// Numbas version: exam_results_page_options
{"name": "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": [{"variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "extensions": [], "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>", "ungrouped_variables": ["a1", "b1", "c1", "F", "G", "H"], "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>", "functions": {}, "parts": [{"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>", "unitTests": [], "customMarkingAlgorithm": "", "showFeedbackIcon": true, "variableReplacements": [], "sortAnswers": false, "type": "gapfill", "scripts": {}, "showCorrectAnswer": true, "gaps": [{"marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "unitTests": [], "mustBeReducedPC": "50", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "customMarkingAlgorithm": "", "type": "numberentry", "scripts": {}, "maxValue": "({F}*{a1}^2-{G}*{a1}+{H})/(({b1}-{a1})*({c1}-{a1}))", "minValue": "({F}*{a1}^2-{G}*{a1}+{H})/(({b1}-{a1})*({c1}-{a1}))", "correctAnswerFraction": true, "mustBeReduced": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true}, {"marks": 1, "variableReplacements": [], "correctAnswerStyle": "plain", "unitTests": [], "mustBeReducedPC": "50", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "customMarkingAlgorithm": "", "type": "numberentry", "scripts": {}, "maxValue": "({F}*{b1}^2-{G}*{b1}+{H})/(({a1}-{b1})*({c1}-{b1}))", "minValue": "({F}*{b1}^2-{G}*{b1}+{H})/(({a1}-{b1})*({c1}-{b1}))", "correctAnswerFraction": true, "mustBeReduced": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true}, {"marks": "1", "variableReplacements": [], "correctAnswerStyle": "plain", "unitTests": [], "mustBeReducedPC": "50", "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "customMarkingAlgorithm": "", "type": "numberentry", "scripts": {}, "maxValue": "({F}*{c1}^2-{G}*{c1}+{H})/(({a1}-{c1})*(-{c1}+{b1}))", "minValue": "({F}*{c1}^2-{G}*{c1}+{H})/(({a1}-{c1})*(-{c1}+{b1}))", "correctAnswerFraction": true, "mustBeReduced": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": false, "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true}], "variableReplacementStrategy": "originalfirst", "marks": 0, "extendBaseMarkingAlgorithm": true}], "rulesets": {}, "type": "question", "variables": {"b1": {"templateType": "randrange", "group": "Ungrouped variables", "name": "b1", "description": "", "definition": "random(6..10#1)"}, "F": {"templateType": "randrange", "group": "Ungrouped variables", "name": "F", "description": "", "definition": "random(1..6#1)"}, "c1": {"templateType": "randrange", "group": "Ungrouped variables", "name": "c1", "description": "", "definition": "random(11..16#1)"}, "a1": {"templateType": "randrange", "group": "Ungrouped variables", "name": "a1", "description": "", "definition": "random(1..5#1)"}, "G": {"templateType": "randrange", "group": "Ungrouped variables", "name": "G", "description": "", "definition": "random(2..10#1)"}, "H": {"templateType": "randrange", "group": "Ungrouped variables", "name": "H", "description": "", "definition": "random(2..12#1)"}}, "name": "Partial Fractions: simple linear factors", "variable_groups": [], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"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": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"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": "Kevin Bohan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3363/"}]}