// Numbas version: exam_results_page_options
{"name": "Maria's copy of 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": [{"variable_groups": [], "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de moivre's theorem", "de Moivre's theorem", "de Moivre's Theorem", "mas1602", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane", "roots of a complex number"], "showQuestionGroupNames": false, "functions": {}, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "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>", "name": "Maria's copy of De Moivre's Theorem: $n$th roots of a complex number", "variables": {"targ1": {"name": "targ1", "group": "Ungrouped variables", "definition": "arg(z1)+adj", "templateType": "anything", "description": ""}, "md1": {"name": "md1", "group": "Ungrouped variables", "definition": "precround((abs(z1))^(1/n),3)", "templateType": "anything", "description": ""}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(5,6,7)", "templateType": "anything", "description": ""}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "a1+b1*i", "templateType": "anything", "description": ""}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "templateType": "anything", "description": ""}, "argrn": {"name": "argrn", "group": "Ungrouped variables", "definition": "precround((targ1+2*pi*(n-1))/n,3)", "templateType": "anything", "description": ""}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "templateType": "anything", "description": ""}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "'The complex number is in the first quadrant.'", "templateType": "anything", "description": ""}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,1,-1)", "templateType": "anything", "description": ""}, "argr1": {"name": "argr1", "group": "Ungrouped variables", "definition": "precround(targ1/n,3)", "templateType": "anything", "description": ""}, "arg1": {"name": "arg1", "group": "Ungrouped variables", "definition": "precround(targ1,3)", "templateType": "anything", "description": ""}, "q4": {"name": "q4", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "templateType": "anything", "description": ""}, "q3": {"name": "q3", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "templateType": "anything", "description": ""}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.001", "templateType": "anything", "description": ""}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "s2*random(1..5)", "templateType": "anything", "description": ""}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..4)", "templateType": "anything", "description": ""}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "s1*random(1..5)", "templateType": "anything", "description": ""}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "templateType": "anything", "description": ""}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=3,-1,1)", "templateType": "anything", "description": ""}, "adj": {"name": "adj", "group": "Ungrouped variables", "definition": "if(arg(z1)<0,2*pi,0)", "templateType": "anything", "description": ""}, "gap": {"name": "gap", "group": "Ungrouped variables", "definition": "precround(2*pi/n,3)", "templateType": "anything", "description": ""}}, "ungrouped_variables": ["q1", "argrn", "q3", "q2", "q4", "md1", "arg1", "s2", "s1", "targ1", "n", "a1", "m1", "t", "tol", "ans1", "gap", "adj", "z1", "argr1", "b1"], "parts": [{"gaps": [{"minValue": "ans1-tol", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "marks": 1, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "precision": "3", "maxValue": "ans1+tol", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry"}, {"minValue": "arg1-tol", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "marks": 1, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "precision": "3", "maxValue": "arg1+tol", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry"}, {"minValue": "n", "integerPartialCredit": 0, "variableReplacementStrategy": "originalfirst", "marks": 0.5, "variableReplacements": [], "maxValue": "n", "integerAnswer": true, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "correctAnswerFraction": false, "allowFractions": false, "type": "numberentry"}, {"minValue": "md1-tol", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "marks": 1, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "precision": "3", "maxValue": "md1+tol", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry"}, {"minValue": "argr1-tol", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "marks": 1, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "precision": "3", "maxValue": "argr1+tol", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry"}, {"minValue": "argrn-tol", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "marks": 1, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "precision": "3", "maxValue": "argrn+tol", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry"}, {"minValue": "gap-tol", "variableReplacementStrategy": "originalfirst", "precisionType": "dp", "marks": 0.5, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "showCorrectAnswer": true, "scripts": {}, "precision": "3", "maxValue": "gap+tol", "allowFractions": false, "correctAnswerFraction": false, "type": "numberentry"}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "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>", "variableReplacements": [], "type": "gapfill"}], "type": "question", "advice": "<p>To be completed.</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "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", "notes": "<p><strong>15/7/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>27/08/2012:</strong></p>\n<p>Added tags.</p>\n<p>Added description.</p>\n<p>Based &nbsp;on question using DM's theorem for positive powers.</p>\n<p><strong>&nbsp;</strong></p>"}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}