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