
\n Input the fraction here: [[0]].
\n You can click on Show steps for help. You will lose 1 mark if you do so.
\n
", "stepsPenalty": 1, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "The formula for adding these expressions is :
\\[\\simplify[std]{a + {s1} * (c / d) = (ad + {s1} * bc) / d}\\]
\n and for this exercise we have $\\simplify{d={a2}x+{d}}$.
\n
"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 2, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a*a2}x+{a*d+c})/({a2}x+{d})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-05, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [10, 11], "checkVariableNames": false, "minlength": {"length": 12, "partialCredit": 0, "message": "Input as a single fraction.
"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}]}, {"name": "Combining algebraic fractions 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "name": "s1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "a", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b*a2/a1)])", "name": "d", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "b", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "name": "c", "description": ""}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "name": "nb", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a2", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "s1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "({a*a2 + c*a1} * x + {a * d + c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))", "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a single fraction.
", "showStrings": false, "partialCredit": 0, "strings": [")-", ")+"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "The formula for {nb} fractions is :
\\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
\n Note that in your answer you do not need to expand the denominator.
", "marks": 0, "scripts": {}}], "prompt": "Express \\[\\simplify{{a} / ({a1}x + {b}) + ({c} / ({a2}x + {d}))}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps if you need help.You will lose one mark if you do so.
\n
", "stepsPenalty": 1}], "statement": "\n Add the following two fractions together and express as a single fraction over a common denominator.
\n
\n ", "tags": ["SFY0001", "algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions", "common denominator"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t5/08/2012:
\n \t\t Added tags.
\n \t\t Added description.
\n \t\t Changed to two questions, for the numerator and denomimator, rather than one as difficult to trap student input for this example. Still some ambiguity however.
\n \t\t 12/08/2012:
\n \t\t Back to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t Used the except feature of ranges to get non-degenerate examples.
\n \t\t Checked calculation.OK.
\n \t\t Improved display in content areas.
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle \\frac{a}{x + b} \\pm \\frac{c}{x + d}$ as an algebraic single fraction over a common denominator.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The formula for {nb} fractions is :
\n \\[\\simplify[std]{a / b + {s1} * (c / d) = (ad + {s1} * bc) / bd}\\]
\n and for this exercise we have $\\simplify{b=x+{b}}$, $\\simplify{d=x+{d}}$.
Hence we have:
\\[\\simplify[std]{{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d})) = ({a} * ({a2}*x + {d}) + {c} * ({a1}*x + {b})) / (({a1}*x + {b}) * ({a2}*x + {d})) = ({a*a2 + c*a1} * x + {a * d + c * b}) / (({a1}*x + {b}) * ({a2}*x + {d}))}\\]
"}, {"name": "Combining algebraic fractions 3.0", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "name": "s1", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b2*a2/c)])", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except a)", "name": "c", "description": ""}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "name": "nb", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b1", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "name": "a", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b2", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a2", "description": ""}}, "ungrouped_variables": ["a", "c", "d", "s1", "a2", "b1", "b2", "nb"], "functions": {}, "parts": [{"customName": "", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "steps": [{"prompt": "Note that:
\\[\\simplify[std]{a + (c / d) = (a*d + c) / d}\\]
", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "useCustomName": false, "customName": "", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "type": "information", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}], "useCustomName": false, "prompt": "\n Express \\[\\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps to get more information. You will lose one mark if you do so.
\n
\n ", "stepsPenalty": 1, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "( {c+b1*a2} * x + {b1 * d + b2 })/ ( ({a2}*x + {d}))", "scripts": {}, "answerSimplification": "std", "useCustomName": false, "checkingType": "absdiff", "valuegenerators": [{"value": "", "name": "x"}], "vsetRange": [10, 11], "showFeedbackIcon": true, "type": "jme", "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 1e-05, "variableReplacements": [], "failureRate": 1, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "customName": "", "checkVariableNames": false, "unitTests": [], "mustmatchpattern": {"message": "Enter your answer as a single fraction.", "pattern": "?/(`!$n)", "partialCredit": 0, "nameToCompare": ""}, "vsetRangePoints": 5, "showPreview": true, "marks": 2}], "type": "gapfill", "unitTests": [], "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Express the following as a single fraction.
", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions"], "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle b+ \\frac{dx+p}{x + q}$ as an algebraic single fraction.
"}, "advice": "The formula for adding these expressions is:
\n \\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]
\n and for this exercise we have $\\simplify{a={b1}}$, $\\simplify{c={c}x+{b2}}$, $\\simplify{d={a2}x+{d}}$.
