// Numbas version: exam_results_page_options
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"<p>$I_{\\var{E1},\\var{E2}}=\\displaystyle \\int_0^{\\pi/2} \\cos^\\var{E1}(x) \\sin^\\var{E2}(x), dx$.</p>"}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "{doublefact}({O1}-1)*{doublefact}({O2}-1)/({doublefact}({O1}+{O2}))", "prompt": "<p>$I_{\\var{O1},\\var{O2}}=\\displaystyle \\int_0^{\\pi/2} \\cos^\\var{O1}(x) \\sin^\\var{O2}(x), dx$.</p>"}, {"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "{doublefact}({E3}-1)*{doublefact}({O3}-1)/({doublefact}({E3}+{O3}))", "prompt": "<p>$I_{\\var{E3},\\var{O3}}=\\displaystyle \\int_0^{\\pi/2} \\cos^\\var{E3}(x) \\sin^\\var{O3}(x), dx$</p>"}], "functions": {"doublefact": {"parameters": [["n", "number"]], "language": "jme", "definition": "//double factorial: descending by 2\n// recursive as I cannot get it to work otherwise\nif(n-1<=0,1,n*doublefact(n-2))", "type": "number"}}, "name": "Using Reduction Formulas", "ungrouped_variables": ["E1", "E2", "E3", "O1r", "O2r", "O3", "O1", "O2"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Using the reduction formulas for products of powers or sin and cosine</p>"}, "extensions": [], "statement": "<p>Define $I_{m,n} =\\displaystyle \\int_0^{\\pi/2} \\cos^m(x) \\sin^n(x), dx$&nbsp;where&nbsp;$m$ and $n$ are non-negative integers.</p>\n<p>Using the results $I_{m,n} = \\dfrac{m-1}{m+n} I_{m-2,n} =\\dfrac{n-1}{m+n} I_{m,n-2}$</p>\n<p>or otherwise, find the exact values of&nbsp;</p>", "advice": "", "variables": {"O1": {"description": "", "definition": "2*O1r+1", "group": "Ungrouped variables", "name": "O1", "templateType": "anything"}, "O2": {"description": "<p>O2</p>", "definition": "2*O2r+1", "group": "Ungrouped variables", "name": "O2", "templateType": "anything"}, "O1r": {"description": "<p>1st odd, lacking a 1</p>", "definition": "random(0..4)", "group": "Ungrouped variables", "name": "O1r", "templateType": "anything"}, "O2r": {"description": "<p>2nd odd, lacks the 1</p>", "definition": "random(1..4 except O1r)", "group": "Ungrouped variables", "name": "O2r", "templateType": "anything"}, "E1": {"description": "<p>first even</p>", "definition": "2*random(1..5)", "group": "Ungrouped variables", "name": "E1", "templateType": "anything"}, "E3": {"description": "<p>3rd even</p>", "definition": "2*random(1..4)", "group": "Ungrouped variables", "name": "E3", "templateType": "anything"}, "E2": {"description": "<p>2nd even</p>", "definition": "2*random(1..5 except E1)", "group": "Ungrouped variables", "name": "E2", "templateType": "anything"}, "O3": {"description": "<p>3rd odd</p>", "definition": "1+2*random(1..3)", "group": "Ungrouped variables", "name": "O3", "templateType": "anything"}}, "contributors": [{"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}]}], "contributors": [{"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}