// Numbas version: exam_results_page_options
{"name": "WM104 in-class assessment : part (i) complex numbers - 25% extra timing", "metadata": {"description": "<p>You are allowed 31 minutes to complete this test.</p><p>You may use rough paper for your calculations.</p>", "licence": "None specified"}, "duration": 1860, "percentPass": "40", "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Y1-A Assessment 1 - part (i)", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Shaheen's copy of Arithmetics of complex numbers II", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "tags": [], "metadata": {"description": "<p>Multiplication and addition of complex numbers. Four parts.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Find the following complex numbers in the form $a+bi\\;$ where $a$ and $b$ are real.</p>\n<p>Input all numbers as fractions or integers (whole numbers). Also do not include brackets in your answers.</p>", "advice": "<p>a)</p>\n  <p>The solution is given by:</p>\n  <p><br/>$\\simplify[std]{{e6*i}}(\\simplify[std]{{a}})=\\simplify[std]{{a*e6*i}}$</p>\n  <p><br/>b)</p>\n  <p>$\\simplify[std]{{a}*{z4}={a*z4}}$</p>\n  <p><br/>c) <br/>\\[ \\begin{eqnarray*} \\simplify[std,!otherNumbers]{{a}*({a3} + {b3} * i + {c3} * i ^ 2 + {d3} * i ^ 3)}&amp;=&amp;\\simplify[std]{{a}*{a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3}}\\\\ &amp;=&amp;\\simplify[std]{{a*(a3 + b3 * i + c3 * i ^ 2 + d3 * i ^ 3)}} \\end{eqnarray*} \\]<br/>d)</p>\n  <p>This can be calculated by using the formula twice, firstly to multiply out the first two sets of parentheses,&nbsp;</p>\n  <p>and then to multiply the result of that calculation by the third set of parentheses.</p>\n  <p>So we obtain:<br/>\\[ \\begin{eqnarray*} (\\var{a})(\\var{z1})(\\var{z3})&amp;=&amp;((\\var{a})(\\var{z1}))(\\var{z3})\\\\ &amp;=&amp;(\\var{a*(z1)})(\\var{z3})\\\\ &amp;=&amp;\\var{a*(z1)*(z3)} \\end{eqnarray*} \\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "s1*random(1..9)+s2*random(1..9)*i", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "s3*random(1..9)+f6*i", "description": "", "templateType": "anything"}, "d3": {"name": "d3", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "s1*random(1..9)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "s2*random(1..9)", "description": "", "templateType": "anything"}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "s6*random(1..9)+e6*i", "description": "", "templateType": "anything"}, "e6": {"name": "e6", "group": "Ungrouped variables", "definition": "s5*random(3..9)", "description": "", "templateType": "anything"}, "z4": {"name": "z4", "group": "Ungrouped variables", "definition": "s6*s2*random(1..9)+s3*s5*random(1..9)*i", "description": "", "templateType": "anything"}, "f6": {"name": "f6", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "templateType": "anything"}, "d6": {"name": "d6", "group": "Ungrouped variables", "definition": "s4*random(1..9)", "description": "", "templateType": "anything"}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "s2*random(1..9)+d6*i", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "f6", "s3", "s2", "s1", "d3", "s6", "s5", "s4", "c3", "a3", "b3", "d6", "e6", "z4", "z1", "z2", "z3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$\\var{e6*i}(\\simplify[std]{{a}})\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{a*e6*i}", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers. Also do not include brackets in your answers.</p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$(\\simplify[std]{{a}})(\\simplify[std]{{z4}})\\;=\\;$[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "({a*z4})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": [".", ")", "("], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as fractions or integers. Also do not include brackets in your answers.</p>"}, "valuegenerators": []}], "sortAnswers": false}]}, {"name": "NEW Shaheen's copy of Complex number arithmetic: multiply and divide", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "tags": [], "metadata": {"description": "<p>This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Given the complex numbers</p>\n<p>\\[a=\\var{a},\\quad b=\\var{b},\\quad c=\\var{c},\\quad d=\\var{d},\\]</p>\n<p>calculate the following. Enter your answers in the form</p>\n<p>\\[x+yi\\]</p>\n<p><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">Note, ensure that you write i (</span><strong style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">not j</strong><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">) at the </span><strong style=\"background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: bold; letter-spacing: normal; min-height: 0px; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">end of the imaginary part</strong><span style=\"display: inline !important; float: none; background-color: transparent; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;\">!