// Numbas version: exam_results_page_options
{"name": "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": [{"parts": [{"variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "scripts": {}, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "prompt": "<p>Write down the inverse Laplace transform</p>\n<p>\\(x(t)=\\)&nbsp;[[0]]</p>", "marks": 0, "unitTests": [], "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "vsetRangePoints": 5, "scripts": {}, "failureRate": 1, "showFeedbackIcon": true, "variableReplacements": [], "expectedVariableNames": [], "checkingType": "absdiff", "type": "jme", "vsetRange": [0, 1], "marks": "2", "unitTests": [], "checkingAccuracy": 0.001, "extendBaseMarkingAlgorithm": true, "answer": "{A}*e^(-{a1}*t)+e^(-{b1}*t)({B}*cos({f}*t)+({C}-{B}*{b1})/{f}*sin({f}*t))", "checkVariableNames": false, "showPreview": true, "showCorrectAnswer": true}], "showCorrectAnswer": true}], "tags": [], "variable_groups": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Having found the partial fraction breakdown:</p>\n<p>\\(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>.</p>", "name": "Laplace transform: irreducible quadratic factor", "preamble": {"css": "", "js": ""}, "functions": {}, "extensions": [], "advice": "<p>\\(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>\\(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>\\(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}}\\)</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>", "variables": {"b1": {"definition": "random(6..12#1)", "description": "", "group": "Ungrouped variables", "name": "b1", "templateType": "randrange"}, "a1": {"definition": "random(2..6#1)", "description": "", "group": "Ungrouped variables", "name": "a1", "templateType": "randrange"}, "A": {"definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "name": "A", "templateType": "randrange"}, "c1": {"definition": "{b1}^2+{f}^2", "description": "", "group": "Ungrouped variables", "name": "c1", "templateType": "anything"}, "f": {"definition": "random(2..8#1)", "description": "", "group": "Ungrouped variables", "name": "f", "templateType": "randrange"}, "B": {"definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "name": "B", "templateType": "randrange"}, "C": {"definition": "random(2..14#1)", "description": "", "group": "Ungrouped variables", "name": "C", "templateType": "randrange"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["A", "a1", "B", "b1", "C", "c1", "f"], "rulesets": {}, "type": "question", "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/"}]}]}], "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/"}]}