// Numbas version: exam_results_page_options
{"name": "Complex Numbers: Modulus, Argument", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Complex Numbers: Modulus, Argument", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p><span style=\"color: #222222; font-family: sans-serif;\"><span style=\"line-height: 20px;\">Finding&nbsp;</span></span>the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!</p>"}, "statement": "<p>Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between <strong>$-\\pi$ and $\\pi$</strong>. If your answer is outside this range, simply add or subtract $2\\pi$ to your answer.</p>\n<p>When calculating the argument pay particular attention to the quadrant in which the complex number lies.</p>\n<p>Input all answers to 3 decimal places.</p>", "variable_groups": [], "preamble": {"css": "", "js": ""}, "advice": "<p>Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi &lt; \\theta \\le \\pi$.</p>\n<p>You have to be careful with using a standard calculator when you are finding the argument of a complex number.</p>\n<p>If $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.</p>\n<p>Using a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi &lt; \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants &ndash; you get the correct value.</p>\n<p>However, The calculator gives the wrong value for complex numbers in the other quadrants.</p>\n<h4>Complex number in the Second Quadrant.</h4>\n<p>Since $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.</p>\n<h4>Complex number in the Third Quadrant.</h4>\n<p>Since $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.</p>\n<h4>a)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z1}|&amp;=&amp;\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &amp;=&amp; \\var{abs(z1)}\\\\ &amp;=&amp;\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m1}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>b)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z2}|&amp;=&amp;\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &amp;=&amp; \\var{abs(z2)}\\\\ &amp;=&amp;\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m2}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>c)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z3}|&amp;=&amp;\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &amp;=&amp; \\var{abs(z3)}\\\\ &amp;=&amp;\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m3}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &amp;=&amp; \\var{arg(z3)}\\\\ &amp;=&amp; \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h4>d)</h4>\n<h5>Modulus</h5>\n<p>\\[ \\begin{eqnarray*} |\\var{z4}|&amp;=&amp;\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &amp;=&amp; \\var{abs(z4)}\\\\ &amp;=&amp;\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.</p>\n<h5>Argument</h5>\n<p><strong>{m4}</strong></p>\n<p>Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &amp;=&amp; \\var{arg(z4)}\\\\ &amp;=&amp; \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>", "variables": {"f": {"name": "f", "description": "", "group": "Ungrouped variables", "definition": "random(1..9)", "templateType": "anything"}, "m1": {"name": "m1", "description": "", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "templateType": "anything"}, "m4": {"name": "m4", "description": "", "group": "Ungrouped variables", "definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "templateType": "anything"}, "ans4": {"name": "ans4", "description": "", "group": "Ungrouped variables", "definition": "precround(abs(z4),3)", "templateType": "anything"}, "arg3": {"name": 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"arg1+tol", "showCorrectAnswer": true, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "mustBeReducedPC": 0, "scripts": {}, "minValue": "arg1-tol", "showFeedbackIcon": true}], "prompt": "\n      <p>$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians</p>\n        <p>Input both answers to 3 decimal places.</p>\n      ", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "marks": 0, "variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "maxValue": "ans2+tol", "showCorrectAnswer": true, "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "type": "numberentry", "marks": 1, "variableReplacements": [], "correctAnswerFraction": false, "mustBeReducedPC": 0, 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"showFeedbackIcon": true, "scripts": {}}], "type": "question", "contributors": [{"name": "John Sheekey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1192/"}]}]}], "contributors": [{"name": "John Sheekey", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1192/"}]}