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