// Numbas version: exam_results_page_options
{"name": "Using Reduction Formulas ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "parts": [{"scripts": {}, "showFeedbackIcon": true, "prompt": "<p>$I_{\\var{E1},\\var{E2}}=\\displaystyle \\int_0^{\\pi/2} \\cos^\\var{E1}(x) \\sin^\\var{E2}(x), dx$.</p>", "type": "jme", "checkingType": "absdiff", "useCustomName": false, "valuegenerators": [], "checkingAccuracy": 0.001, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "variableReplacements": [], "vsetRangePoints": 5, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "showPreview": true, "customName": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "answer": "{doublefact}({E1}-1)*{doublefact}({E2}-1)/({doublefact}({E1}+{E2}))*Pi/2", "failureRate": 1, "customMarkingAlgorithm": ""}, {"scripts": {}, "showFeedbackIcon": true, "prompt": "<p>$I_{\\var{O1},\\var{O2}}=\\displaystyle \\int_0^{\\pi/2} \\cos^\\var{O1}(x) \\sin^\\var{O2}(x), dx$.</p>", "type": "jme", "checkingType": "absdiff", "useCustomName": false, "valuegenerators": [], "checkingAccuracy": 0.001, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "variableReplacements": [], "vsetRangePoints": 5, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "showPreview": true, "customName": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "answer": "{doublefact}({O1}-1)*{doublefact}({O2}-1)/({doublefact}({O1}+{O2}))", "failureRate": 1, "customMarkingAlgorithm": ""}, {"scripts": {}, "showFeedbackIcon": true, "prompt": "<p>$I_{\\var{E3},\\var{O3}}=\\displaystyle \\int_0^{\\pi/2} \\cos^\\var{E3}(x) \\sin^\\var{O3}(x), dx$</p>", "type": "jme", "checkingType": "absdiff", "useCustomName": false, "valuegenerators": [], "checkingAccuracy": 0.001, "vsetRange": [0, 1], "marks": 1, "unitTests": [], "variableReplacements": [], "vsetRangePoints": 5, "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "showPreview": true, "customName": "", "showCorrectAnswer": true, "extendBaseMarkingAlgorithm": true, "answer": "{doublefact}({E3}-1)*{doublefact}({O3}-1)/({doublefact}({E3}+{O3}))", "failureRate": 1, "customMarkingAlgorithm": ""}], "functions": {"doublefact": {"language": "jme", "type": "number", "parameters": [["n", "number"]], "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))"}}, "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["E1", "E2", "E3", "O1r", "O2r", "O3", "O1", "O2"], "advice": "", "name": "Using Reduction Formulas ", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Using the reduction formulas for products of powers or sin and cosine</p>"}, "variables": {"E2": {"name": "E2", "templateType": "anything", "group": "Ungrouped variables", "definition": "2*random(1..5 except E1)", "description": "<p>2nd even</p>"}, "O2": {"name": "O2", "templateType": "anything", "group": "Ungrouped variables", "definition": "2*O2r+1", "description": "<p>O2</p>"}, "O2r": {"name": "O2r", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4 except O1r)", "description": "<p>2nd odd, lacks the 1</p>"}, "O1": {"name": "O1", "templateType": "anything", "group": "Ungrouped variables", "definition": "2*O1r+1", "description": ""}, "O1r": {"name": "O1r", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "<p>1st odd, lacking a 1</p>"}, "E1": {"name": "E1", "templateType": "anything", "group": "Ungrouped variables", "definition": "2*random(1..5)", "description": "<p>first even</p>"}, "E3": {"name": "E3", "templateType": "anything", "group": "Ungrouped variables", "definition": "2*random(1..4)", "description": "<p>3rd even</p>"}, "O3": {"name": "O3", "templateType": "anything", "group": "Ungrouped variables", "definition": "1+2*random(1..3)", "description": "<p>3rd odd</p>"}}, "tags": ["integration", "reduction formula"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "type": "question", "contributors": [{"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}]}], "contributors": [{"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}]}