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