// Numbas version: exam_results_page_options
{"name": "simple linear factors partial fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "simple linear factors partial fractions", "tags": [], "metadata": {"description": "<p>Manipulation of&nbsp;algebraic fractions</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Determine the partial fraction breakdown of the following expression:</p>\n<p>&nbsp; &nbsp; &nbsp;\\(I(s)=\\frac{\\var{R}s+\\var{T}}{(s+\\var{a})(s+\\var{b})(s+\\var{c})}\\)</p>", "advice": "<p>\\(\\frac{\\var{R}s+\\var{T}}{(s+\\var{a})(s+\\var{b})(s+\\var{c})}=\\frac{A}{s+\\var{a}}+\\frac{B}{s+\\var{b}}+\\frac{C}{s+\\var{c}}\\)</p>\n<p>Mutiply across by the denominator \\((s+\\var{a})(s+\\var{b})(s+\\var{c})\\) to get</p>\n<p>\\(\\var{R}s+\\var{T}=A(s+\\var{b})(s+\\var{c})+B(s+\\var{a})(s+\\var{c})+C(s+\\var{a})(s+\\var{b})\\)</p>\n<p>let s = \\(-\\var{a}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{a}}+{T}}=\\simplify{(-{a}+{b})*(-{a}+{c})}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{{{R}*{-{a}}+{T}}/((-{a}+{b})*(-{a}+{c}))}\\)</p>\n<p>let s = \\(-\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{b}}+{T}}=\\simplify{(-{b}+{a})*(-{b}+{c})}B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{{{R}*{-{b}}+{T}}/((-{b}+{a})*(-{b}+{c}))}\\)</p>\n<p>let s = \\(-\\var{c}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{c}}+{T}}=\\simplify{(-{c}+{a})*(-{c}+{b})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\simplify{{{R}*{-{c}}+{T}}/((-{c}+{b})*(-{c}+{a}))}\\)</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(9 .. 15#1)", "description": "", "templateType": "randrange", "can_override": false}, "T": {"name": "T", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 4#1)", "description": "", "templateType": "randrange", "can_override": false}, "k2": {"name": "k2", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "random(2 .. 12#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(5 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2 .. 16#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "T", "R", "k", "k2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Express your answer as a sum of three fractions:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(I(s) =\\)&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "(({T}-{a}*{R})/(({b}-{a})*({c}-{a})))/(s+{a})+(({T}-{b}*{R})/(({a}-{b})*({c}-{b})))/(s+{b})+(({T}-{c}*{R})/(({a}-{c})*({b}-{c})))/(s+{c})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "s", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "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/"}]}