// Numbas version: exam_results_page_options
{"name": "Christian'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": [{"parts": [{"customMarkingAlgorithm": "gap_variables:\n  dict(map([x,eval(expression(x))],x,filter(x in question_definitions[\"variables\"],x,[\"gap\",\"number\",\"answer\",\"index\"])))\n\ngap_feedback (Feedback on each of the gaps):\n  map(\n    let(result,let(gap_variables,submit_part(gaps[gap_number][\"path\"],answer)),\n      feedback(translate('part.gapfill.feedback header',[\"index\":index]));\n      concat_feedback(result[\"feedback\"], if(marks>0,result[\"marks\"]/marks,1));\n      result\n    ),\n    [gap_number,answer,index],\n    gap_variables;zip(gap_order,studentAnswer,list(1..len(gaps)))\n  )\n\nmarked_original_order (Mark the gaps in the original order, 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"variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "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>"}], "tags": [], "extensions": [], "variable_groups": [], "preamble": {"js": "", "css": ""}, "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>"}, "advice": "<p>Please consult the relevant examples in the lecture notes.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "ungrouped_variables": ["q1", "argrn", "q3", "q2", "q4", "md1", "arg1", "s2", "s1", "targ1", "n", "a1", "m1", "t", "tol", "ans1", "gap", "adj", "z1", "argr1", "b1"], "functions": {}, "name": "Christian's copy of De Moivre's Theorem: $n$th roots of a complex number", "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": {"n": {"group": "Ungrouped variables", "description": "", "definition": "random(3,4,5,6,7)", "templateType": "anything", "name": "n"}, "t": {"group": "Ungrouped variables", "description": "", "definition": "random(1..4)", "templateType": "anything", "name": "t"}, "adj": {"group": "Ungrouped variables", "description": "", "definition": "if(arg(z1)<0,2*pi,0)", "templateType": "anything", "name": "adj"}, "a1": {"group": "Ungrouped variables", "description": "", "definition": "s1*random(1..5)", "templateType": "anything", "name": "a1"}, "tol": {"group": "Ungrouped variables", "description": "", "definition": "0.001", "templateType": "anything", "name": "tol"}, "q2": {"group": "Ungrouped variables", "description": "", "definition": "'The complex number is in the second quadrant.'", "templateType": "anything", "name": "q2"}, "md1": {"group": "Ungrouped variables", "description": "", "definition": "precround((abs(z1))^(1/n),3)", "templateType": "anything", "name": "md1"}, "m1": {"group": "Ungrouped variables", "description": "", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "templateType": "anything", "name": "m1"}, "q4": {"group": "Ungrouped variables", "description": "", "definition": "'The complex number is in the fourth quadrant.'", "templateType": "anything", "name": "q4"}, "gap": {"group": "Ungrouped variables", "description": "", "definition": "precround(2*pi/n,3)", "templateType": "anything", "name": "gap"}, "targ1": {"group": "Ungrouped variables", "description": "", "definition": "arg(z1)+adj", "templateType": "anything", "name": "targ1"}, "z1": {"group": "Ungrouped variables", "description": "", "definition": "a1+b1*i", "templateType": "anything", "name": "z1"}, "argrn": {"group": "Ungrouped variables", "description": "", "definition": "precround((targ1+2*pi*(n-1))/n,3)", "templateType": "anything", "name": "argrn"}, "arg1": {"group": "Ungrouped variables", "description": "", "definition": "precround(targ1,3)", "templateType": "anything", "name": "arg1"}, "b1": {"group": "Ungrouped variables", "description": "", "definition": "s2*random(1..5)", "templateType": "anything", "name": "b1"}, "q3": {"group": "Ungrouped variables", "description": "", "definition": "'The complex number is in the third quadrant.'", "templateType": "anything", "name": "q3"}, "q1": {"group": "Ungrouped variables", "description": "", "definition": "'The complex number is in the first quadrant.'", "templateType": "anything", "name": "q1"}, "argr1": {"group": "Ungrouped variables", "description": "", "definition": "precround(targ1/n,3)", "templateType": "anything", "name": "argr1"}, "s1": {"group": "Ungrouped variables", "description": "", "definition": "switch(t=1,1,t=4,1,-1)", "templateType": "anything", "name": "s1"}, "ans1": {"group": "Ungrouped variables", "description": "", "definition": "precround(abs(z1),3)", "templateType": "anything", "name": "ans1"}, "s2": {"group": "Ungrouped variables", "description": "", "definition": "switch(t=1,-1,t=3,-1,1)", "templateType": "anything", "name": "s2"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "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/"}]}