</span></p>", "advice": "<p>\\[z_1+z_2=(\\var{z1})+(\\var{z2})=\\simplify{{z1}+{z2}}\\]</p>\n<p>\\[z_1-z_2=(\\var{z1})-(\\var{z2})=\\simplify{{z1}-{z2}}\\]</p>\n<p>\\[z_1z_3=(\\var{z1})(\\var{z3})=\\var{z1z3}\\]</p>\n<p>\\[\\frac{z_3}{z_4}=\\frac{\\var{z3}}{\\var{z4}}=\\frac{(\\var{z3})(\\var{z4bar})}{(\\var{z4})(\\var{z4bar})}=\\simplify{{z3z4bar}/{modz4sq}}\\approx \\var{re_z3onz4}+\\var{im_z3onz4}\\]</p>", "rulesets": {}, "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}, "twocans": {"name": "twocans", "group": "Ungrouped variables", "definition": "(a*conj(b))", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "twocans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>$ab=$</p>", "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Expand the brackets $(\\var{a})(\\var{b})$.</p>"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Simplify using the fact $i^2=-1$.</p>"}], "answer": "{a*b}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>No brackets are required or allowed.</p>"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>For the question below, please&nbsp;give your answers for both the real and imaginary parts either as whole numbers or decimals correct to 4 decimal places, as appropriate:</p>\n<p>$\\frac{b}{d}=$&nbsp;</p>", "answer": "{b/d}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "<p>No brackets are required or allowed.</p>"}, "valuegenerators": []}]}, {"name": "Rectangular to Polar Conversion - Degrees", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Peter Johnston", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/771/"}], "advice": "<p>Firstly, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$</p>\n<p>Secondly, since the real part of $z$ is {rez} and the imaginary part of $z$ is {imz}, {quad} So,</p>\n<p>$\\arg(z) = \\var{dpformat(argz,2)}^\\circ$.</p>\n<p><br/>&nbsp;</p>", "statement": "", "variable_groups": [], "variables": {"q1": {"name": "q1", "description": "", "definition": "'The complex number is in the first quadrant.'", "templateType": "anything", "group": "Ungrouped variables"}, "s2": {"name": "s2", "description": "", "definition": "switch(t=4,-1,t=3,-1,1)", "templateType": "anything", "group": "Ungrouped variables"}, "q3": {"name": "q3", "description": "", "definition": "'The complex number is in the third quadrant.'", "templateType": "anything", "group": "Ungrouped variables"}, "t": {"name": "t", "description": "", "definition": "random(1..4)", "templateType": "anything", "group": "Ungrouped variables"}, "b1": {"name": "b1", "description": "", "definition": "s2*random(1..10)", "templateType": "anything", "group": "Ungrouped variables"}, "a1": {"name": "a1", "description": "", "definition": "s1*random(1..10)", "templateType": "anything", "group": "Ungrouped variables"}, "imz": {"name": "imz", "description": "<p>im</p>", "definition": "if (b1>=0,\"Positive\",\"Negative\")", "templateType": "anything", "group": "Ungrouped variables"}, "modz": {"name": "modz", "description": "", "definition": "abs(z1)", "templateType": "anything", "group": "Ungrouped variables"}, "q4": {"name": "q4", "description": "", "definition": "'The complex number is in the fourth quadrant.'", "templateType": "anything", "group": "Ungrouped variables"}, "tol": {"name": "tol", "description": "", "definition": "0.01", "templateType": "anything", "group": "Ungrouped variables"}, "rez": {"name": "rez", "description": "", "definition": "if (a1>=0,\"Positive\",\"Negative\")", "templateType": "anything", "group": "Ungrouped variables"}, "z1": {"name": "z1", "description": "", "definition": "a1+b1*i", "templateType": "anything", "group": "Ungrouped variables"}, "q2": {"name": "q2", "description": "", "definition": "'The complex number is in the second quadrant.'", "templateType": "anything", "group": "Ungrouped variables"}, "argz": {"name": "argz", "description": "", "definition": "if(arg(z1)<0,degrees(arg(z1)+2*pi),degrees(arg(z1)))", "templateType": "anything", "group": "Ungrouped variables"}, "quad": {"name": "quad", "description": "", "definition": "switch(t=1,q1,t=2,q2,t=3,q3,q4)", "templateType": "anything", "group": "Ungrouped variables"}, "s1": {"name": "s1", "description": "", "definition": "switch(t=1,1,t=4,1,-1)", "templateType": "anything", "group": "Ungrouped