// Numbas version: exam_results_page_options
{"name": "De Moivre's Theorem - Square Root", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculating the square root of a complex number using De Moivre.</p>"}, "tags": [], "functions": {}, "name": "De Moivre's Theorem - Square Root", "parts": [{"prompt": "<p>\\(\\sqrt{\\var{x}+\\var{y}i}=\\pm(A+Bi)\\)</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>", "marks": 0, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "gaps": [{"precision": "1", "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "maxValue": "{x2}+0.1", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "showPrecisionHint": true, "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "precisionType": "dp", "marks": 1, "minValue": "{x2}-0.1", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"]}, {"precision": "1", "allowFractions": false, "precisionMessage": "You have not given your answer to the correct precision.", "correctAnswerStyle": "plain", "maxValue": "{y2}+0.1", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "scripts": {}, "showCorrectAnswer": true, "variableReplacements": [], "showPrecisionHint": true, "showFeedbackIcon": true, "correctAnswerFraction": false, "type": "numberentry", "precisionType": "dp", "marks": 1, "minValue": "{y2}-0.1", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"]}], "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}}], "extensions": [], "rulesets": {}, "ungrouped_variables": ["x", "y", "n", "theta", "mod", "x2", "y2"], "statement": "<p>Use De Moivre's theorem to find a square root of \\(Z=\\var{x}+\\var{y}i\\)</p>", "advice": "<p>To evaluate \\(\\sqrt{\\var{x}+\\var{y}i}\\) we must first express&nbsp;\\(Z=\\var{x}+\\var{y}i\\) &nbsp;in polar form.</p>\n<p>The modulus of \\(\\var{x}+\\var{y}i\\) = \\(\\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^{\\var{n}}=|Z|^{\\var{n}}\\left(\\cos(\\var{n}*\\theta)+i\\sin(\\var{n}*\\theta)\\right)\\)</p>\n<p>\\((\\simplify{{x}+{y}i})^\\var{n}=\\left(\\sqrt{\\simplify{{x^2}+{y}^2}}\\right)^{\\var{n}}\\left(\\cos(\\simplify{{n}*{theta}})+i\\sin(\\simplify{{n}*{theta}})\\right)\\)</p>\n<p style=\"padding-left: 90px;\">=\\(\\simplify{{x2}+{y2}i}\\)</p>\n<p style=\"padding-left: 90px;\">\\(A=\\var{x2}\\) &nbsp;and &nbsp;\\(B=\\var{y2}\\)</p>\n<p></p>", "preamble": {"css": "", "js": ""}, "variables": {"mod": {"name": "mod", "group": "Ungrouped variables", "definition": "sqrt({x}^2+{y}^2)", "templateType": "anything", "description": ""}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "{mod}^{n}*sin({n}*{theta})", "templateType": "anything", "description": ""}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "{mod}^{n}*cos({n}*{theta})", "templateType": "anything", "description": ""}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "0.5", "templateType": "number", "description": ""}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(1..12#1)", "templateType": "randrange", "description": ""}, "theta": {"name": "theta", "group": "Ungrouped variables", "definition": "arctan({y}/{x})", "templateType": "anything", "description": ""}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "random(1..15#1)", "templateType": "randrange", "description": ""}}, "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/"}]}