// Numbas version: exam_results_page_options {"name": "Laplace: Inverse Laplace Completing the Square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Laplace: Inverse Laplace Completing the Square", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

\n

Determine the inverse Laplace Transform of the following using completion of the square.

\n

\n

\n

\\(F(s)=\\dfrac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\\)

\n

\n

", "advice": "

\\(Q(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\\)

\n

\\(Q(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}}\\)

\n

\\(Q(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}}\\)

\n

\\(q(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)\\)

", "rulesets": {}, "extensions": [], "variables": {"C": {"name": "C", "group": "Ungrouped variables", "definition": "random(2 .. 14#1)", "description": "", "templateType": "randrange"}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2 .. 6#1)", "description": "", "templateType": "randrange"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "{b1}^2+{f}^2", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(6 .. 12#1)", "description": "", "templateType": "randrange"}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange"}}, "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, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Write down the inverse Laplace transform

\n

\\(\\mathscr{L}^{-1}\\{F(s)\\}=\\) [[0]]

", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "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, "valuegenerators": [{"name": "t", "value": ""}]}], "sortAnswers": false}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}