// Numbas version: exam_results_page_options {"name": "25% EXTRA TIME - WM104: In-class Assessment - part (i) - complex numbers", "metadata": {"description": "

AEP Y1 - Engineering Mathematics Assessment 1(i)

Note that part(i) counts 20% towards the final Y1 Engineering Mathematics module grade.

", "licence": "None specified"}, "duration": 1860, "percentPass": "40", "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Shaheen's copy of Complex number arithmetic: add and subtract", "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": "

This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Given the complex numbers

\n

\\[a=\\var{z1},\\quad b=\\var{z2},\\quad c=\\var{z3}\\]

\n

calculate the following. Enter your answers in the form

\n

\\[x+yi\\]

\n

Note, ensure that you write i ( not j ) at the end of the imaginary part!

", "advice": "

\\[z_1+z_2=(\\var{z1})+(\\var{z2})=\\simplify{{z1}+{z2}}\\]

\n

\\[z_1-z_2=(\\var{z1})-(\\var{z2})=\\simplify{{z1}-{z2}}\\]

\n

\\[z_1z_3=(\\var{z1})(\\var{z3})=\\var{z1z3}\\]

\n

\\[\\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}\\]

", "rulesets": {}, "variables": {"z1": {"name": "z1", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}, "z3": {"name": "z3", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["z1", "z2", "z3"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$a+b=$

", "answer": "{z1+z2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

No brackets are required or allowed.

"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$a+c=$

", "answer": "{z1+z3}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

No brackets are required or allowed.

"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$a-b=$

", "answer": "{z1-z2}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

No brackets are required or allowed.

"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

$a-c=$

", "answer": "{z1-z3}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["(", ")"], "showStrings": false, "partialCredit": 0, "message": "

No brackets are required or allowed.

"}, "valuegenerators": []}]}, {"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": "

This question provides practice at adding, subtracting, dividing and multiplying complex numbers in rectangular form.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

Given the complex numbers

\n

\\[a=\\var{a},\\quad b=\\var{b},\\quad c=\\var{c},\\quad d=\\var{d},\\]

\n

calculate the following. Enter your answers in the form

\n

\\[x+yi\\]

\n

Note, ensure that you write i (not j) at the end of the imaginary part!

", "advice": "

\\[z_1+z_2=(\\var{z1})+(\\var{z2})=\\simplify{{z1}+{z2}}\\]

\n

\\[z_1-z_2=(\\var{z1})-(\\var{z2})=\\simplify{{z1}-{z2}}\\]

\n

\\[z_1z_3=(\\var{z1})(\\var{z3})=\\var{z1z3}\\]

\n

\\[\\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}\\]

", "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": "

$ab=$

", "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": "

Expand the brackets $(\\var{a})(\\var{b})$.

"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

Simplify using the fact $i^2=-1$.

"}], "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": "

No brackets are required or allowed.

"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "

For the question below, please give your answers for both the real and imaginary parts either as whole numbers or decimals correct to 4 decimal places, as appropriate:

\n

$\\frac{b}{d}=$ 

", "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": "

No brackets are required or allowed.

"}, "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": "

Firstly, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$

\n

Secondly, since the real part of $z$ is {rez} and the imaginary part of $z$ is {imz}, {quad} So,

\n

$\\arg(z) = \\var{dpformat(argz,2)}^\\circ$.

\n


 

", "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": "

im

", "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": "

Find the modulus and argument (in degrees) of the complex number $z=\\var{z1}$.

\n

$|z| = $ [[0]]

\n

$\\arg(z) = $ [[1]]$^\\circ$.

", "variableReplacements": [], "scripts": {}, "type": "gapfill", "showCorrectAnswer": true}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "tags": [], "metadata": {"description": "

Practice to decide which quadrant a complex number lies in.

", "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": "

Practice to decide which quadrant a complex number lies in.

"}, "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": "

Find the modulus and argument (in radians) of the complex number $z=\\var{z1}$.

\n

$|z| = $ [[0]]

\n

$\\arg(z) = $ [[1]] radians.

