// Numbas version: exam_results_page_options
{"name": "Ugur's copy of Laplace transform: irreducible quadratic factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Laplace transform: irreducible quadratic factor", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>The Laplace Transfor of a function $x(t)$ is given by&nbsp;</p>\n<p>\\[X(s)= <br/>\\frac{\\simplify{{A} + {B}}s^2 + \\simplify{{A}*{b1}*2 +{B}*{a1} + {C}}s + \\simplify{{A}*{c1}+{C}*{a1}}}{s^3 +\\simplify{{a1}+{{b1}*2}}s^2 +\\simplify{{c1}+{{a1}*{b1}*2}}s + \\simplify{{a1}*{c1}}}.\\]</p>", "advice": "<p>a) Observe that&nbsp;</p>\n<p>\\[\\frac{\\simplify{{A}+{B}}s^2+\\simplify{{A}*{b1}*2 + {B}*{a1} + {C}}s + \\simplify{{A}*{c1}&nbsp; + {C}*{a1}}}{s^3 +\\simplify{{a1}+{{b1}*2}}s^2 +\\simplify{{c1}+{{a1}*{b1}*2}}s + \\simplify{{a1}*{c1}}} =\\frac{\\var{A}s^2+\\simplify{{A}*{b1}*2}s + \\simplify{{A}*{c1}} +\\var{B}s^2 + \\simplify{{B}*{a1}}s +\\var{C}s + \\simplify{{C}*{a1}}}{s^3 +\\simplify{{a1}+{{b1}*2}}s^2 +\\simplify{{c1}+{{a1}*{b1}*2}}s + \\simplify{{a1}*{c1}}}= \\\\[6mm]<br/>=\\frac{\\var{A}(s^2+\\simplify{{b1}*2}s+\\var{c1}) +(s+\\var{a1})(\\var{B}s+\\var{C})}{(s+\\var{a1})(s^2+\\simplify{{b1}*2}s+\\var{c1})} =\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\\]<br/><br/><br/><br/>b)<br/>We established&nbsp;<br/>\\[X(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\\]</p>\n<p>in the previous part.</p>\n<p>By Completing the square we get:</p>\n<p>\\[X(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}}.\\]</p>\n<p>Making it further resemble a transform in the Laplace Transfrom table we need to further manipulate the numerator of the second fraction, and then decompose it fruther:</p>\n<p>\\[X(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}(s+\\var{b1})-\\simplify{{B}*{b1}-{C}}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} = \\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}(s+\\var{b1})}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}} -\\frac{\\simplify{{B}*{b1}-{C}}}{\\sqrt{\\simplify{{c1}-{b1}^2}}}\\cdot\\frac{\\sqrt{\\simplify{{c1}-{b1}^2}}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}}.\\]</p>\n<p>Now, reading from the Fourier Transform table:</p>\n<p>\\[x(t)=\\var{A}e^{\\var{a1}t}+\\var{B}e^{-\\var{b1}t}\\cos\\left(\\sqrt{\\simplify{{c1}-{b1}^2}}t\\right)+\\frac{-\\simplify{{B}*{b1}-{C}}}{\\sqrt{\\simplify{{c1}-{b1}^2}}}e^{-\\var{b1}t}\\sin\\left(\\sqrt{\\simplify{{c1}-{b1}^2}}t\\right)\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(6 .. 12#1)", "description": "", "templateType": "randrange", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "{b1}^2+{f}^2", "description": "", "templateType": "anything", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "random(2 .. 14#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "a1", "B", "b1", "C", "c1", "f"], "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>Find the partial fraction decomposition of $X(s)$.</p>\n<p>$X(s) = $&nbsp;</p>\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({A}/(s+{a1})) +({B}*s+{C})/((s^2 + ({b1}*2)s)+{c1})", "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}, {"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>Write down the inverse Laplace transform</p>\n<p>\\(x(t)=\\)&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{A}*e^(-{a1}*t)+e^(-{b1}*t)({B}*cos({f}*t)+({C}-{B}*{b1})/{f}*sin({f}*t))", "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": "t", "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/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}