// Numbas version: exam_results_page_options
{"name": "Ugur's copy of Laplace transform #2 - Trig functions", "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 #2 - Trig functions", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>What is the Laplace transform for the function:</p>\n<p style=\"padding-left: 30px;\">\\(x(t)=\\var{a}\\sin({\\var{n}t})+\\var{b}\\cos({\\var{k}t}).\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "advice": "<p>By the linearity of Laplace Transform:</p>\n<p>\\[X(s)=\\mathcal L\\{\\var{a}\\sin({\\var{n}t})+\\var{b}\\cos({\\var{k}t})\\} = \\var{a}\\mathcal L\\{\\sin({\\var{n}t})\\}+\\var{b}\\mathcal L\\{\\cos({\\var{k}t})\\}.\\]</p>\n<p>From Laplace Transform table one can read&nbsp;</p>\n<p>\\[\\mathcal L\\{\\sin({\\var{n}t})\\} = \\frac{\\var{n}}{s^2 + \\var{n}^2} \\quad \\mbox{ and } \\quad \\mathcal L\\{\\cos({\\var{n}t})\\} = \\frac{s}{s^2 + \\var{k}^2}.\\]&nbsp;</p>\n<p>Hence&nbsp;</p>\n<p>\\[X(s)=\\mathcal L\\{\\var{a}\\sin({\\var{n}t})+\\var{b}\\cos({\\var{k}t})\\} = \\var{a}\\mathcal L\\{\\sin({\\var{n}t})\\}+\\var{b}\\mathcal L\\{\\cos({\\var{k}t})\\} = \\frac{\\simplify{{a}*{n}}}{s^2 + \\simplify{{n}^2}} + \\frac{\\var{b}s}{s^2 + \\simplify{{k}^2}}.\\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2 .. 5#1)", "description": "", "templateType": "randrange", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2 .. 12#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 11#1)", "description": "", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "k", "n"], "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>\\(X(s)=\\)&nbsp;[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{n}*{a}/(s^2+{n}^2)+{b}s/(s^2+{k}^2)", "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/"}, {"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/"}]}