variables"}}, "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["b1", "q1", "q3", "q2", "q4", "s2", "s1", "quad", "a1", "z1", "t", "rez", "imz", "modz", "argz", "tol"], "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "gaps": [{"correctAnswerStyle": "plain", "precisionType": "dp", "showFeedbackIcon": true, "scripts": {}, "showPrecisionHint": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "modz-tol", "marks": 1, "precision": "2", "precisionPartialCredit": 0, "variableReplacements": [], "allowFractions": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "maxValue": "modz+tol"}, {"correctAnswerStyle": "plain", "precisionType": "dp", "showFeedbackIcon": true, "scripts": {}, "showPrecisionHint": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "notationStyles": ["plain", "en", "si-en"], "minValue": "argz-tol", "marks": 1, "precision": "2", "precisionPartialCredit": 0, "variableReplacements": [], "allowFractions": false, "correctAnswerFraction": false, "showCorrectAnswer": true, "maxValue": "argz+tol"}], "prompt": "<p>Find the modulus and argument (in degrees) of the complex number $z=\\var{z1}$.</p>\n<p>$|z| = $ [[0]]</p>\n<p>$\\arg(z) = $ [[1]]$^\\circ$.</p>", "variableReplacements": [], "scripts": {}, "type": "gapfill", "showCorrectAnswer": true}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "tags": [], "metadata": {"description": "<p>Practice to decide which quadrant a complex number lies in.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "type": "question"}, {"name": "Rectangular to Polar Conversion - Radians", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Peter Johnston", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/771/"}], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Practice to decide which quadrant a complex number lies in.</p>"}, "tags": [], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "ungrouped_variables": ["b1", "q1", "q3", "q2", "q4", "s2", "s1", "quad", "a1", "z1", "t", "rez", "imz", "modz", "argz", "tol"], "statement": "", "parts": [{"gaps": [{"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "marks": 1, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "modz-tol", "scripts": {}, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "modz+tol", "showFeedbackIcon": true}, {"precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "2", "showPrecisionHint": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "strictPrecision": false, "precisionType": "dp", "marks": 1, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "minValue": "argz-tol", "scripts": {}, "allowFractions": false, "correctAnswerFraction": false, "maxValue": "argz+tol", "showFeedbackIcon": true}], "prompt": "<p>Find the modulus and argument (in radians) of the complex number $z=\\var{z1}$.</p>\n<p>$|z| = $ [[0]]</p>\n<p>$\\arg(z) = $ [[1]] radians.</p>", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacements": []}], "advice": "<p>Firstly, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$</p>\n<p>Secondly, since the real part of $z$ is {rez} and the imaginary part of $z$ is {imz}, {quad} So,</p>\n<p>$\\arg(z) = \\var{dpformat(argz,2)}$ radians.</p>\n<p><br/>&nbsp;</p>", "preamble": {"css": "", "js": ""}, "variable_groups": [], "variables": {"q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "name": "q3", "description": ""}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "t", "description": ""}, "imz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (b1>=0,\"Positive\",\"Negative\")", "name": "imz", "description": "<p>im</p>"}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,1,-1)", "name": "s1", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..10)", "name": "b1", "description": ""}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "name": "z1", "description": ""}, "quad": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q1,t=2,q2,t=3,q3,q4)", "name": "quad", "description": ""}, "argz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(arg(z1)<0,arg(z1)+2*pi,arg(z1))", "name": "argz", "description": ""}, "q1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the first quadrant.'", "name": "q1", "description": ""}, "q4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "name": "q4", "description": ""}, "modz": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(z1)", "name": "modz", "description": ""}, "rez": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if (a1>=0,\"Positive\",\"Negative\")", "name": "rez", "description": ""}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "name": "q2", "description": ""}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "name": "tol", "description": ""}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=4,-1,t=3,-1,1)", "name": "s2", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..10)", "name": "a1", "description": ""}}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}, {"name": "Shaheen's copy of De Moivre's Theorem: Positive Powers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "tags": [], "metadata": {"description": "<p>Find modulus and argument of two complex numbers. </p><p>Then use De Moivre's Theorem to find powers of the complex numbers.