// Numbas version: exam_results_page_options
{"name": "De Moivre's theorem 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>To evaluate \\((\\var{x}+\\var{y}j)^{\\frac{1}{3}}\\) we must first express&nbsp;\\(Z=\\var{x}+\\var{y}j\\) &nbsp;in polar form.</p>\n<p>The modulus of \\(\\var{x}+\\var{y}j\\) = \\(\\sqrt{\\var{x}^2+(\\var{y})^2}=\\sqrt{\\simplify{{x^2}+{y}^2}}\\)</p>\n<p>The argument of the complex number is given by \\(\\theta=tan^{-1}\\left(\\frac{\\var{y}}{\\var{x}}\\right)=\\var{theta}\\)</p>\n<p>According to De Moivre's theorem \\(Z^{\\frac{1}{3}}=|Z|^{\\frac{1}{3}}\\left(cos(\\frac{\\theta+2n\\pi}{3})+jsin(\\frac{\\theta+2n\\pi}{3})\\right)\\)</p>\n<p>By letting \\(\\theta\\) take the values 0, 1 and 2 we find each of the three cube roots.</p>\n<p>Below is the third cube root which can be found by setting \\(\\theta=1\\).</p>\n<p>\\((\\simplify{{x}+{y}j})^{\\frac{1}{3}}=\\left(\\sqrt{\\simplify{{x^2}+{y}^2}}\\right)^{\\frac{1}{3}}\\left(cos(\\simplify{{n}*({theta}+2*pi)})+jsin(\\simplify{{n}*({theta}+2pi)})\\right)\\)</p>\n<p style=\"padding-left: 90px;\">=\\(\\simplify{{x4}+{y4}j}\\)</p>\n<p style=\"padding-left: 90px;\">\\(A=\\var{x4}\\) &nbsp;and &nbsp;\\(B=\\var{y4}\\)</p>\n<p></p>", "parts": [{"prompt": "<p>\\(Z=\\var{x}+\\var{y}j\\) has three cube roots.</p>\n<p>\\(Z=\\var{x2}+\\var{y2}j\\) &nbsp; and &nbsp; \\(Z=\\var{x3}+\\var{y3}j\\) are two of the roots.</p>\n<p>The third cube root is \\(Z=A+Bj\\)</p>\n<p>Calculate \\(A\\) and \\(B\\)</p>\n<p></p>\n<p style=\"padding-left: 30px;\">\\(A\\) =&nbsp;[[0]]</p>\n<p style=\"padding-left: 30px;\">\\(B\\) =&nbsp;[[1]]</p>", "showFeedbackIcon": true, "marks": 0, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"correctAnswerStyle": "plain", "precisionType": "dp", "marks": 1, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showFeedbackIcon": true, "type": "numberentry", "showPrecisionHint": true, "correctAnswerFraction": false, "precision": "1", "showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "minValue": "{x4}-0.1", "strictPrecision": false, "maxValue": "{x4}+0.1", "precisionMessage": "You have not given your answer to the correct precision."}, {"correctAnswerStyle": "plain", "precisionType": "dp", "marks": 1, "allowFractions": false, "notationStyles": ["plain", "en", "si-en"], "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "showFeedbackIcon": true, "type": "numberentry", "showPrecisionHint": true, "correctAnswerFraction": false, "precision": "1", "showCorrectAnswer": true, "scripts": {}, "variableReplacements": [], "minValue": "{y4}-0.1", "strictPrecision": false, "maxValue": "{y4}+0.1", "precisionMessage": "You have not given your answer to the correct precision."}]}], "variable_groups": [], "statement": "<p>This question asks you to use De Moivre's theorem to find the cube&nbsp;roots of \\(Z=\\var{x}+\\var{y}j\\)</p>", "preamble": {"css": "", "js": ""}, "rulesets": {}, "tags": [], "name": "De Moivre's theorem 3", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculate the cube roots of a complex number using De Moivre.</p>"}, "ungrouped_variables": ["x", "y", "n", "theta", "mod", "x2", "y2", "x3", "y3", "x4", "y4"], "extensions": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "variables": {"y3": {"definition": "{mod}^{n}*sin({n}*({theta}+2*pi))", "description": "", "name": "y3", "templateType": "anything", "group": "Ungrouped variables"}, "y2": {"definition": "{mod}^{n}*sin({n}*{theta})", "description": "", "name": "y2", "templateType": "anything", "group": "Ungrouped variables"}, "x4": {"definition": "{mod}^{n}*cos({n}*({theta}+4*pi))", "description": "", "name": "x4", "templateType": "anything", "group": "Ungrouped variables"}, "y4": {"definition": "{mod}^{n}*sin({n}*({theta}+4*pi))", "description": "", "name": "y4", "templateType": "anything", "group": "Ungrouped variables"}, "y": {"definition": "random(1..15#1)", "description": "", "name": "y", "templateType": "randrange", "group": "Ungrouped variables"}, "mod": {"definition": "sqrt({x}^2+{y}^2)", "description": "", "name": "mod", "templateType": "anything", "group": "Ungrouped variables"}, "x3": {"definition": "{mod}^{n}*cos({n}*({theta}+2*pi))", "description": "", "name": "x3", "templateType": "anything", "group": "Ungrouped variables"}, "n": {"definition": "1/3", "description": "", "name": "n", "templateType": "anything", "group": "Ungrouped variables"}, "x": {"definition": "random(1..12#1)", "description": "", "name": "x", "templateType": "randrange", "group": "Ungrouped variables"}, "x2": {"definition": "{mod}^{n}*cos({n}*{theta})", "description": "", "name": "x2", "templateType": "anything", "group": "Ungrouped variables"}, "theta": {"definition": "arctan({y}/{x})", "description": "", "name": "theta", "templateType": "anything", "group": "Ungrouped variables"}}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}