// Numbas version: finer_feedback_settings
{"name": "simple linear factors partial fractions and delta function", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "extensions": [], "tags": [], "ungrouped_variables": ["a", "b", "c", "P", "R", "k", "k2"], "functions": {}, "name": "simple linear factors partial fractions and delta function", "advice": "<p>\\(X(s)=\\frac{\\var{R}s+\\var{P}}{(s+\\var{a})(s+\\var{b})(s+\\var{c})}+\\frac{\\var{k}e^{-\\var{k2}s}}{(s+\\var{a})(s+\\var{b})}\\)</p>\n<p>Take the first fraction and mutiply across by the denominator \\((s+\\var{a})(s+\\var{b})(s+\\var{c})\\) to get</p>\n<p>\\(\\var{R}s+\\var{P}=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}}+{P}}=\\simplify{(-{a}+{b})*(-{a}+{c})}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{{{R}*{-{a}}+{P}}/((-{a}+{b})*(-{a}+{c}))}\\)</p>\n<p>let s = \\(-\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{b}}+{P}}=\\simplify{(-{b}+{a})*(-{b}+{c})}B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{{{R}*{-{b}}+{P}}/((-{b}+{a})*(-{b}+{c}))}\\)</p>\n<p>let s = \\(-\\var{c}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{R}*{-{c}}+{P}}=\\simplify{(-{c}+{a})*(-{c}+{b})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\simplify{{{R}*{-{c}}+{P}}/((-{c}+{b})*(-{c}+{a}))}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>Take the second fraction, ignore the exponential function and&nbsp;mutiply across by the denominator \\((s+\\var{a})(s+\\var{b})\\) to get</p>\n<p>\\(\\var{k}=A(s+\\var{b})+B(s+\\var{a})\\)</p>\n<p>let s = \\(-\\var{a}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{k}=A(\\simplify{-{a}+{b}})+B(0)\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{{k}/({b}-{a})}\\)</p>\n<p>let s = \\(-\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{k}=A(0)+B(\\simplify{{a}-{b}})\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{{k}/({a}-{b})}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"text-align: right;\"></p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\"></p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variable_groups": [], "variables": {"b": {"definition": "random(5..8#1)", "templateType": "randrange", "name": "b", "group": "Ungrouped variables", "description": ""}, "R": {"definition": "random(2..12#1)", "templateType": "randrange", "name": "R", "group": "Ungrouped variables", "description": ""}, "a": {"definition": "random(1..4#1)", "templateType": "randrange", "name": "a", "group": "Ungrouped variables", "description": ""}, "k": {"definition": "random(2..16#1)", "templateType": "randrange", "name": "k", "group": "Ungrouped variables", "description": ""}, "P": {"definition": "random(1..20#1)", "templateType": "randrange", "name": "P", "group": "Ungrouped variables", "description": ""}, "k2": {"definition": "random(2..6#1)", "templateType": "randrange", "name": "k2", "group": "Ungrouped variables", "description": ""}, "c": {"definition": "random(9..15#1)", "templateType": "randrange", "name": "c", "group": "Ungrouped variables", "description": ""}}, "statement": "<p>Determine the partial fraction breakdown of the following expression:</p>\n<p>&nbsp; &nbsp; &nbsp;\\(X(s)=\\frac{\\var{R}s+\\var{P}}{(s+\\var{a})(s+\\var{b})(s+\\var{c})}+\\frac{\\var{k}e^{-\\var{k2}s}}{(s+\\var{a})(s+\\var{b})}\\)</p>\n<p>.</p>", "parts": [{"gaps": [{"expectedvariablenames": [], "checkingaccuracy": 0.001, "variableReplacements": [], "showCorrectAnswer": true, "checkvariablenames": false, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "answer": "(({P}-{a}*{R})/(({b}-{a})*({c}-{a})))/(s+{a})+(({P}-{b}*{R})/(({a}-{b})*({c}-{b})))/(s+{b})+(({P}-{c}*{R})/(({a}-{c})*({b}-{c})))/(s+{c})+e^((-{k2})*s)({k}/({b}-{a})/(s+{a})+({k}/({a}-{b})/(s+{b})))", "checkingtype": "absdiff", "showpreview": true, "scripts": {}, "showFeedbackIcon": true, "type": "jme", "vsetrange": [0, 1], "marks": "2"}], "type": "gapfill", "scripts": {}, "prompt": "<p>Express your answer as a sum of five fractions:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(X(s) =\\)&nbsp;[[0]]</p>", "variableReplacements": [], "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0}], "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "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/"}]}