// Numbas version: exam_results_page_options
{"name": "even power #1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "parts": [{"gaps": [{"strictPrecision": false, "precision": "3", "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "showPrecisionHint": true, "marks": "2", "correctAnswerFraction": false, "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "variableReplacements": [], "type": "numberentry", "precisionPartialCredit": 0, "maxValue": "{m}*{b}/{8}-{m}*{a}/{8}+{m}*sin(4*{k}*{a})/(32*{k})-{m}*sin(4*{k}*{b})/(32*{k})", "showFeedbackIcon": true, "allowFractions": false, "minValue": "{m}*{b}/{8}-{m}*{a}/{8}+{m}*sin(4*{k}*{a})/(32*{k})-{m}*sin(4*{k}*{b})/(32*{k})", "correctAnswerStyle": "plain", "mustBeReduced": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "type": "gapfill", "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>\\(Answer =\\)&nbsp;[[0]]</p>", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}], "name": "even power #1", "variablesTest": {"maxRuns": 100, "condition": ""}, "statement": "<p>Evaluate the following definite integral correct to three decimal places:</p>\n<p style=\"padding-left: 30px;\">\\(\\int_\\var{a}^\\var{b}\\var{m}sin^2(\\var{k}t)cos^2(\\var{k}t)dt\\)</p>", "ungrouped_variables": ["m", "a", "b", "n", "k"], "variables": {"n": {"definition": "random(2..7#1)", "description": "", "group": "Ungrouped variables", "name": "n", "templateType": "randrange"}, "b": {"definition": "random(1..1.7#0.1)", "description": "", "group": "Ungrouped variables", "name": "b", "templateType": "randrange"}, "m": {"definition": "random(2..9#1)", "description": "", "group": "Ungrouped variables", "name": "m", "templateType": "randrange"}, "k": {"definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "name": "k", "templateType": "randrange"}, "a": {"definition": "random(0..0.6#0.1)", "description": "", "group": "Ungrouped variables", "name": "a", "templateType": "randrange"}}, "rulesets": {}, "variable_groups": [], "functions": {}, "advice": "<p></p>\n<p>\\(\\int_\\var{a}^\\var{b}\\var{m}sin^2(\\var{k}t)cos^2(\\var{k}t)dt\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\var{m}\\left(\\frac{1}{2}(1-cos(\\simplify{2*{k}}t))\\right)\\left(\\frac{1}{2}(1+cos(\\simplify{2*{k}}t))\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{4}\\left(1-cos(\\simplify{2*{k}}t)+cos(\\simplify{2*{k}}t)-cos^2(\\simplify{2*{k}}t)\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{4}\\left(1-cos^2(\\simplify{2*{k}}t)\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{4}sin^2(\\simplify{2*{k}}t)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\int_\\var{a}^\\var{b}\\frac{\\var{m}}{4}\\frac{1}{2}\\left(1-cos(\\simplify{4*{k}}t)\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\var{m}}{8}t-\\frac{\\var{m}}{\\simplify{32*{k}}}sin(\\simplify{4*{k}}t)\\Big|_\\var{a}^\\var{b}\\)</p>\n<p style=\"padding-left: 30px;\">\\(=\\frac{\\simplify{{m}*{b}}}{8}-\\frac{\\var{m}}{\\simplify{32*{k}}}sin(\\simplify{4*{k}*{b}})-\\frac{\\simplify{{m}*{a}}}{8}+\\frac{\\var{m}}{\\simplify{32*{k}}}sin(\\simplify{4*{k}*{a}})\\)</p>\n<p></p>\n<p></p>", "extensions": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "tags": [], "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/"}]}