// Numbas version: exam_results_page_options
{"name": "Partial fraction breakdown A,B & C: simple irreducible quadratic factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>Find the partial fraction breakdown of the compound fraction:</p>\n<p style=\"padding-left: 30px;\">\\(Q(s)=\\frac{\\var{F}s+\\var{H}}{(s+\\var{a1})(s^2+\\simplify{{b1}^2})}\\)</p>", "ungrouped_variables": ["a1", "b1", "F", "H"], "functions": {}, "variables": {"a1": {"group": "Ungrouped variables", "definition": "random(1..5#1)", "name": "a1", "templateType": "randrange", "description": ""}, "H": {"group": "Ungrouped variables", "definition": "random(2..12#1)", "name": "H", "templateType": "randrange", "description": ""}, "F": {"group": "Ungrouped variables", "definition": "random(1..6#1)", "name": "F", "templateType": "randrange", "description": ""}, "b1": {"group": "Ungrouped variables", "definition": "random(6..10#1)", "name": "b1", "templateType": "randrange", "description": ""}}, "extensions": [], "preamble": {"css": "", "js": ""}, "tags": [], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"gaps": [{"marks": "1", "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "(-{F}*{a1}+{H})/(({b1}^2+{a1}^2))", "minValue": "(-{F}*{a1}+{H})/(({b1}^2+{a1}^2))", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "mustBeReducedPC": 0, "mustBeReduced": false}, {"marks": 1, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "({F}*{a1}-{H})/(({b1}^2+{a1}^2))", "minValue": "({F}*{a1}-{H})/(({b1}^2+{a1}^2))", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "mustBeReducedPC": 0, "mustBeReduced": false}, {"marks": 1, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "maxValue": "({H}*{a1}+{F}*{b1}^2)/({b1}^2+{a1}^2)", "minValue": "({H}*{a1}+{F}*{b1}^2)/({b1}^2+{a1}^2)", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "variableReplacements": [], "mustBeReducedPC": 0, "mustBeReduced": false}], "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "prompt": "<p>If the partial fraction breakdown is given by:</p>\n<p style=\"padding-left: 30px;\">\\(Q(s) =\\frac{A}{s+\\var{a1}}+\\frac{Bs+C}{s^2+\\simplify{{b1}^2}}\\)</p>\n<p>Calculate the values of \\(A, B\\) and \\(C\\) and give your answers as fractions</p>\n<p style=\"padding-left: 30px;\">\\(A=\\) [[0]]</p>\n<p style=\"padding-left: 30px;\">\\(B=\\)&nbsp;[[1]]</p>\n<p style=\"padding-left: 30px;\">\\(C=\\)&nbsp;[[2]]</p>", "scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "showFeedbackIcon": true}], "name": "Partial fraction breakdown A,B & C: simple irreducible quadratic factor", "advice": "<p>\\(Q(s)=\\frac{\\var{F}s+\\var{H}}{(s+\\var{a1})(s^2+\\simplify{{b1}^2})}=\\frac{A}{s+\\var{a1}}+\\frac{Bs+C}{s^2+\\simplify{{b1}^2}}\\)</p>\n<p></p>\n<p>Multiply across by \\((s+\\var{a1})(s^2+\\simplify{{b1}^2})\\)</p>\n<p>\\(\\var{F}s+\\var{H}=A(s^2+\\simplify{{b1}^2})+Bs(s+\\var{a1})+C(s+\\var{a1})\\)</p>\n<p>let \\(s=-\\var{a1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}(-\\var{a1})+\\var{H}=A(\\simplify{{b1}^2+{a1}^2})+B(0)+C(0)\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{F}*(-{a1})+{H}}=\\simplify{({b1}^2+{a1}^2)}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\frac{\\simplify{-{F}*{a1}+{H}}}{\\simplify{({b1}^2+{a1}^2)}}\\)</p>\n<p>let \\(s=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{H}=A(\\simplify{{b1}^2})+B(0)+C(\\var{a1})\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{H}=(\\frac{\\simplify{-{F}*{a1}+{H}}}{\\simplify{({b1}^2+{a1}^2)}})*\\simplify{{b1}^2}+\\var{a1}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{\\simplify{{a1}^2*{H}+{b1}^2*{a1}*{F}}}{\\simplify{({b1}^2+{a1}^2)}}=\\var{a1}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\frac{\\simplify{{a1}*{H}+{b1}^2*{F}}}{\\simplify{({b1}^2+{a1}^2)}}\\)</p>\n<p>Compare coefficients of \\(s^2\\)</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=\\frac{\\simplify{{F}*{a1}-{H}}}{\\simplify{({b1}^2+{a1}^2)}}\\)</p>", "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/"}]}