// Numbas version: exam_results_page_options
{"name": "Modulus and argument of a complex number II", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"absz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1/z2),3)", "description": "", "name": "absz"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "argz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z1/z2),3)", "description": "", "name": "argz"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "b-b*i", "description": "", "name": "z2"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a+a*i", "description": "", "name": "z1"}}, "ungrouped_variables": ["a", "argz", "b", "absz", "tol", "z1", "z2"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Modulus and argument of a complex number II", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "absz+tol", "minValue": "absz-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\lvert z \\rvert=$ [[0]] (Enter your answer to 3 d.p.)</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "argz+tol", "minValue": "argz-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>$\\theta=$ [[0]] (Enter your answer to 3 d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Find the modulus $\\lvert z \\rvert$ and argument $\\theta$ (with $-\\pi&lt;\\theta\\leqslant\\pi$) of the complex number</p>\n  <p>\\[\\var{z}=\\frac{\\var{z1}}{\\var{z2}}.\\]</p>", "tags": ["MAS2103", "checked2015"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>15/7/2012:</strong></p>\n<p>Added tags.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Modulus and argument of a single complex number $z=z_1/z_2$, where $\\mathrm{Re}(z_1)=\\mathrm{Im}(z_1)$ and $\\mathrm{Re}(z_2)=-\\mathrm{Im}(z_2)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>For a complex number $z=\\frac{a+ai}{b+bi}$, the modulus is given by</p>\n<p>\\[\\lvert z \\rvert=\\frac{\\sqrt{a^2+a^2}}{\\sqrt{b^2+b^2}}=\\frac{a\\sqrt{2}}{b\\sqrt{2}}=\\frac{a}{b}\\]</p>\n<p>In this part $a=\\var{a}$ and $b=\\var{b}$, so $\\lvert z \\rvert=\\frac{\\var{a}}{\\var{b}}=\\var{absz}$ to 3 d.p.</p>\n<h4>b)</h4>\n<p>To calculate the argument of a complex number $z=\\frac{a+ai}{b-bi}$, with $a&gt;0$, $b&gt;0$, first write $z$ in the form $z=c+di$.</p>\n<p>To do this, multiply&nbsp;$z=\\frac{a+ai}{b-bi}$ by&nbsp;$\\frac{b+bi}{b+bi}$, so that</p>\n<p>\\[z=\\frac{(a+ai)(b+bi)}{(b-bi)(b+bi)}=\\frac{2abi}{2b^2}=\\frac{a}{b}i\\]</p>\n<p>So, given that $a$ and $b$ are both positive, $\\mathrm{Im}(z)&gt;0$, and because $\\mathrm{Re}(z)=0$, the argument is $\\theta=\\frac{\\pi}{2}$.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}