// Numbas version: exam_results_page_options
{"name": "Maria's copy of De Moivre's Theorem: Positive Powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"marks": 0, "prompt": "<p>Find the modulus and argument of $\\var{z1}$ to 3 decimal places.</p>\n      <p>(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n      <p>(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.</p>\n      <p>Hence find:</p>\n      <p>(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]</p>\n      <p>Input as a complex number, with real and imaginary parts integral values.</p>", "gaps": [{"minValue": "ans1-tol", "marks": 1, "showCorrectAnswer": true, "maxValue": "ans1+tol", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "type": "numberentry"}, {"minValue": "arg1-tol", "marks": 1, "showCorrectAnswer": true, "maxValue": "arg1+tol", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "type": "numberentry"}, {"showCorrectAnswer": true, "answersimplification": "std", "type": "jme", "showpreview": true, "scripts": {}, "checkvariablenames": false, "checkingaccuracy": 0.001, "marks": 2, "answer": "{a3}+{b3}*i", "vsetrangepoints": 5, "checkingtype": "absdiff", "vsetrange": [0, 1], "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "showCorrectAnswer": true}, {"marks": 0, "prompt": "<p>Find the modulus and argument of $\\var{z2}$ to 3 decimal places.</p>\n      <p>(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.</p>\n      <p>(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.</p>\n      <p>Hence find:</p>\n      <p>(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]</p>\n      <p>Input as a complex number, with real and imaginary parts integral values.</p>", "gaps": [{"minValue": "ans2-tol", "marks": 1, "showCorrectAnswer": true, "maxValue": "ans2+tol", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "type": "numberentry"}, {"minValue": "arg2-tol", "marks": 1, "showCorrectAnswer": true, "maxValue": "arg2+tol", "correctAnswerFraction": false, "allowFractions": false, "showPrecisionHint": false, "scripts": {}, "type": "numberentry"}, {"showCorrectAnswer": true, "answersimplification": "std", "type": "jme", "showpreview": true, "scripts": {}, "checkvariablenames": false, "checkingaccuracy": 0.001, "marks": 2, "answer": "{a4}+{b4}*i", "vsetrangepoints": 5, "checkingtype": "absdiff", "vsetrange": [0, 1], "expectedvariablenames": []}], "type": "gapfill", "scripts": {}, "showCorrectAnswer": true}], "variables": {"ans1": {"definition": "precround(abs(z1),3)", "templateType": "anything", "name": "ans1", "description": "", "group": "Ungrouped variables"}, "s5": {"definition": "switch(t=3,-1,1)", "templateType": "anything", "name": "s5", "description": "", "group": "Ungrouped variables"}, "ans3": {"definition": "precround(abs(z3),3)", "templateType": "anything", "name": "ans3", "description": "", "group": "Ungrouped variables"}, "q3": {"definition": "'The complex number is in the third quadrant.'", "templateType": "anything", "name": "q3", "description": "", "group": "Ungrouped variables"}, "arg1": {"definition": "precround(arg(z1),3)", "templateType": "anything", "name": "arg1", "description": "", "group": "Ungrouped variables"}, "s6": {"definition": "switch(t=1,-1,t=4,-1,1)", "templateType": "anything", "name": "s6", "description": "", "group": "Ungrouped variables"}, "z5": {"definition": "a4+b4*i", "templateType": "anything", "name": "z5", "description": "", "group": "Ungrouped variables"}, "ans4": {"definition": "precround(abs(z4),3)", "templateType": "anything", "name": "ans4", "description": "", "group": "Ungrouped variables"}, "s7": {"definition": "switch(t=2,-1,t=3,1,-1)", "templateType": "anything", "name": "s7", "description": "", "group": "Ungrouped variables"}, "s3": {"definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "templateType": "anything", "name": "s3", "description": "", "group": "Ungrouped variables"}, "b2": {"definition": "s5*random(1..3)", "templateType": "anything", "name": "b2", "description": "", "group": "Ungrouped variables"}, "tol": {"definition": "0.001", "templateType": "anything", "name": "tol", "description": "", "group": "Ungrouped variables"}, "tb4": {"definition": "(abs(z2)^n4)*sin(n4*arg(z2))", "templateType": "anything", "name": "tb4", "description": "", "group": "Ungrouped variables"}, "d4": {"definition": "s5*random(1..9)", "templateType": "anything", "name": "d4", "description": "", "group": "Ungrouped variables"}, "q2": {"definition": "'The complex number is in the second quadrant.'", "templateType": "anything", "name": "q2", "description": "", "group": "Ungrouped variables"}, "a4": {"definition": "precround(ta4,3)", "templateType": "anything", "name": "a4", "description": "", "group": "Ungrouped