// Numbas version: exam_results_page_options
{"name": "De Moivre's Theorem: $n$th roots of a complex number $(-\\pi\\leq \\theta \\leq \\pi)$", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["q1", "argrn", "q3", "q2", "q4", "md1", "arg1", "s2", "s1", "targ1", "n", "a1", "m1", "t", "tol", "ans1", "gap", "adj", "z1", "argr1", "b1"], "extensions": [], "variables": {"s1": {"name": "s1", "templateType": "anything", "definition": "switch(t=1,1,t=4,1,-1)", "description": "", "group": "Ungrouped variables"}, "tol": {"name": "tol", "templateType": "anything", "definition": "0.001", "description": "", "group": "Ungrouped variables"}, "q1": {"name": "q1", "templateType": "anything", "definition": "'The complex number is in the first quadrant.'", "description": "", "group": "Ungrouped variables"}, "arg1": {"name": "arg1", "templateType": "anything", "definition": "precround(targ1,3)", "description": "", "group": "Ungrouped variables"}, "argrn": {"name": "argrn", "templateType": "anything", "definition": "precround((targ1+2*pi*(n-1))/n,3)", "description": "", "group": "Ungrouped variables"}, "n": {"name": "n", "templateType": "anything", "definition": "random(5,6,7)", "description": "", "group": "Ungrouped variables"}, "argr1": {"name": "argr1", "templateType": "anything", "definition": "precround(targ1/n,3)", "description": "", "group": "Ungrouped variables"}, "m1": {"name": "m1", "templateType": "anything", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "group": "Ungrouped variables"}, "a1": {"name": "a1", "templateType": "anything", "definition": "s1*random(1..5)", "description": "", "group": "Ungrouped variables"}, "z1": {"name": "z1", "templateType": "anything", "definition": "a1+b1*i", "description": "", "group": "Ungrouped variables"}, "b1": {"name": "b1", "templateType": "anything", "definition": "s2*random(1..5)", "description": "", "group": "Ungrouped variables"}, "md1": {"name": "md1", "templateType": "anything", "definition": "precround((abs(z1))^(1/n),3)", "description": "", "group": "Ungrouped variables"}, "ans1": {"name": "ans1", "templateType": "anything", "definition": "precround(abs(z1),3)", "description": "", "group": "Ungrouped variables"}, "targ1": {"name": "targ1", "templateType": "anything", "definition": "arg(z1)+adj", "description": "", "group": "Ungrouped variables"}, "q3": {"name": "q3", "templateType": "anything", "definition": "'The complex number is in the third quadrant.'", "description": "", "group": "Ungrouped variables"}, "gap": {"name": "gap", "templateType": "anything", "definition": "precround(2*pi/n,3)", "description": "", "group": "Ungrouped variables"}, "q4": {"name": "q4", "templateType": "anything", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "group": "Ungrouped variables"}, "s2": {"name": "s2", "templateType": "anything", "definition": "switch(t=1,-1,t=3,-1,1)", "description": "", "group": "Ungrouped variables"}, "t": {"name": "t", "templateType": "anything", "definition": "random(1..4)", "description": "", "group": "Ungrouped variables"}, "adj": {"name": "adj", "templateType": "anything", "definition": "if(arg(z1)<-pi,2*pi,0)", "description": "", "group": "Ungrouped variables"}, "q2": {"name": "q2", "templateType": "anything", "definition": "'The complex number is in the second quadrant.'", "description": "", "group": "Ungrouped variables"}}, "advice": "<p>To be completed.</p>", "variable_groups": [], "functions": {}, "statement": "<p>Find the $\\var{n}$th roots of $\\var{z1}$.&nbsp;</p>\n<p>Important: When calculating the roots, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.</p>\n<p><strong>For the purposes of this question all arguments of complex numbers are between $- \\pi$ and $\\pi$ radians.</strong></p>", "parts": [{"gaps": [{"strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "minValue": "ans1-tol", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "ans1+tol", "correctAnswerStyle": "plain", "allowFractions": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}, {"strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "minValue": "arg1-tol", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "arg1+tol", "correctAnswerStyle": "plain", "allowFractions": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}, {"correctAnswerFraction": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "minValue": "n", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "n", "scripts": {}, "allowFractions": false, "integerAnswer": true, "correctAnswerStyle": "plain", "marks": 0.5, "integerPartialCredit": 0}, {"strictPrecision": true, "correctAnswerFraction": false, "variableReplacements": [], "minValue": "md1-tol", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "md1+tol", "correctAnswerStyle": "plain", "allowFractions": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}, {"strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "minValue": "argr1-tol", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "argr1+tol", "correctAnswerStyle": "plain", "allowFractions": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}, {"strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "minValue": "argrn-tol", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "argrn+tol", "correctAnswerStyle": "plain", "allowFractions": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": false, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}, {"strictPrecision": false, "correctAnswerFraction": false, "variableReplacements": [], "minValue": "gap-tol", "notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "type": "numberentry", "maxValue": "gap+tol", "correctAnswerStyle": "plain", "allowFractions": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "precision": "3", "showPrecisionHint": false, "marks": 0.5, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision."}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "prompt": "<p>Find the modulus and argument of $\\var{z1}$ to 3 decimal places.</p>\n<p><strong>(i)</strong> $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n<p><strong>(ii)</strong> $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.&nbsp;</p>\n<p>Hence find the following $\\var{n}$th roots of $\\var{z1}$ i.e. solve for $z$, $z^\\var{n}=\\var{z1}$.</p>\n<p>How many roots are there? [[2]]</p>\n<p>All the roots have the same modulus.</p>\n<p>Input the modulus here: [[3]] (to 3 decimal places).</p>\n<p>What is the argument of the root with the least argument? [[4]] radians (to 3 decimal places)</p>\n<p>What is the argument of the root with the greatest argument? [[5]] radians (to 3 decimal places).</p>\n<p>If the roots are ordered in terms of their increasing arguments, what is the angle between successive roots? [[6]] radians (to 3 decimal places).</p>", "scripts": {}}], "name": "De Moivre's Theorem: $n$th roots of a complex number $(-\\pi\\leq \\theta \\leq \\pi)$", "metadata": {"description": "<p>Find modulus and argument of the complex number $z_1$ and find the $n$th roots of $z_1$ where $n=5,\\;6$ or $7$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}]}]}], "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}]}