\n Hence we have:
\\[\\begin{eqnarray*} \\simplify[std]{{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \\simplify{(({b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({b1*a2}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ( {b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Combining algebraic fractions 3.1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,-1,1)", "name": "s1", "description": ""}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1", "name": "a2", "description": ""}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,round(b2*a2/c)])", "name": "d", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9 except a)", "name": "c", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b1", "description": ""}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(c<0,'taking away','adding')", "name": "nb", "description": ""}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "name": "b2", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "name": "a", "description": ""}}, "ungrouped_variables": ["a", "c", "d", "s1", "a2", "b1", "b2", "nb"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "({a*a2}*x^2 + {c+b1*a2+a*d} * x + {b1 * d + b2 })/ ( ({a2}*x + {d}))", "vsetrange": [10, 11], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a single fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["+(", "-(", ")+", ")-"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "Note that:
\\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]
\n
", "marks": 0, "scripts": {}}], "prompt": "Express \\[\\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})}\\] as a single fraction.
\n Input the fraction here: [[0]].
\n Click on Show steps to get more information. You will lose one mark if you do so.
\n
", "stepsPenalty": 1}], "statement": "Express the following as a single fraction.
\n
", "tags": ["SFY0001", "algebra", "algebraic fractions", "algebraic manipulation", "checked2015", "combining algebraic fractions"], "rulesets": {"std": ["all", "fractionNumbers", "!noLeadingMinus", "!collectNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "18/08/2012:
\n Added tags.
\n Added description.
\n Modified copy of Combining algebraic fractions 3.
\n Checked calculations.OK
\n
\n
", "licence": "Creative Commons Attribution 4.0 International", "description": "Express $\\displaystyle ax+b+ \\frac{dx+p}{x + q}$ as an algebraic single fraction.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The formula for adding these expressions is:
\n \\[\\simplify[std]{a + (c / d) = (ad + c) / d}\\]
\n and for this exercise we have $\\simplify{a={a}x+{b1}}$, $\\simplify{c={c}x+{b2}}$, $\\simplify{d={a2}x+{d}}$.
\n Hence we have:
\\[\\begin{eqnarray*} \\simplify[std]{{a}x+{b1} } +\\simplify[std]{ ({c}x+{b2}) / ({a2}x + {d})} &=& \\simplify{(({a}x+{b1}) * ({a2}*x + {d}) + ({c}x+{b2}) ) / ( ({a2}*x + {d}))}\\\\ &=&\\simplify[std]{ (({a*a2} * x^2 + {b1*a2+ a*d}x+{b1*d})+{c}x+{b2}) / ( ({a2}*x + {d}))}\\\\&=&\\simplify[std]{ ({a*a2} * x^2 + {a * d +b1*a2+ c }x+{b1*d+b2}) / (({a2}*x + {d}))}\\end{eqnarray*}\\]
"}, {"name": "Combining fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "name": "s1"}, "dosomething": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(s1<0,'Take away', 'Add')", "description": "", "name": "dosomething"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(f=2,1,f=3,random(1,2),f=4,random(1,3),f=5, random(1..4),f=6,random(1,5),f=7,random(1..6),f=8,random(1,3,5,7),f=9,random(1,2,4,5,7,8),f=10,random(1,3,7,9),f=11,random(1..10))", "description": "", "name": "b"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(g=2,random(3..11#2),g=3,random(2,4,5,7,8,10,11),g=4,random(3,5,7,9,11),g=5, random(2,3,4,6,7,8,9,11),g=6,random(5,7,11),g=7,random(2,3,4,5,6,8,9,10,11),g=8,random(3,5,7,9,11),g=9,random(2,4,5,7,8,10,11),g=10,random(3,7,9),g=11,random(2..10),g=12,random(5,7,11))", "description": "", "name": "f"}, "action1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(s1<0,'taking away', 'adding')", "description": "", "name": "action1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..11)", "description": "", "name": "a"}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(a=1, random(2..11),a=2,random(3..11#2),a=3,random(4,5,7,8,10,11),a=4,random(5,7,9,11),a=5, random(6,7,8,9,11),a=6,random(7,11),a=7,random(8,9,10,11),a=8,random(9,11),a=9,random(10,11),a=10,11,a=11,12)", "description": "", "name": "g"}, "action": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(s1<0,'Taking away', 'Adding')", "description": "", "name": "action"}}, "ungrouped_variables": ["a", "b", "g", "dosomething", "f", "s1", "action1", "action"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"stepsPenalty": 0.5, "scripts": {}, "gaps": [{"answer": "{a*f+s1*b*g}/{g*f}", "musthave": {"message": "You must write your answer in the form p/q for integers p and q
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", "showStrings": false, "partialCredit": 0, "strings": ["+", ".", "(", ")", "1-", "2-", "3-", "4-", "5-", "6-", "7-", "8-", "9-"]}, "showpreview": true, "maxlength": {"length": 7, "message": "answer too long
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "checkvariablenames": false, "type": "jme", "answersimplification": "std", "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "\\[\\simplify[std]{{a} / {g} + ({s1*b} / {f})}\\]
Input your answer here: [[0]]
\n No decimal numbers allowed.