", "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "variableReplacements": []}], "advice": "

Firstly, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$

\n

Secondly, since the real part of $z$ is {rez} and the imaginary part of $z$ is {imz}, {quad} So,

\n

$\\arg(z) = \\var{dpformat(argz,2)}$ radians.

\n


 

", "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": "

im

"}, "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": "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/"}], "variable_groups": [], "variables": {"c4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(a4=f,f+1,f)", "description": "", "name": "c4"}, "b3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tb3,3)", "description": "", "name": "b3"}, "s5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=3,-1,1)", "description": "", "name": "s5"}, "arg4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z4),3)", "description": "", "name": "arg4"}, "b4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tb4,3)", "description": "", "name": "b4"}, "n4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..10)", "description": "", "name": "n4"}, "s7": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=2,-1,t=3,1,-1)", "description": "", "name": "s7"}, "arg2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z2),3)", "description": "", "name": "arg2"}, "m1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "name": "m1"}, "arg3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z3),3)", "description": "", "name": "arg3"}, "q4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "name": "q4"}, "ans1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "description": "", "name": "ans1"}, "z3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c2+d2*i", "description": "", "name": "z3"}, "z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1+b1*i", "description": "", "name": "z1"}, "z4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a3+b3*i", "description": "", "name": "z4"}, "tb4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(abs(z2)^n4)*sin(n4*arg(z2))", "description": "", "name": "tb4"}, "ans3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z3),3)", "description": "", "name": "ans3"}, "ta3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(z1)^n2*cos(n2*arg(z1))", "description": "", "name": "ta3"}, "a2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(co=a1,co+1,co)", "description": "", "name": "a2"}, "s4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "name": "s4"}, "b2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..3)", "description": "", "name": "b2"}, "ans4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z4),3)", "description": "", "name": "ans4"}, "z6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "c4+d4*i", "description": "", "name": "z6"}, "m2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q2,t=2,q1,t=3,q4,q2)", "description": "", "name": "m2"}, "m4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q1,t=2,q3,t=3,q2,q4)", "description": "", "name": "m4"}, "q1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the first quadrant.'", "description": "", "name": "q1"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "t"}, "s1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,1,-1)", "description": "", "name": "s1"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.001", "description": "", "name": "tol"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s1*random(1..3)", "description": "", "name": "a1"}, "co": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s4*random(1..3)", "description": "", "name": "co"}, "tb3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(z1)^n2*sin(n2*arg(z1))", "description": "", "name": "tb3"}, "s2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=3,-1,1)", "description": "", "name": "s2"}, "s3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "description": "", "name": "s3"}, "m3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,q3,t=2,q4,t=3,q1,q3)", "description": "", "name": "m3"}, "n2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(7..10)", "description": "", "name": "n2"}, "d2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "description": "", "name": "d2"}, "c2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s6*random(1..9)", "description": "", "name": "c2"}, "z5": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a4+b4*i", "description": "", "name": "z5"}, "d4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "description": "", "name": "d4"}, "f": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "f"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a2+b2*i", "description": "", "name": "z2"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "s2*random(1..3)", "description": "", "name": "b1"}, "s8": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,-1,t=3,1,-1)", "description": "", "name": "s8"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "description": "", "name": "q3"}, "ans2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(abs(z2),3)", "description": "", "name": "ans2"}, "ta4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(abs(z2)^n4)*cos(n4*arg(z2))", "description": "", "name": "ta4"}, "a3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ta3,3)", "description": "", "name": "a3"}, "a4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(ta4,3)", "description": "", "name": "a4"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "description": "", "name": "q2"}, "s6": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "name": "s6"}, "arg1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(arg(z1),3)", "description": "", "name": "arg1"}}, "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"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans1+tol", "minValue": "ans1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "arg1+tol", "minValue": "arg1-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{a3}+{b3}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

Find the modulus and argument of $\\var{z1}$ to 3 decimal places.

\n

(i) $|\\var{z1}|\\;=\\;$ [[0]], to 3 decimal places.

