// Numbas version: exam_results_page_options
{"name": "Partial Fractions: Simple Linear Factors [L5]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "name": "Partial Fractions: Simple Linear Factors [L5]", "statement": "\n    <p>&nbsp;</p>\n  <p>&nbsp;</p>\n    ", "preamble": {"js": "", "css": ""}, "tags": ["L5"], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Split $\\displaystyle \\frac{ax+b}{(cx + d)(px+q)}$ into partial fractions.</p>"}, "variable_groups": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "functions": {}, "advice": "<p>a)</p>\n<p>We use partial fractions to find $A$ and $B$ such that:&nbsp;<br/>\\[ \\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})}\\]</p>\n<p>Dividing&nbsp;both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1}x+{b})({a2}x+{d})&nbsp;)}\\;\\;$ we obtain:</p>\n<p>$\\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \\Rightarrow \\simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Constant term: $\\simplify[std]{ {d}*A+{b}*B={a*d+c*b}&nbsp;}$</p>\n<p>Coefficent $x$: $ \\simplify[std]{ {a2}A+{a1}B&nbsp;= {a*a2+c*a1}&nbsp;}$&nbsp;</p>\n<p>On solving these equations we obtain $A = \\var{a}$ and $B=\\var{c}$</p>\n<p>Which 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})}\\]</p>\n<p></p>\n<p>b)</p>\n<p>We use partial fractions to find $A$ and $B$ such that:&nbsp;<br/>\\[ \\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_})}\\]</p>\n<p>Dividing&nbsp;both sides of the equation by $\\displaystyle \\simplify[std]{1/( ({a1_}x+{b_})({a2_}x+{d_})&nbsp;)}\\;\\;$ we obtain:</p>\n<p>$\\simplify[std]{A*({a2_}x+{d_})+B*({a1_}x+{b_}) = {a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}} \\Rightarrow \\simplify[std]{({a2_}A+{a1_}B)*x+{d_}*A+{b_}*B={a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}}$</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Constant term: $\\simplify[std]{ {d_}*A+{b_}*B={a_*d_+c_*b_}&nbsp;}$</p>\n<p>Coefficent $x$: $ \\simplify[std]{ {a2_}A+{a1_}B&nbsp;= {a_*a2_+c_*a1_}&nbsp;}$&nbsp;</p>\n<p>On solving these equations we obtain $A = \\var{a_}$ and $B=\\var{c_}$</p>\n<p>Which 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_})}\\]</p>", "parts": [{"type": "gapfill", "unitTests": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "prompt": "<p>Split \\[\\simplify{({a*a2 + &nbsp;c*a1} * x + {a * d + &nbsp;c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetRange": [10, 11], "type": "jme", "unitTests": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answer": "{a} / ({a1}*x + {b}) + ({c} / ({a2}*x + {d}))", "checkingType": "absdiff", "marks": 2, "checkVariableNames": false, "vsetRangePoints": 5, "notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "partialCredit": 0, "strings": [")(", ")*("]}, "expectedVariableNames": [], "showFeedbackIcon": true, "variableReplacements": [], "checkingAccuracy": 1e-05, "variableReplacementStrategy": "originalfirst", "showPreview": true, "answerSimplification": "std", "showCorrectAnswer": true, "customMarkingAlgorithm": ""}], "showCorrectAnswer": true, "customMarkingAlgorithm": ""}, {"type": "gapfill", "unitTests": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "prompt": "<p>Split \\[\\simplify{({a_*a2_ + &nbsp;c_*a1_} * x + {a_ * d_ + &nbsp;c_ * b_})/ (({a1_}*x + {b_}) * ({a2_}*x + {d_}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "sortAnswers": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetRange": [10, 11], "type": "jme", "unitTests": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "failureRate": 1, "answer": "{a_} / ({a1_}*x + {b_}) + ({c_} / ({a2_}*x + {d_}))", "checkingType": "absdiff", "marks": 2, "checkVariableNames": false, "vsetRangePoints": 5, "notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "partialCredit": 0, "strings": [")(", ")*("]}, "expectedVariableNames": [], "showFeedbackIcon": true, "variableReplacements": [], "checkingAccuracy": 1e-05, "variableReplacementStrategy": "originalfirst", "showPreview": true, "answerSimplification": "std", "showCorrectAnswer": true, "customMarkingAlgorithm": ""}], "showCorrectAnswer": true, "customMarkingAlgorithm": ""}], "ungrouped_variables": ["a", "c", "b", "d", "nb", "a1", "a2", "s1", "a_", "c_", "b_", "d_", "nb_", "a1_", "a2_", "s1_"], "variables": {"d_": {"templateType": "anything", "group": "Ungrouped variables", "name": "d_", "description": "", "definition": "random(-9..9 except [0,round(b*a2/a1)])"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": "", "definition": "random(-9..9 except [0,round(b*a2/a1)])"}, "c_": {"templateType": "anything", "group": "Ungrouped variables", "name": "c_", "description": "", "definition": "random(-9..9 except 0)"}, "nb_": {"templateType": "anything", "group": "Ungrouped variables", "name": "nb_", "description": "", "definition": "if(c_<0,'taking away','adding')"}, "a_": {"templateType": "anything", "group": "Ungrouped variables", "name": "a_", "description": "", "definition": "random(1..9)"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": "", "definition": "random(1..9)"}, "a1_": {"templateType": "anything", "group": "Ungrouped variables", "name": "a1_", "description": "", "definition": "1"}, "s1_": {"templateType": "anything", "group": "Ungrouped variables", "name": "s1_", "description": "", "definition": "if(c_<0,-1,1)"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": "", "definition": " 1"}, "nb": {"templateType": "anything", "group": "Ungrouped variables", "name": "nb", "description": "", "definition": "if(c<0,'taking away','adding')"}, "a2_": {"templateType": "anything", "group": "Ungrouped variables", "name": "a2_", "description": "", "definition": "1"}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": "", "definition": "if(c<0,-1,1)"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": "", "definition": "random(-9..9 except 0)"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": "", "definition": "random(-9..9 except 0)"}, "b_": {"templateType": "anything", "group": "Ungrouped variables", "name": "b_", "description": "", "definition": "random(-9..9 except 0)"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": "", "definition": "1"}}, "type": "question", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Abbi Mullins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2466/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": 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