\n Do not include brackets in your answer.
\n You can get help by clicking on Steps. If you do so you will lose 1/2 mark.
", "steps": [{"type": "information", "prompt": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
", "showCorrectAnswer": true, "scripts": {}, "marks": 0}], "showCorrectAnswer": true, "marks": 0}], "statement": "\n {dosomething} the following fractions and reduce the\n \n resulting fraction to lowest form.
Input your answer as a fraction and not\n \n as a decimal.
\n \n ", "tags": ["adding fractions", "checked2015", "combining fractions", "Fractions", "fractions", "lowest form of a fraction", "mas1601", "MAS1601", "subtracting fractions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "Putting something here so Loughborough doesn't break.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Add/subtract fractions and reduce to lowest form.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "The rule for {action1} fractions is \\[\\simplify{a/b+ {s1}*(c/d)=(a*d+{s1}*b*c)/(b*d)}.\\]
In this case we have:
\\[\\simplify[std,!unitFactor]{{a} / {g} + ({s1*b} / {f}) = ({a} * {f} + {g} * {s1*b}) / ({g} * {f}) ={a*f+s1*g*b}/{g*f}}.\\]
Note that this fraction is in its lowest form as there are no common factors in the denominator and the numerator.
"}, {"name": "Fractions: Lowest form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "variable_groups": [], "variables": {"d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(c[x]*a[x],x,0..3)", "name": "d", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(-9..-5),random(5..9),random(11..19),random(35..61)]", "name": "a", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(chcp(a[x],2,9,random(2..9)),x,0..2)+chcp(a[3],40,80,random(40..80))", "name": "b", "description": ""}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "[random(2..9),random(2..9),random(2..9),random(40..70)]", "name": "c", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "map(c[x]*b[x],x,0..3)", "name": "f", "description": ""}}, "ungrouped_variables": ["a", "c", "b", "d", "f"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {"chcp": {"type": "number", "language": "jme", "definition": "if(gcd(a,d)=1,d,chcp(a,b,c,random(b..c)))", "parameters": [["a", "number"], ["b", "number"], ["c", "number"], ["d", "number"]]}}, "showQuestionGroupNames": false, "parts": [{"marks": 0, "scripts": {}, "gaps": [{"answer": "{a[0]}/{b[0]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form without brackets.
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", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form without brackets.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 3, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"answer": "{a[2]}/{b[2]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form. Do not include brackets in your answer.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 4, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 0.5, "vsetrangepoints": 5}, {"answer": "{a[3]}/{b[3]}", "musthave": {"message": "Input as a fraction.
", "showStrings": false, "partialCredit": 0, "strings": ["/"]}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "Input as a fraction in lowest form. Do not include brackets in your answer.
", "showStrings": false, "partialCredit": 0, "strings": ["(", "."]}, "showpreview": true, "maxlength": {"length": 5, "message": "Input as a fraction in lowest form by cancelling common factors in the denominator and numerator.
", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1.5, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "steps": [{"type": "information", "showCorrectAnswer": true, "prompt": "Given a fraction $\\displaystyle \\frac{a}{b}$ then it is in lowest form if $a$ and $b$ have no common factors.
\n If $c$ was a common factor then we could cancel the $c$ and we have converted the fraction into a fraction with smaller numbers.
\n For example the fraction $\\displaystyle \\frac{18}{24}=\\frac{9 \\times 2}{12 \\times 2} = \\frac{9}{12}$ as we can cancel the common factor $2$.
\n But we are not yet finished as $\\displaystyle \\frac{9}{12}=\\frac{3 \\times 3}{4 \\times 3} = \\frac{3}{4}$ on cancelling the common factor $3$. We cannot go any further as $3$ and $4$ have no common factors (other than $1$, which is never considered as a factor).
\n Of course we could have spotted that $6$ was a common factor as $\\displaystyle \\frac{18}{24}=\\frac{3 \\times 6}{4 \\times 6}=\\frac{3}{4}$ , but it is perfectly OK to do it in stages as we did above. Just make sure that your final fraction does not have common factors.
\n
", "marks": 0, "scripts": {}}], "prompt": "$\\displaystyle \\simplify[noc]{{d[0]}/{f[0]}}\\;=$[[0]],$\\;\\;\\displaystyle \\simplify[noc]{{d[1]}/{f[1]}}\\;=$[[1]],$\\;\\;\\displaystyle \\simplify[noc]{{d[2]}/{f[2]}}\\;=$[[2]],$\\;\\;\\displaystyle \\simplify[noc]{{d[3]}/{f[3]}}\\;=$[[3]]
\n Input as fractions and do not include brackets in your answer.
\n You can click on Show steps for help. You will not lose any marks if you do.
