// Numbas version: finer_feedback_settings
{"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": [{"functions": {}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "tags": [], "extensions": [], "name": "De Moivre's Theorem: $n$th roots of a complex number", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variablesTest": {"condition": "", "maxRuns": 100}, "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>", "parts": [{"unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "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>", "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "variableReplacements": [], "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "gaps": [{"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "unitTests": [], "minValue": "ans1-tol", "extendBaseMarkingAlgorithm": true, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "variableReplacements": [], "mustBeReducedPC": 0, "precision": "3", "showCorrectAnswer": true, "marks": 1, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "maxValue": "ans1+tol", "precisionType": "dp", "mustBeReduced": false, "correctAnswerFraction": false, "allowFractions": false, "customMarkingAlgorithm": ""}, {"precisionPartialCredit": 0, "strictPrecision": false, "variableReplacementStrategy": "originalfirst", 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the fourth quadrant.'", "description": "", "templateType": "anything"}, "b1": {"group": "Ungrouped variables", "name": "b1", "definition": "s2*random(1..5)", "description": "", "templateType": "anything"}, "n": {"group": "Ungrouped variables", "name": "n", "definition": "random(3,4,5,6,7)", "description": "", "templateType": "anything"}, "delang": {"group": "Ungrouped variables", "name": "delang", "definition": "precround(2*pi/n,3)", "description": "", "templateType": "anything"}, "targ1": {"group": "Ungrouped variables", "name": "targ1", "definition": "arg(z1)+adj", "description": "", "templateType": "anything"}, "argr1": {"group": "Ungrouped variables", "name": "argr1", "definition": "precround(targ1/n,3)", "description": "", "templateType": "anything"}, "tol": {"group": "Ungrouped variables", "name": "tol", "definition": "0.001", "description": "", "templateType": "anything"}, "argrn": {"group": "Ungrouped variables", "name": "argrn", "definition": 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{"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"}, "advice": "<p>Please consult the relevant examples in the lecture notes.</p>", "ungrouped_variables": ["q1", "argrn", "q3", "q2", "q4", "md1", "arg1", "s2", "s1", "targ1", "n", "a1", "m1", "t", "tol", "ans1", "delang", "adj", "z1", "argr1", "b1"], "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Peter Johnston", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/771/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Peter Johnston", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/771/"}]}