// Numbas version: exam_results_page_options {"name": "Maths Support: Partial fractions", "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "preventleave": false, "browse": true, "showfrontpage": false, "showresultspage": "never"}, "duration": 0.0, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "timing": {"timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "shufflequestions": false, "questions": [], "percentpass": 0.0, "allQuestions": true, "pickQuestions": 0, "type": "exam", "feedback": {"showtotalmark": true, "advicethreshold": 0.0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Partial Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}], "functions": {}, "ungrouped_variables": ["a", "c", "b", "d"], "tags": ["rebelmaths"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "
Factorise $\\simplify{ x^2 + {b+d}x + {b*d} }$
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "(x+{b})(x+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "Express $\\frac{\\simplify{{a+c}*x+{a*d+c*b}}}{\\simplify{ x^2 + {b+d}x + {b*d} }}$ in partial fractions.
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a}/(x+{b})+{c}/(x+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-5..5 except b 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}}, "metadata": {"description": "Splitting an algebraic fraction into partial fractions
\nrebelmaths
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Partial Fractions", "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/"}], "functions": {}, "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "advice": "\nWe use partial fractions to find $A$ and $B$ such that:
\\[ \\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))} \\;\\;\\;=\\simplify[std]{ A/({a1}x+{b})+B/({a2}x+{d})}\\]
Multiplying both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1}x+{b})({a2}x+{d}) )}\\;\\;$ we obtain:
\n$\\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \\Rightarrow \\simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$
\nIdentifying coefficients:
\nConstant term: $\\simplify[std]{ {d}*A+{b}*B={a*d+c*b} }$
\nCoefficent $x$: $ \\simplify[std]{ {a2}A+{a1}B = {a*a2+c*a1} }$
\nOn solving these equations we obtain $A = \\var{a}$ and $B=\\var{c}$
\nWhich gives:\\[ \\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))}\\;\\;= \\simplify[std]{{a}/({a1}x+{b})+{c}/({a2}x+{d})}\\]
\n ", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\nSplit \\[\\simplify{({a*a2 + c*a1} * x + {a * d + c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}\\] into partial fractions.
\nInput the partial fractions here: [[0]].
\n\n
\n ", "gaps": [{"notallowed": {"message": "
Input as the sum of partial fractions.
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\n ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..9)", "name": "a"}, "c": {"definition": "random(-9..9 except 0)", "name": "c"}, "b": {"definition": "random(-9..9 except 0)", "name": "b"}, "d": {"definition": "random(-9..9 except [0,round(b*a2/a1)])", "name": "d"}, "nb": {"definition": "if(c<0,'taking away','adding')", "name": "nb"}, "a1": {"definition": "random(1..8)", "name": "a1"}, "a2": {"definition": "random(1..8)", "name": "a2"}, "s1": {"definition": "if(c<0,-1,1)", "name": "s1"}}, "metadata": {"notes": "\n \t\t \t\t
5/08/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tAdded description.
\n \t\t \t\tChanged 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\t12/08/2012:
\n \t\t \t\tBack to one input of a fraction and trapped input in Forbidden Strings.
\n \t\t \t\tUsed the except feature of ranges to get non-degenerate examples.
\n \t\t \t\tChecked calculation.OK.
\n \t\t \t\tImproved display in content areas.
\n \t\t \n \t\t", "description": "Split $\\displaystyle \\frac{ax+b}{(cx + d)(px+q)}$ into partial fractions.
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