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Use de Moivre's theorem to write the following complex number in the form $a+bi$.</p>\n<p>Remember that the argument of a complex numbers lies in the range $-\\pi \\lt \\theta \\le \\pi$.</p>\n<p><span style=\"text-decoration: underline;\">Important</span></p>\n<p>When calculating the final answer <strong>in part (iii),&nbsp;</strong><strong>you must use whole number values (integers)</strong>, otherwise the final answer will not be marked as correct.</p>", "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*} \\] <strong>to 3 decimal places</strong>.</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*} \\] <strong>to 3 decimal places</strong>.</p>\n<p>We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ <strong>radians</strong>.</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*} \\] <strong>to 3 decimal places</strong>.</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*} \\] <strong>to 3 decimal places</strong>.</p>\n<p>We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ ra<strong>dians</strong>.</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>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "variables": {"c4": {"name": "c4", "group": "Ungrouped variables", "definition": "if(a4=f,f+1,f)", "description": "", "templateType": "anything"}, "b3": {"name": "b3", "group": "Ungrouped variables", "definition": "precround(tb3,3)", "description": "", "templateType": "anything"}, "s5": {"name": "s5", "group": "Ungrouped variables", "definition": "switch(t=3,-1,1)", "description": "", "templateType": "anything"}, "arg4": {"name": "arg4", "group": "Ungrouped variables", "definition": "precround(arg(z4),3)", "description": "", "templateType": "anything"}, "b4": {"name": "b4", "group": "Ungrouped variables", "definition": "precround(tb4,3)", "description": "", "templateType": "anything"}, "n4": {"name": "n4", "group": "Ungrouped variables", "definition": "random(7..10)", "description": "", "templateType": "anything"}, "s7": {"name": "s7", "group": "Ungrouped variables", "definition": "switch(t=2,-1,t=3,1,-1)", "description": "", "templateType": "anything"}, "arg2": {"name": "arg2", "group": "Ungrouped variables", "definition": "precround(arg(z2),3)", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "templateType": "anything"}, "arg3": {"name": "arg3", "group": "Ungrouped variables", "definition": "precround(arg(z3),3)", "description": "", "templateType": "anything"}, "q4": {"name": "q4", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "templateType": "anything"}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "c2+d2*i", "description": "", "templateType": "anything"}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "a1+b1*i", "description": "", "templateType": "anything"}, "z4": {"name": "z4", "group": "Ungrouped variables", "definition": "a3+b3*i", "description": "", "templateType": "anything"}, "tb4": {"name": "tb4", "group": "Ungrouped variables", "definition": "(abs(z2)^n4)*sin(n4*arg(z2))", "description": "", "templateType": "anything"}, "ans3": {"name": "ans3", "group": "Ungrouped variables", "definition": "precround(abs(z3),3)", "description": "", "templateType": "anything"}, "ta3": {"name": "ta3", "group": "Ungrouped variables", "definition": "abs(z1)^n2*cos(n2*arg(z1))", "description": "", "templateType": "anything"}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "if(co=a1,co+1,co)", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "s5*random(1..3)", "description": "", "templateType": "anything"}, "ans4": {"name": "ans4", "group": "Ungrouped variables", "definition": "precround(abs(z4),3)", "description": "", "templateType": "anything"}, "z6": {"name": "z6", "group": "Ungrouped variables", "definition": "c4+d4*i", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Ungrouped variables", "definition": "switch(t=1,q2,t=2,q1,t=3,q4,q2)", "description": "", "templateType": "anything"}, "m4": {"name": "m4", "group": "Ungrouped variables", "definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "description": "", "templateType": "anything"}, "q1": 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"Ungrouped variables", "definition": "switch(t=1,-1,t=3,-1,1)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "Ungrouped