// Numbas version: finer_feedback_settings
{"name": "Repeated linear factor partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b": {"templateType": "randrange", "definition": "random(5..8#1)", "description": "", "name": "b", "group": "Ungrouped variables"}, "T": {"templateType": "randrange", "definition": "random(1..20#1)", "description": "", "name": "T", "group": "Ungrouped variables"}, "a": {"templateType": "randrange", "definition": "random(1..4#1)", "description": "", "name": "a", "group": "Ungrouped variables"}, "R": {"templateType": "randrange", "definition": "random(2..12#1)", "description": "", "name": "R", "group": "Ungrouped variables"}}, "metadata": {"description": "<p>Manipulation of&nbsp;algebraic fractions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "T", "R"], "parts": [{"type": "gapfill", "marks": 0, "prompt": "<p>Express your answer as a sum of three fractions:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(I(s) =\\)&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "marks": "3", "answer": "(({T}-{a}*{R})/(({b}-{a})^2))/(s+{a})+((-{T}+{a}*{R})/(({b}-{a})^2))/(s+{b})+(({T}-{b}*{R})/(({a}-{b})))/(s+{b})^2", "expectedvariablenames": [], "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "checkingaccuracy": 0.001, "vsetrangepoints": 5, "showpreview": true, "checkvariablenames": false, "vsetrange": [0, 1], "showFeedbackIcon": true, "checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}}], "variablesTest": {"condition": "", "maxRuns": 100}, "extensions": [], "tags": [], "variable_groups": [], "advice": "<p>\\(\\frac{\\var{R}s+\\var{T}}{(s+\\var{a})(s+\\var{b})^2}=\\frac{A}{s+\\var{a}}+\\frac{B}{s+\\var{b}}+\\frac{C}{(s+\\var{b})^2}\\)</p>\n<p>Mutiply across by the denominator \\((s+\\var{a})(s+\\var{b})^2\\) to get</p>\n<p>\\(\\var{R}s+\\var{T}=A(s+\\var{b})^2+B(s+\\var{a})(s+\\var{b})+C(s+\\var{a})\\)</p>\n<p>let s = \\(-\\var{a}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{a}}+{T}}=\\simplify{(-{a}+{b})*(-{a}+{b})}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{{{R}*{-{a}}+{T}}/((-{a}+{b})*(-{a}+{b}))}\\)</p>\n<p>let s = \\(-\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{b}}+{T}}=\\simplify{(-{b}+{a})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\simplify{{{R}*{-{b}}+{T}}/((-{b}+{a}))}\\)</p>\n<p>coefficient of&nbsp;\\(s^2\\) =&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(0=A+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=-A\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{{{R}*{{a}}-{T}}/((-{a}+{b})*(-{a}+{b}))}\\)</p>", "preamble": {"css": "", "js": ""}, "functions": {}, "rulesets": {}, "statement": "<p>Determine the partial fraction breakdown of the following expression:</p>\n<p>&nbsp; &nbsp; &nbsp;\\(I(s)=\\frac{\\var{R}s+\\var{T}}{(s+\\var{a})(s+\\var{b})^2}\\)</p>", "name": "Repeated linear factor partial fractions", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}