// Numbas version: exam_results_page_options
{"name": "Partial Fractions 3 - double root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "nb", "s1", "a_", "c_", "b_", "nb_", "a1_", "a2_", "s1_", "new", "a1", "a2", "a3", "d", "d_", "b1", "b2", "b3", "p", "q", "c1", "c2", "c3", "p1", "q1"], "name": "Partial Fractions 3 - double root", "tags": ["algebra", "algebraic fractions", "algebraic manipulation", "combining algebraic fractions", "common denominator"], "preamble": {"css": "", "js": ""}, "advice": "<p>a)</p>\n<p>We use partial fractions to find $A$, $B$ and $C$ such that:&nbsp;<br/>$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{A/(x+{a})+B/(x+{a})^2+C/(x+{b})}$</p>\n<p>Dividing&nbsp;both sides of the equation by $\\displaystyle \\simplify[std]{1/( (x+{a})^2(x+{b})&nbsp;)}\\;\\;$ we obtain:</p>\n<p>$ \\simplify{A(x+{a})(x+{b})+B(x+{b})+C(x+{a})^2 = {a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}$</p>\n<p>$\\Rightarrow&nbsp;\\simplify[std]{(A+C)x^2+({a+b}A+B+{2a}C)x+({a*b}A+{b}B+{a*a}C)={a1+a3}*x^2+{a1*a+a1*b+a2+2*a*a3}*x + {a1*a*b + a2*b + a3*a^2}}$</p>\n<p><strong>Identifying coefficients:</strong></p>\n<p>Coefficient $x^2$: $\\simplify[std]{A+C={a1+a3}&nbsp;}$</p>\n<p>Coefficent $x$: $ \\simplify[std]{ {a+b}A+B+{2a}C&nbsp;= {a1*a+a1*b+a2+2*a*a3}&nbsp;}$&nbsp;</p>\n<p>Constant term: $\\simplify{{a*b}A+{b}B+{a*a}C ={a1*a*b + a2*b + a3*a^2}}$</p>\n<p>On solving these equations we obtain $A = \\var{a1}$, $B=\\var{a2}$ and $C=\\var{a3}$</p>\n<p>Which gives:$\\simplify{({a1+a3}x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))} \\;\\;\\;=\\simplify{{a1}/(x+{a})+{a2}/(x+{a})^2+{a3}/(x+{b})}$</p>\n<p></p>\n<p>Apply same method to solve b) and c)</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>Split \\[\\simplify{(({a1+a3})x^2+{a1*a+a1*b+a2+2*a*a3} * x + {a1*a*b + a2*b + a3*a^2})/ ((x + {a})^2 * (x + {b}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "strings": [")(", ")*("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 1e-05, "vsetrange": [10, 11], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{a1} / (x + {a}) + {a2}/(x + {a})^2 + {a3} / (x + {b})", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Split \\[\\simplify{(({b1+b3})x^2+{b1*p+b1*q+b2+2*p*b3} * x + {b1*p*q + b2*q + b3*p^2})/ ((x + {p})^2 * (x + {q}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "strings": [")(", ")*("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 1e-05, "vsetrange": [10, 11], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{b1} / (x + {p}) + {b2}/(x + {p})^2 + {b3} / (x + {q})", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>Split \\[\\simplify{(({c1+c3})x^2+{c1*p1+c1*q1+c2+2*p1*c3} * x + {c1*p1*q1 + c2*q1 + c3*p1^2})/ ((x + {p1})^2 * (x + {q1}))}\\] into partial fractions.</p>\n<p>Input the partial fractions here: [[0]].</p>\n<p>&nbsp;</p>\n<p>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input as the sum of partial fractions.</p>", "showStrings": false, "strings": [")(", ")*("], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 1e-05, "vsetrange": [10, 11], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{c1} / (x + {p1}) + {c2}/(x + {p1})^2 + {c3} / (x + {q1})", "marks": 2, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "\n    <p>&nbsp;</p>\n  <p>&nbsp;</p>\n    ", "variable_groups": [], "variablesTest": {"maxRuns": "100000", "condition": "b1+b3=0\n"}, "variables": {"b1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "q1": {"definition": "random(-5..5 except 0 except p1)", "templateType": "anything", "group": "Ungrouped variables", "name": "q1", "description": ""}, "nb_": {"definition": "if(c_<0,'taking away','adding')", "templateType": "anything", "group": "Ungrouped variables", "name": "nb_", "description": ""}, "d_": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "d_", "description": ""}, "nb": {"definition": "if(c<0,'taking away','adding')", "templateType": "anything", "group": "Ungrouped variables", "name": "nb", "description": ""}, "c_": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c_", "description": ""}, "new": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "new", "description": ""}, "a_": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a_", "description": ""}, "s1": {"definition": "if(c<0,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1", "description": ""}, "a1": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "p1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p1", "description": ""}, "c3": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "c2": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "a": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(-9..9 except 0 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "a2_": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a2_", "description": ""}, "q": {"definition": "random(-5..5 except 0 except p)", "templateType": "anything", "group": "Ungrouped variables", "name": "q", "description": ""}, "p": {"definition": "random(-5..5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p", "description": ""}, "a2": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "s1_": {"definition": "if(c_<0,-1,1)", "templateType": "anything", "group": "Ungrouped variables", "name": "s1_", "description": ""}, "b_": {"definition": "random(-9..9 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "b_", "description": ""}, "a1_": {"definition": "1", "templateType": "anything", "group": "Ungrouped variables", "name": "a1_", "description": ""}}, "metadata": {"notes": "<p><strong>5/08/2012:</strong></p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>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.</p>\n<p><strong>12/08/2012:</strong></p>\n<p>Back to one input of a fraction and trapped input in Forbidden Strings.</p>\n<p>Used the except feature of ranges to get non-degenerate examples.</p>\n<p>Checked calculation.OK.</p>\n<p>Improved display in content areas.</p>", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}