variables", "definition": "switch(t=1,q3,t=2,q4,t=3,q1,q3)", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Ungrouped variables", "definition": "random(7..10)", "description": "", "templateType": "anything"}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "templateType": "anything"}, "z5": {"name": "z5", "group": "Ungrouped variables", "definition": "a4+b4*i", "description": "", "templateType": "anything"}, "d4": {"name": "d4", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "a2+b2*i", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "s2*random(1..3)", "description": "", "templateType": "anything"}, "s8": {"name": "s8", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,-1,t=3,1,-1)", "description": "", "templateType": "anything"}, "q3": {"name": "q3", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "description": "", "templateType": "anything"}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "precround(abs(z2),3)", "description": "", "templateType": "anything"}, "ta4": {"name": "ta4", "group": "Ungrouped variables", "definition": "(abs(z2)^n4)*cos(n4*arg(z2))", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "precround(ta3,3)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "precround(ta4,3)", "description": "", "templateType": "anything"}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "description": "", "templateType": "anything"}, "s6": {"name": "s6", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "templateType": "anything"}, "arg1": {"name": "arg1", "group": "Ungrouped variables", "definition": "precround(arg(z1),3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "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"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Find the modulus and argument of $\\var{z1}$ <strong>&nbsp;correct to 3 decimal places</strong>.</p>\n<p>(i) $|\\var{z1}|\\;=\\;$ [[0]], <strong>correct&nbsp;</strong><strong>to 3 decimal places</strong>.</p>\n<p>(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] <strong>radians</strong>, <strong>correct&nbsp;to 3 decimal places</strong>.</p>\n<p>Hence find:</p>\n<p>(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]</p>\n<p>Input as a complex number, with integer values for the real and imaginary parts.</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "ans1-tol", "maxValue": "ans1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, 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true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "warnifunattempted", "message": "<p>There are answers missing - please check before continuing!</p>"}, "preventleave": true, "startpassword": "Y1Ai"}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "<p>The complex numbers assessment has now ended.</p>"}, "timedwarning": {"action": "warn", "message": "<p>You have 5 minutes left to complete this complex numbers assessment.</p>"}}, "feedback": {"showactualmark": false, "showtotalmark": true, "showanswerstate": false, "allowrevealanswer": false, "advicethreshold": 0, "intro": "<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"text-decoration: underline;\">WM104 in-class assessment: part (i) - complex numbers</span></p>\n<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\"></span></p>\n<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">Content assessed : </span></p>\n<ul>\n<li style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">complex arithmetic</span></li>\n<li style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">argument and modulus of complex numbers</span></li>\n<li style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">De Moivre's theorem<o:p></o:p></span></li>\n</ul>\n<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\"></span></p>\n<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">This complex numbers in-class assesment counts 20% towards your final maths grade for the WM104 module.<o:p></o:p></span></p>\n<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">Note, although questions are randomised for each student, all questions test the same learning outcomes at the same level for each student.&nbsp;&nbsp;<o:p></o:p></span></p>\n<p style=\"background: white; margin: 0cm 0cm 7.9pt 0cm;\"><span style=\"font-size: 11.5pt; font-family: 'Helvetica',sans-serif; color: #555555;\">If you have any questions during the test, please put up your hand to alert the invigilator that you need attention.&nbsp;<o:p></o:p></span></p>", "feedbackmessages": []}, "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "extensions": [], 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