// 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": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Calculating the square root of a complex number using De Moivre.</p>"}, "tags": [], "name": "De Moivre's theorem - Square Root", "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>", "extensions": [], "parts": [{"gaps": [{"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1", "showPrecisionHint": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "marks": 1, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "{x2}-0.1", "scripts": {}, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{x2}+0.1", "showFeedbackIcon": true}, {"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1", "showPrecisionHint": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "marks": 1, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "{y2}-0.1", "scripts": {}, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{y2}+0.1", "showFeedbackIcon": true}], "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>", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacements": []}], "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": ""}, "variable_groups": [], "variables": {"y2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{mod}^{n}*sin({n}*{theta})", "name": "y2", "description": ""}, "x2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{mod}^{n}*cos({n}*{theta})", "name": "x2", "description": ""}, "theta": {"templateType": "anything", "group": "Ungrouped variables", "definition": "arctan({y}/{x})", "name": "theta", "description": ""}, "y": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1..15#1)", "name": "y", "description": ""}, "n": {"templateType": "number", "group": "Ungrouped variables", "definition": "0.5", "name": "n", "description": ""}, "mod": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sqrt({x}^2+{y}^2)", "name": "mod", "description": ""}, "x": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(1..12#1)", "name": "x", "description": ""}}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "contributors": [{"name": "Peter Johnston", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/771/"}]}]}], "contributors": [{"name": "Peter Johnston", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/771/"}]}