variables"}, "z1": {"definition": "a1+b1*i", "templateType": "anything", "name": "z1", "description": "", "group": "Ungrouped variables"}, "z3": {"definition": "c2+d2*i", "templateType": "anything", "name": "z3", "description": "", "group": "Ungrouped variables"}, "ta3": {"definition": "abs(z1)^n2*cos(n2*arg(z1))", "templateType": "anything", "name": "ta3", "description": "", "group": "Ungrouped variables"}, "f": {"definition": "random(1..9)", "templateType": "anything", "name": "f", "description": "", "group": "Ungrouped variables"}, "s8": {"definition": "switch(t=1,1,t=4,-1,t=3,1,-1)", "templateType": "anything", "name": "s8", "description": "", "group": "Ungrouped variables"}, "t": {"definition": "random(1..4)", "templateType": "anything", "name": "t", "description": "", "group": "Ungrouped variables"}, "m4": {"definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "templateType": "anything", "name": "m4", "description": "", "group": "Ungrouped variables"}, "m1": {"definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "templateType": "anything", "name": "m1", "description": "", "group": "Ungrouped variables"}, "m3": {"definition": "switch(t=1,q3,t=2,q4,t=3,q1,q3)", "templateType": "anything", "name": "m3", "description": "", "group": "Ungrouped variables"}, "b1": {"definition": "s2*random(1..3)", "templateType": "anything", "name": "b1", "description": "", "group": "Ungrouped variables"}, "co": {"definition": "s4*random(1..3)", "templateType": "anything", "name": "co", "description": "", "group": "Ungrouped variables"}, "a1": {"definition": "s1*random(1..3)", "templateType": "anything", "name": "a1", "description": "", "group": "Ungrouped variables"}, "c2": {"definition": "s6*random(1..9)", "templateType": "anything", "name": "c2", "description": "", "group": "Ungrouped variables"}, "z2": {"definition": "a2+b2*i", "templateType": "anything", "name": "z2", "description": "", "group": "Ungrouped variables"}, "a2": {"definition": "if(co=a1,co+1,co)", "templateType": "anything", "name": "a2", "description": "", "group": "Ungrouped variables"}, "z6": {"definition": "c4+d4*i", "templateType": "anything", "name": "z6", "description": "", "group": "Ungrouped variables"}, "s2": {"definition": "switch(t=1,-1,t=3,-1,1)", "templateType": "anything", "name": "s2", "description": "", "group": "Ungrouped variables"}, "arg4": {"definition": "precround(arg(z4),3)", "templateType": "anything", "name": "arg4", "description": "", "group": "Ungrouped variables"}, "s1": {"definition": "switch(t=1,1,t=4,1,-1)", "templateType": "anything", "name": "s1", "description": "", "group": "Ungrouped variables"}, "m2": {"definition": "switch(t=1,q2,t=2,q1,t=3,q4,q2)", "templateType": "anything", "name": "m2", "description": "", "group": "Ungrouped variables"}, "s4": {"definition": "switch(t=1,-1,t=4,-1,1)", "templateType": "anything", "name": "s4", "description": "", "group": "Ungrouped variables"}, "arg2": {"definition": "precround(arg(z2),3)", "templateType": "anything", "name": "arg2", "description": "", "group": "Ungrouped variables"}, "ans2": {"definition": "precround(abs(z2),3)", "templateType": "anything", "name": "ans2", "description": "", "group": "Ungrouped variables"}, "c4": {"definition": "if(a4=f,f+1,f)", "templateType": "anything", "name": "c4", "description": "", "group": "Ungrouped variables"}, "arg3": {"definition": "precround(arg(z3),3)", "templateType": "anything", "name": "arg3", "description": "", "group": "Ungrouped variables"}, "q1": {"definition": "'The complex number is in the first quadrant.'", "templateType": "anything", "name": "q1", "description": "", "group": "Ungrouped variables"}, "n2": {"definition": "random(7..10)", "templateType": "anything", "name": "n2", "description": "", "group": "Ungrouped variables"}, "d2": {"definition": "s7*random(1..9)", "templateType": "anything", "name": "d2", "description": "", "group": "Ungrouped variables"}, "ta4": {"definition": "(abs(z2)^n4)*cos(n4*arg(z2))", "templateType": "anything", "name": "ta4", "description": "", "group": "Ungrouped variables"}, "b4": {"definition": "precround(tb4,3)", "templateType": "anything", "name": "b4", "description": "", "group": "Ungrouped variables"}, "tb3": {"definition": "abs(z1)^n2*sin(n2*arg(z1))", "templateType": "anything", "name": "tb3", "description": "", "group": "Ungrouped variables"}, "b3": {"definition": "precround(tb3,3)", "templateType": "anything", "name": "b3", "description": "", "group": "Ungrouped variables"}, "a3": {"definition": "precround(ta3,3)", "templateType": "anything", "name": "a3", "description": "", "group": "Ungrouped variables"}, "n4": {"definition": "random(7..10)", "templateType": "anything", "name": "n4", "description": "", "group": "Ungrouped