// Numbas version: exam_results_page_options
{"name": "Binomial Expansion 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a1", "b1", "c1", "c2", "d1", "d3", "d2", "a2", "b2", "a3", "b3", "c3", "c4", "d4", "e1", "e2", "e3", "e4", "f1", "f2", "f3", "f4"], "name": "Binomial Expansion 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Recall the binomial theorem&nbsp;<img src=\"http://www.purplemath.com/modules/prob/binom01.gif\" width=\"182\" height=\"50\" border=\"0\" alt=\"(a + b)^n = sum[k=0,n][(n over k)a^(n-k)b^k]\"/>&nbsp;and use this to answer each part<span color=\"#FFFFFF\" face=\"Arial\" size=\"1\" style=\"color: #ffffff; font-family: Arial; font-size: xx-small;\"></span></p>\n<p>Alternatively, take out a factor of $a^n$ and recall the formula for $(1+\\frac{b}{a}x)^n$ to find the expansion as in part d)</p>", "rulesets": {}, "parts": [{"notallowed": {"message": "", "showStrings": false, "strings": [")^"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$(\\var{a2}+x)^{\\var{a1}}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "{a2}^{a1}+{a2}^({a1}-1){a1}x+{a1}{a1-1}/2 {a2}^({a1}-2)x^2 +{a1}{a1-1}{a1-2}/6 {a2}^({a1}-3)x^3 + {a1}{a1-1}{a1-2}{a1-3}/24{a2}^({a1}-4)x^4 + {a1}{a1-1}{a1-2}{a1-3}{a1-4}/120{a2}^({a1}-5)x^5", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "", "showStrings": false, "strings": ["^2"], "partialCredit": 0}}, {"notallowed": {"message": "", "showStrings": false, "strings": [")^"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$(\\var{b2}+x)^{\\var{b1}}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "{b2}^{b1}+{b2}^({b1}-1){b1}x+{b1}{b1-1}/2 {b2}^({b1}-2)x^2 +{b1}{b1-1}{b1-2}/6 {b2}^({b1}-3)x^3 + {b1}{b1-1}{b1-2}{b1-3}/24{b2}^({b1}-4)x^4 + {b1}{b1-1}{b1-2}{b1-3}{b1-4}/120{b2}^({b1}-5)x^5", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "", "showStrings": false, "strings": ["^2"], "partialCredit": 0}}, {"notallowed": {"message": "", "showStrings": false, "strings": [")^"], "partialCredit": 0}, "vsetrangepoints": 5, "prompt": "<p>$(\\var{c3}-\\var{c2}x)^{\\var{c1}}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "({c3})^{c1}-{c1}{c2}({c3})^({c1}-1)x+{c1}{c1-1}/2 ({c3})^({c1}-2)({c2}x)^2 -{c1}{c1-1}{c1-2}/6 ({c3})^({c1}-3)({c2}x)^3 + {c1}{c1-1}{c1-2}{c1-3}/24({c3})^({c1}-4)({c2}x)^4 - {c1}{c1-1}{c1-2}{c1-3}{c1-4}/120({c3})^({c1}-5)({c2}x)^5 ", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "", "showStrings": false, "strings": ["^2"], "partialCredit": 0}}, {"vsetrangepoints": 5, "prompt": "<p>$(\\var{d4}+\\frac{\\var{d3}}{2}x)^{{\\var{d1}}}$</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "{d4}^{d1}(1+{d1}{d2}x+{d1}{d1-1}/2 ({d2}x)^2 +{d1}{d1-1}{d1-2}/6 ({d2}x)^3 + {d1}{d1-1}{d1-2}{d1-3}/24({d2}x)^4 + {d1}{d1-1}{d1-2}{d1-3}{d1-4}/120({d2}x)^5 )", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "", "showStrings": false, "strings": ["^2"], "partialCredit": 0}}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>$\\simplify{({e2}x+{e3}y)^{{e1}}}$</p>", "expectedvariablenames": ["x", "y"], "checkingaccuracy": "0.01", "vsetrange": ["0.1", 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>Hint: If you wish to submit an answer with xy in it, then type x*y</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all,fractionnumbers", "scripts": {}, "marks": 1, "answer": "({e3}y)^{e1}+{e1}{e2}(x*({e3}y)^({e1}-1))+{e1}{e1-1}/2 ({e2}x)^2*({e3}y)^({e1}-2) +{e1}{e1-1}{e1-2}/6 ({e2}x)^3*({e3}y)^({e1}-3) + {e1}{e1-1}{e1-2}{e1-3}/24({e2}x)^4*({e3}y)^({e1}-4) + {e1}{e1-1}{e1-2}{e1-3}{e1-4}/120({e2}x)^5", "showCorrectAnswer": true, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "", "showStrings": false, "strings": ["^2"], "partialCredit": 0}}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "<p>$\\simplify{({f2}x-{f3}y)^{{f1}}}$</p>", "expectedvariablenames": ["x", "y"], "checkingaccuracy": "0.01", "vsetrange": ["0.1", 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>Hint: If you wish to submit an answer with xy in it, then type x*y</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "answersimplification": "all,fractionnumbers", "scripts": {}, "marks": 1, "answer": "(-{f3}y)^{f1}+{f1}{f2}(x*(-{f3}y)^({f1}-1))+{f1}{f1-1}/2 ({f2}x)^2*(-{f3}y)^({f1}-2) +{f1}{f1-1}{f1-2}/6 ({f2}x)^3*(-{f3}y)^({f1}-3) + {f1}{f1-1}{f1-2}{f1-3}/24({f2}x)^4*(-{f3}y)^({f1}-4) + {f1}{f1-1}{f1-2}{f1-3}{f1-4}/120({f2}x)^5", "showCorrectAnswer": true, "checkvariablenames": true, "checkingtype": "absdiff", "type": "jme", "musthave": {"message": "", "showStrings": false, "strings": ["^2"], "partialCredit": 0}}], "statement": "<p>Using the binomial theorem, or otherwise, expand the following</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"f1": {"definition": "random(2..5 except e1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "f2": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "f3": {"definition": "random(1..3 except f2)", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}, "f4": {"definition": "f2/f3", "templateType": "anything", "group": "Ungrouped variables", "name": "f4", "description": ""}, "b1": {"definition": "random(3..5 except a1)", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "b2": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b2", "description": ""}, "b3": {"definition": "1/b2", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "d4": {"definition": "random(2..5 except d3)", "templateType": "anything", "group": "Ungrouped variables", "name": "d4", "description": ""}, "d2": {"definition": "d3/(2d4)", "templateType": "anything", "group": "Ungrouped variables", "name": "d2", "description": ""}, "d3": {"definition": "random(1,3,5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d3", "description": ""}, "d1": {"definition": "random(3..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}, "e2": {"definition": "random(1..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "e2", "description": ""}, "a1": {"definition": "random(3..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "a3": {"definition": "1/a2", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "a2": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "a2", "description": ""}, "c3": {"definition": "random(2..9 except c2)", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "e4": {"definition": "e2/e3", "templateType": "anything", "group": "Ungrouped variables", "name": "e4", "description": ""}, "c1": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "e1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "e1", "description": ""}, "e3": {"definition": "random(1..3 except e2)", "templateType": "anything", "group": "Ungrouped variables", "name": "e3", "description": ""}, "c4": {"definition": "c2/c3", "templateType": "anything", "group": "Ungrouped variables", "name": "c4", "description": ""}, "c2": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}