// Numbas version: exam_results_page_options
{"name": "Ashley's copy of Partial Fractions: simple linear factors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"variableReplacements": [], "type": "gapfill", "customMarkingAlgorithm": "", "prompt": "<p>If the partial fraction breakdown is given by:</p>\n<p style=\"padding-left: 30px;\">\\(Q(s) =\\frac{A}{x+\\var{a1}}+\\frac{B}{x+\\var{b1}}+\\frac{C}{x+\\var{c1}}\\)</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>", "showFeedbackIcon": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"allowFractions": true, "variableReplacements": [], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "maxValue": "({F}*{a1}^2-{G}*{a1}+{H})/(({b1}-{a1})*({c1}-{a1}))", "showFeedbackIcon": false, "unitTests": [], "minValue": "({F}*{a1}^2-{G}*{a1}+{H})/(({b1}-{a1})*({c1}-{a1}))", "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": "50", "showCorrectAnswer": true, "correctAnswerFraction": true, "mustBeReduced": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1}, {"allowFractions": true, "variableReplacements": [], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "maxValue": "({F}*{b1}^2-{G}*{b1}+{H})/(({a1}-{b1})*({c1}-{b1}))", "showFeedbackIcon": false, "unitTests": [], "minValue": "({F}*{b1}^2-{G}*{b1}+{H})/(({a1}-{b1})*({c1}-{b1}))", "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": "50", "showCorrectAnswer": true, "correctAnswerFraction": true, "mustBeReduced": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1}, {"allowFractions": true, "variableReplacements": [], "type": "numberentry", "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "maxValue": "({F}*{c1}^2-{G}*{c1}+{H})/(({a1}-{c1})*(-{c1}+{b1}))", "showFeedbackIcon": false, "unitTests": [], "minValue": "({F}*{c1}^2-{G}*{c1}+{H})/(({a1}-{c1})*(-{c1}+{b1}))", "notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "mustBeReducedPC": "50", "showCorrectAnswer": true, "correctAnswerFraction": true, "mustBeReduced": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": "1"}], "marks": 0}], "variable_groups": [], "ungrouped_variables": ["a1", "b1", "c1", "F", "G", "H", "F2"], "advice": "<p>\\(\\frac{\\var{F}s^2+\\var{G}s+\\var{H}}{(S+\\var{a1})(s+\\var{b1})(s+\\var{c1})}=\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{s+\\var{c1}}\\)</p>\n<p>Multiply across by \\((s+\\var{a1})(s+\\var{b1})(s+\\var{c1})\\)</p>\n<p>\\(\\var{F}s^2+\\var{G}s+\\var{H}=A(s+\\var{b1})(s+\\var{c1})+B(s+\\var{a1})(s+\\var{c1})+C(s+\\var{a1})(s+\\var{b1})\\)</p>\n<p>let \\(s=-\\var{a1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}(-\\var{a1})^2+\\var{G}(-\\var{a1})+\\var{H}=A(\\simplify{{b1}-{a1}})(\\simplify{{c1}-{a1}})+B(0)+C(0)\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{F}*{a1}^2-{G}*{a1}+{H}}=\\simplify{({b1}-{a1})*({c1}-{a1})}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\frac{\\simplify{{F}*{a1}^2-{G}*{a1}+{H}}}{\\simplify{({b1}-{a1})*({c1}-{a1})}}\\)</p>\n<p>let \\(s=-\\var{b1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}(-\\var{b1})^2+\\var{G}(-\\var{b1})+\\var{H}=A(0)+B(\\simplify{-{b1}+{a1}})(\\simplify{{c1}-{b1}})+C(0)\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{F}*{b1}^2-{G}*{b1}+{H}}=\\simplify{(-{b1}+{a1})*({c1}-{b1})}B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\frac{\\simplify{{F}*{b1}^2-{G}*{b1}+{H}}}{\\simplify{(-{b1}+{a1})*({c1}-{b1})}}\\)</p>\n<p>let \\(s=-\\var{c1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}(-\\var{c1})^2+\\var{G}(-\\var{c1})+\\var{H}=A(0)+B(0)+C(\\simplify{-{c1}+{a1}})(\\simplify{-{c1}+{b1}})\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{F}*{c1}^2-{G}*{c1}+{H}}=\\simplify{(-{c1}+{a1})*(-{c1}+{b1})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\frac{\\simplify{{F}*{c1}^2-{G}*{c1}+{H}}}{\\simplify{(-{c1}+{a1})*(-{c1}+{b1})}}\\)</p>", "variables": {"F2": {"name": "F2", "templateType": "anything", "definition": "random(-5..5) ", "description": "", "group": "Ungrouped variables"}, "c1": {"name": "c1", "templateType": "randrange", "definition": "random(11..16#1)", "description": "", "group": "Ungrouped variables"}, "G": {"name": "G", "templateType": "anything", "definition": "random(filter(x<>(F|H),x,[-3,-2,-1,1,2,3,4,5,6]))", "description": "", "group": "Ungrouped variables"}, "F": {"name": "F", "templateType": "randrange", "definition": "random(-5..5#2)", "description": "", "group": "Ungrouped variables"}, "b1": {"name": "b1", "templateType": "randrange", "definition": "random(6..10#1)", "description": "", "group": "Ungrouped variables"}, "a1": {"name": "a1", "templateType": "randrange", "definition": "random(1..5#1)", "description": "", "group": "Ungrouped variables"}, "H": {"name": "H", "templateType": "anything", "definition": "random(filter(x<>(F),x,[-3,-2,-1,1,2,3,4,5,6]))", "description": "", "group": "Ungrouped variables"}}, "name": "Ashley's copy of Partial Fractions: simple linear factors", "statement": "<p>Find the partial fraction breakdown of the compound fraction:</p>\n<p style=\"padding-left: 30px;\">\\(Q(s)=\\frac{\\var{F}x^2+\\var{G}x+\\var{H}}{(x+\\var{a1})(x+\\var{b1})(x+\\var{c1})}\\)</p>", "preamble": {"css": "", "js": ""}, "rulesets": {}, "functions": {}, "tags": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "type": "question", "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Ashley Cusack", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1818/"}]}]}], "contributors": [{"name": "Angharad Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/315/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Ashley Cusack", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1818/"}]}