variables"}, "z4": {"definition": "a3+b3*i", "templateType": "anything", "name": "z4", "description": "", "group": "Ungrouped variables"}, "q4": {"definition": "'The complex number is in the fourth quadrant.'", "templateType": "anything", "name": "q4", "description": "", "group": "Ungrouped variables"}}, "variable_groups": [], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["co", "ans1", "arg2", "ans3", "ans4", "tb4", "b4", "b1", "b2", "b3", "d4", "d2", "z6", "q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "m3", "arg1", "z3", "tb3", "ta4", "arg3", "tol", "ta3", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "ans2", "z1", "z2", "c4", "f", "a2", "t", "n2", "n4", "c2"], "statement": "<p>Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.</p>\n  <p>Note that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n  <p>Important: When calculating the final answer in part (iii) of each question, you must use non-truncated values for the modulus and argument calculated in parts (i) and (ii) and not the approximated values, otherwise the final answer may not be correct.</p>", "showQuestionGroupNames": false, "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "pickQuestions": 0, "name": ""}], "name": "Maria's copy of De Moivre's Theorem: Positive Powers", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "<p><strong>15/7/2015:</strong></p>\n<p>Added tags.</p>\n<p><strong>5/07/2012:</strong></p>\n<p>Added tags.</p>\n<p>The question doesn't really make sense. In the instruction we are asked to find modulus and argument of (a+i*b)^n but the question that is displayed is to find the modulus and argument of (a+i*b). Does the question need to be rewrittten to avoid this conflicting instruction?</p>\n<p><strong>9/07/2012:</strong></p>\n<p>Changed prompt instructions to make this question clearer.</p>\n<p>Corrected request from 2dps to 3 dps for last question.</p>\n<p>Also set new tolerance variable, tol=0.001 for all numeric answers.</p>\n<p><strong>13/07/2012:</strong></p>\n<p><strong><br /></strong>Not a good question as can be done without using de Moivre. Needs to be recast.</p>", "description": "<p>Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find positive powers of the complex numbers.</p>"}, "functions": {}, "advice": "<p>Given a complex number $z=r(\\cos(\\theta)+i\\sin(\\theta))$ de Moivre's theorem states that $z^n=r^n(\\cos(n\\theta)+i\\sin(n\\theta))$ for an integer power $n$.<br/>So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.</p>\n  <p>As usual, you must be careful that the argument is calculated correctly by paying attention to the quadrant of the complex plane in which the complex number lies.</p>\n  <p>Also remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n  <p>With the above in mind we can now answer the questions:</p>\n  <p>a)</p>\n  <h4>Modulus</h4>\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  <p>Note that $r^{\\var{n2}}=|(\\var{z1})^{\\var{n2}}| =\\var{abs(z1)}^{\\var{n2}}=\\var{abs(z1)^n2}$ which we will use in the calculation for $(\\var{z1})^{\\var{n2}}$</p>\n  <h4>Argument</h4>\n  <p>{m1}.<br/>Hence we see that: <br/>\\[\\begin{eqnarray*} \\arg(\\var{z1}) &amp;=&amp; \\var{arg(z1)}\\\\ &amp;=&amp; \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.</p>\n  <p>Hence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &amp;=&amp; \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &amp;=&amp; \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &amp;=&amp; \\simplify[std]{{a3}+{b3}i}. \\end{eqnarray*} \\]&nbsp;</p>\n  <p>b)</p>\n  <h4>Modulus</h4>\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  <p>Note that $r^{\\var{n4}}=|(\\var{z2})^{\\var{n4}}| =\\var{abs(z2)}^{\\var{n4}}=\\var{abs(z2)^n4}$ which we will use in the calculation for $(\\var{z2})^{\\var{n4}}$</p>\n  <h4>Argument</h4>\n  <p>{m2}.<br/>Hence we see that: <br/>\\[\\begin{eqnarray*} \\arg(\\var{z2}) &amp;=&amp; \\var{arg(z2)}\\\\ &amp;=&amp; \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.</p>\n  <p>We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.</p>\n  <p>Hence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &amp;=&amp; \\var{abs(z2)^n4}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &amp;=&amp; \\var{abs(z2)^n4}\\cos(\\var{n4*arg(z2)})+\\var{abs(z2)^n4}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &amp;=&amp; \\simplify[std]{{a4}+{b4}i}. \\end{eqnarray*} \\]&nbsp;</p>\n  <p>&nbsp;</p>", "tags": ["arctan", "argument of a complex number", "argument of complex number", "argument of complex numbers", "checked2015", "complex numbers", "de Moivre's theorem", "de Moivre's Theorem", "de moivre's theorem", "MAS1602", "modulus of complex numbers", "quadrants", "quadrants in the complex plane"], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}