// Numbas version: exam_results_page_options
{"name": "Polar form of a complex number", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "argz", "b", "absz", "tol", "z"], "parts": [{"marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "type": "numberentry", "correctAnswerFraction": false, "scripts": {}, "showCorrectAnswer": true, "maxValue": "absz+tol", "showPrecisionHint": false, "minValue": "absz-tol"}], "type": "gapfill", "prompt": "<p>$r=$ [[0]] (Enter your answer to 3 d.p.)</p>", "scripts": {}, "showCorrectAnswer": true}, {"marks": 0, "gaps": [{"allowFractions": false, "marks": 1, "type": "numberentry", "correctAnswerFraction": false, "scripts": {}, "showCorrectAnswer": true, "maxValue": "argz+tol", "showPrecisionHint": false, "minValue": "argz-tol"}], "type": "gapfill", "prompt": "<p>$\\theta=$ [[0]] (Enter your answer to 3 d.p.)</p>", "scripts": {}, "showCorrectAnswer": true}], "variables": {"argz": {"name": "argz", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "precround(arg(z),3)"}, "tol": {"name": "tol", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "0.001"}, "b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0..5 except a)*sign(random(-1,1))"}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "random(0..5)*sign(random(-1,1))"}, "z": {"name": "z", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "a+b*i"}, "absz": {"name": "absz", "templateType": "anything", "group": "Ungrouped variables", "description": "", "definition": "precround(abs(z),3)"}}, "question_groups": [{"name": "", "questions": [], "pickQuestions": 0, "pickingStrategy": "all-ordered"}], "type": "question", "name": "Polar form of a complex number", "rulesets": {}, "functions": {}, "variable_groups": [], "metadata": {"notes": "<p><strong>15/7/2012:</strong></p>\n<p>Added tags.</p>\n<p></p>\n<p>This question combines the original questions 5 and 6 from MAS2103 CBA 1.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Polar form of a complex number.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>To write a complex number $z=a+bi$ in polar form $z=r\\mathrm{e}^{i\\theta}$, we calculate the modulus $r = \\lvert z \\rvert$ and argument $\\theta = \\arg(z)$.</p>\n<p>Hence</p>\n<p>\\[r=\\lvert z \\rvert=\\sqrt{a^2+b^2}=\\sqrt{(\\var{a})^2+(\\var{b})^2}=\\var{absz}\\;\\text{to 3d.p.}\\]</p>\n<p>and, in general,</p>\n<p>\\[\\theta=\\arg(z)=\\arctan\\left(\\frac{b}{a}\\right).\\]</p>\n<p>If $a=0$, however, then $\\mathrm{Re}(z)=0$, so $\\arg(z)=\\pm\\frac{\\pi}{2}$, depending on whether $\\mathrm{Im}(z)$ is positive or negative.</p>\n<p>In this case $a=\\var{a}$, and $b=\\var{b}$, so $\\arg(z)=\\var{argz}$.</p>", "tags": ["MAS2103", "checked2015"], "statement": "<p>Write the complex number $z=\\var{z}$ in polar form $z=r\\mathrm{e}^{i\\theta}$, with $-\\pi&lt;\\theta\\leqslant\\pi$, by calculating $r$ and $\\theta$.</p>", "preamble": {"css": "", "js": ""}, "showQuestionGroupNames": false, "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/"}]}