// Numbas version: exam_results_page_options
{"name": "De Moivre's Theorem: $n$th roots of a complex number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "De Moivre's Theorem: $n$th roots of a complex number", "parts": [{"showCorrectAnswer": true, "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": {}, "marks": 0, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "gaps": [{"showCorrectAnswer": true, "strictPrecision": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "precisionPartialCredit": 0, "minValue": "ans1-tol", "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "ans1+tol", "showFeedbackIcon": true, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "strictPrecision": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "precisionPartialCredit": 0, "minValue": "arg1-tol", "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "arg1+tol", "showFeedbackIcon": true, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"marks": 0.5, "correctAnswerStyle": "plain", "type": "numberentry", "scripts": {}, "showFeedbackIcon": true, "allowFractions": false, "showCorrectAnswer": true, "minValue": "n", "variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "maxValue": "n", "correctAnswerFraction": false, "variableReplacements": []}, {"showCorrectAnswer": true, "strictPrecision": true, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "precisionPartialCredit": 0, "minValue": "md1-tol", "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "md1+tol", "showFeedbackIcon": true, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "strictPrecision": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "precisionPartialCredit": 0, "minValue": "argr1-tol", "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "argr1+tol", "showFeedbackIcon": true, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "strictPrecision": false, "marks": 1, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "precisionPartialCredit": 0, "minValue": "argrn-tol", "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "argrn+tol", "showFeedbackIcon": true, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain"}, {"showCorrectAnswer": true, "strictPrecision": false, "marks": 0.5, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "type": "numberentry", "scripts": {}, "allowFractions": false, "variableReplacementStrategy": "originalfirst", "precision": "3", "precisionPartialCredit": 0, "minValue": "gap-tol", "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "maxValue": "gap+tol", "showFeedbackIcon": true, "correctAnswerFraction": false, "precisionType": "dp", "correctAnswerStyle": "plain"}]}], "functions": {}, "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variable_groups": [], "extensions": [], "advice": "<p>Please consult the relevant examples in the lecture notes.</p>", "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 $0$ and $2\\pi$ radians.</strong></p>", "variables": {"s2": {"definition": "switch(t=1,-1,t=3,-1,1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s2"}, "arg1": {"definition": "precround(targ1,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "arg1"}, "q4": {"definition": "'The complex number is in the fourth quadrant.'", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "q4"}, "tol": {"definition": "0.001", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "tol"}, "z1": {"definition": "a1+b1*i", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "z1"}, "targ1": {"definition": "arg(z1)+adj", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "targ1"}, "ans1": {"definition": "precround(abs(z1),3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "ans1"}, "q3": {"definition": "'The complex number is in the third quadrant.'", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "q3"}, "argr1": {"definition": "precround(targ1/n,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "argr1"}, "q2": {"definition": "'The complex number is in the second quadrant.'", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "q2"}, "m1": {"definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "m1"}, "b1": {"definition": "s2*random(1..5)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "b1"}, "n": {"definition": "random(3,4,5,6,7)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "n"}, "a1": {"definition": "s1*random(1..5)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "a1"}, "md1": {"definition": "precround((abs(z1))^(1/n),3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "md1"}, "gap": {"definition": "precround(2*pi/n,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "gap"}, "adj": {"definition": "if(arg(z1)<0,2*pi,0)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "adj"}, "q1": {"definition": "'The complex number is in the first quadrant.'", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "q1"}, "argrn": {"definition": "precround((targ1+2*pi*(n-1))/n,3)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "argrn"}, "t": {"definition": "random(1..4)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "t"}, "s1": {"definition": "switch(t=1,1,t=4,1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": "", "name": "s1"}}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "ungrouped_variables": ["q1", "argrn", "q3", "q2", "q4", "md1", "arg1", "s2", "s1", "targ1", "n", "a1", "m1", "t", "tol", "ans1", "gap", "adj", "z1", "argr1", "b1"], "type": "question", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}]}