", "stepsPenalty": 0}], "statement": "Reduce the following fractions to their lowest form.
", "tags": ["Fractions", "SFY0001", "cancellation", "cancelling", "cancelling ", "checked2015", "common factor", "denominator", "lowest form", "numerator"], "rulesets": {"noc": ["std", "!simplifyFractions"], "std": ["all", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "11/08/2012:
\n Added tags.
\n Added description.
\n Function chcp(a,b,c,d) gives number coprime to a in the range b..c, d is usually random(b..c) for redundant reasons!
\n Note that the answer is constrained by max length as well as requiring / and no brackets.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Reducing fractions to their lowest form by cancelling common factors in the numerator and denominator. There are 4 questions.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "We have:
\n $\\displaystyle \\simplify[noc]{{d[0]}/{f[0]}}=\\simplify[]{({a[0]}*{c[0]})/({b[0]}*{c[0]})}=\\simplify[all]{{a[0]}/{b[0]}}$. Common factor $\\var{c[0]}$.
\n $\\displaystyle \\simplify[noc]{{d[1]}/{f[1]}}=\\simplify[]{({a[1]}*{c[1]})/({b[1]}*{c[1]})}=\\simplify[all]{{a[1]}/{b[1]}}$. Common factor $\\var{c[1]}$.
\n $\\displaystyle \\simplify[noc]{{d[2]}/{f[2]}}=\\simplify[]{({a[2]}*{c[2]})/({b[2]}*{c[2]})}=\\simplify[all]{{a[2]}/{b[2]}}$. Common factor $\\var{c[2]}$.
\n $\\displaystyle \\simplify[noc]{{d[3]}/{f[3]}}=\\simplify[]{({a[3]}*{c[3]})/({b[3]}*{c[3]})}=\\simplify[all]{{a[3]}/{b[3]}}$. Common factor $\\var{c[3]}$.
"}, {"name": "Simplifying fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}], "parts": [{"scripts": {}, "gaps": [{"answer": "{g*f}/{a*f+s1*b*g}", "musthave": {"showStrings": false, "message": "You must write your answer in the form p/q for integers p and q
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", "partialCredit": 0}, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "std", "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "\\[\\simplify{{g} / ({a} + {s1} * ({b * g} / {f}))}\\]
Input your answer here: [[0]]
\n Your answer must be of the form a/b for suitable integers a and b. No decimal numbers allowed.
\n Do not include brackets in your answer.
", "marks": 0}], "variables": {"s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a*f+s*b*g=1,-s,s)", "name": "s1", "description": ""}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(f=2,1,f=3,random(1,2),f=4,random(1,3),f=5, random(1..4),f=6,random(1,5),f=7,random(1..5),f=8,random(1,3,5),f=9,random(1,2,4,5),f=10,random(1,3),f=11,random(1..5))", "name": "b", "description": ""}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(g=2,random(3..7#2),g=3,random(2,4,5),g=4,random(3,5),g=5, random(2,3,4),g=6,random(5,7),g=7,random(2,3,4),g=8,random(3,5,7),g=9,random(2,4,5),g=10,random(3,7),g=11,random(2..5))", "name": "f", "description": ""}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "name": "s", "description": ""}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..7)", "name": "a", "description": ""}, "g": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(a=1, random(2..7),a=2,random(3..7#2),a=3,random(4,5,7),a=4,random(5,7),a=5, random(6,7,8),a=6,random(7,11),a=7,random(8,9))", "name": "g", "description": ""}}, "ungrouped_variables": ["a", "b", "g", "f", "s1", "s"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Write the following expression as a single fraction in its lowest form:
", "tags": ["checked2015", "Fractions", "fractions", "lowest form", "mas1601", "MAS1601", "simplifying fractions"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "5/08/2012:
\n Added description.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\frac{a} {b + \\frac{c}{d}}$ as a single fraction in the form $\\displaystyle \\frac{p}{q}$ for integers $p$ and $q$.
"}, "advice": "We have:
\\[\\simplify[std]{{g} / ({a} + {s1} * ({b * g} / {f})) = {g} / (({a} * {f} + {s1} * {b * g}) / {f}) ={g} / (({a * f + s1 * b * g}) / {f})= ({f}*{g}) / ({a * f + s1 * b * g}) = ({g * f} / {(a * f + s1 * b * g)})}\\]
Here we use the result that dividing by a fraction $\\frac{a}{b}$ is the same as multiplying by $\\frac{b}{a}$.
The resulting fraction is in lowest form i.e. the top and bottom do not have a common factor.
"}], "pickQuestions": 0, "pickingStrategy": "all-ordered"}], "metadata": {"description": "Questions used in a university course titled \"Foundation mathematics\"", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "name": "Maria's copy of Foundation mathematics", "showQuestionGroupNames": false, "variable_groups": [], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}], "extensions": [], "custom_part_types": [], "resources": []}