\n

(ii) $\\arg(\\var{z1})\\;=\\;$[[1]] radians, to 3 decimal places.

\n

Hence find:

\n

(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]

\n

Input as a complex number, with real and imaginary parts integral values.

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ans2+tol", "minValue": "ans2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "arg2+tol", "minValue": "arg2-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"answer": "{a4}+{b4}*i", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 2, "vsetrangepoints": 5}], "type": "gapfill", "prompt": "

Find the modulus and argument of $\\var{z2}$ to 3 decimal places.

\n

(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.

\n

(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.

\n

Hence find:

\n

(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]

\n

Input as a complex number, with real and imaginary parts integral values.

", "showCorrectAnswer": true, "marks": 0}], "statement": "

Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.

\n

Note that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.

\n

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.

", "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"], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "

15/7/2015:

\n

Added tags.

\n

5/07/2012:

\n

Added tags.

\n

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?

\n

9/07/2012:

\n

Changed prompt instructions to make this question clearer.

\n

Corrected request from 2dps to 3 dps for last question.

\n

Also set new tolerance variable, tol=0.001 for all numeric answers.

\n

13/07/2012:

\n


Not a good question as can be done without using de Moivre. Needs to be recast.

", "licence": "Creative Commons Attribution 4.0 International", "description": "

Find modulus and argument of two complex numbers. Then use De Moivre's Theorem to find positive powers of the complex numbers.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

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 $n$.
So if we know the modulus $r$ and the argument $\\theta$ for $z$ then the theorem provides a way of calculating $z^n$.

\n

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.

\n

Also remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.

\n

With the above in mind we can now answer the questions:

\n

a)

\n

Modulus

\n

\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.

\n

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 $(\\var{z1})^{\\var{n2}}$

\n

Argument

\n

{m1}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

\n

We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.

\n

Hence we have \\[\\begin{eqnarray*}(\\var{z1})^{\\var{n2}} &=& \\var{abs(z1)^n2}(\\cos(\\var{n2*arg(z1)})+\\sin(\\var{n2*arg(z1)})i)\\\\ &=& \\var{abs(z1)^n2}\\cos(\\var{n2*arg(z1)})+\\var{abs(z1)^n2}\\times\\sin(\\var{n2*arg(z1)})i\\\\ &=& \\simplify[std]{{a3}+{b3}i}. \\end{eqnarray*} \\] 

\n

b)

\n

Modulus

\n

\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.

\n

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 $(\\var{z2})^{\\var{n4}}$

\n

Argument

\n

{m2}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

\n

We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.

\n

Hence we have \\[\\begin{eqnarray*}(\\var{z2})^{\\var{n4}} &=& \\var{abs(z2)^n4}(\\cos(\\var{n4*arg(z2)})+\\sin(\\var{n4*arg(z2)})i)\\\\ &=& \\var{abs(z2)^n4}\\cos(\\var{n4*arg(z2)})+\\var{abs(z2)^n4}\\times\\sin(\\var{n4*arg(z2)})i\\\\ &=& \\simplify[std]{{a4}+{b4}i}. \\end{eqnarray*} \\] 

\n

 

"}]}], "navigation": {"allowregen": false, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "showresultspage": "oncompletion", "onleave": {"action": "warnifunattempted", "message": "

You have not answered this question!

"}, "preventleave": true, "startpassword": "y1ai"}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "

This complex numbers assessment has now ended. 

"}, "timedwarning": {"action": "warn", "message": "

You have 5 minutes left to complete this complex numbers assessment.

"}}, "feedback": {"showactualmark": false, "showtotalmark": true, "showanswerstate": false, "allowrevealanswer": false, "advicethreshold": 0, "intro": "

You are allowed 31 minutes to complete this complex numbers assessment.

\n

This assessment is worth 20% of your overall marks for the WM104 module.

\n

Content assessed: complex arithmetic, polar and cartesian/rectangular form, modulus and argument, de moivre's theorem

", "feedbackmessages": []}, "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}], "extensions": [], "custom_part_types": [], "resources": []}