// Numbas version: exam_results_page_options {"name": "Complex Numbers Quiz V2", "metadata": {"description": "Complex number quiz", "licence": "All rights reserved"}, "duration": 4800, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Part I", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questions": [{"name": "Real and imaginary parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Kariane Ouellet", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4505/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

For each of the following, simplify the number in rectangular form, and state the real and imaginary parts.

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a) $z= \\simplify{ {{re1}}+{{im1}}i}$

\n

\n

$Re(z)=$[[0]]

\n

$Im(z)=$[[1]]

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$z=\\simplify[basic]{ {{co1}}+{{co2}}i+{{co3}}i^2}$

\n

$Re(z)=$[[0]]

\n

$Im(z)=$[[1]]

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$z=i^\\var{power}$

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What are the solutions of: $\\simplify {x^2-{{co1}}x+{{co2}}=0}$?

\n

Write your answers in exact values. (for fractions, use the /, and for square root, write sqrt(number)) 

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The solution from the negative root is $x_1$=[[0]]

\n

The solution from the positive root is $x_2$=[[1]]

\n

*Do not forget to write i if there is in your answers*

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Simplify the following expression and write your answer in rectangular form. (input: a+bi, with a and b real numbers)

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$\\simplify[basic]{ z=({{re_a}}+{{im_a}}i)({{re_b}}+{{im_b}}i) }$

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Two parts.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Express the following in rectangular form.

\n

Input $a$ and $b$ as fractions or integers (whole numbers) and not as decimals.

", "advice": "\n \n \n

Division of two complex numbers can be performed by mutiplying both the numerator and denominator by the conjugate of the denominator.
Suppose that \\[ z = \\frac{a+bi}{c+di},\\;\\; c+di \\neq 0\\] then we have:
\\[\\begin{eqnarray*}\n \n z&=&\\frac{a+bi}{c+di}\\\\\n \n &=&\\frac{(a+bi)(c-di)}{(c+di)(c-di)}\\\\\n \n &=&\\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\\\\\n \n &=&\\frac{ac+bd}{c^2+d^2}+\\frac{bc-ad}{c^2+d^2}i\n \n \\end{eqnarray*}\n \n \\]
Although this is a formula for the inverse, the best way to find these complex numbers is to remember to multiply top and bottom by the conjugate of the denominator.
(a)
\\[\\begin{eqnarray*}\\simplify[std]{{c1}/{z1}} &=&\\simplify[std]{({c1}*{conj(z1)})/({z1}*{conj(z1)})}\\\\\n \n &=&\\simplify[std]{{c1*conj(z1)}/{abs(z1)^2}}\\\\\n \n &=& \\simplify[std]{{c1*re(z1)}/{abs(z1)^2}-{c1*im(z1)}/{abs(z1)^2}*i}\n \n \\end{eqnarray*} \\]
(b)
\\[\\begin{eqnarray*}\\simplify[std]{{c2}/{z2}} &=&\\simplify[std]{({c2}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{c2*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{c2*re(z2)}/{abs(z2)^2}-{c2*im(z2)}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]
(c)
\\[\\begin{eqnarray*}\\simplify[std]{{z1}/{z3}} &=&\\simplify[std]{({z1}*{conj(z3)})/({z3}*{conj(z3)})}\\\\\n \n &=&\\simplify[std]{{z1*conj(z3)}/{abs(z3)^2}}\\\\\n \n &=& \\simplify[std]{{re(z1*conj(z3))}/{abs(z3)^2}+{im(z1*conj(z3))}/{abs(z3)^2}*i}\n \n \\end{eqnarray*} \\]
(d)
\\[\\begin{eqnarray*}\\simplify[std]{{z3}/{z2}} &=&\\simplify[std]{({z3}*{conj(z2)})/({z2}*{conj(z2)})}\\\\\n \n &=&\\simplify[std]{{z3*conj(z2)}/{abs(z2)^2}}\\\\\n \n &=& \\simplify[std]{{re(z3*conj(z2))}/{abs(z2)^2}+{im(z3*conj(z2))}/{abs(z2)^2}*i}\n \n \\end{eqnarray*} \\]

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$\\displaystyle \\simplify[std]{{c1}/{z1}} = $ [[0]]

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$\\displaystyle \\simplify[std]{{z1}/{z3}}\\;=\\;$[[0]].

\n

Do not include brackets in your answer.

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Finding the modulus and argument (in radians) of four complex numbers; the arguments between $-\\pi$ and $\\pi$ and careful with quadrants!

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

Find the modulus and argument (in radians) of the following complex numbers, where the argument lies between $-\\pi$ and $\\pi$. For example, if you find the angle is $4\\pi/3$, you should input $-2.094$ as your answer (found by doing -($2\\pi-4\\pi/3$))

\n

\n

When calculating the argument pay particular attention to the quadrant in which the complex number lies.

\n

Input all answers to 3 decimal places (not in exact value).

", "advice": "

Note that the arguments $\\theta$ of the complex numbers are in radians and have to be in the range $-\\pi < \\theta \\le \\pi$.

\n

You have to be careful with using a standard calculator when you are finding the argument of a complex number.

\n

If $z=a+bi=r(\\cos(\\theta)+i\\sin(\\theta))$ then we have:$r\\cos(\\theta)=a,\\;\\;r\\sin(\\theta)=b$ and so $\\tan(\\theta) = b/a$.

\n

Using a calculator to find the argument via $\\arctan(b/a)$ works in the range $-\\pi < \\theta \\le \\pi$ when the complex number is in the first or fourth quadrants – you get the correct value.

\n

However, The calculator gives the wrong value for complex numbers in the other quadrants.

\n

Complex number in the Second Quadrant.

\n

Since $\\arctan(b/a)$ does not distinguish between the second and fourth quadrants and the calculator gives the argument for the fourth quadrant you have to add $\\pi$ onto the calculator value.

\n

Complex number in the Third Quadrant.

\n

Since $\\arctan(b/a)$ does not distinguish between the first and third quadrants and the calculator gives the argument for the first quadrant you have to take away $\\pi$ from the calculator value.

\n

a)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

Argument.

\n

{m1}

\n

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

\n

b)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

Argument.

\n

{m2}

\n

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

\n

c)Modulus.

\n

\\[ \\begin{eqnarray*} |\\var{z3}|&=&\\sqrt{(\\var{c2})^2+(\\var{d2})^2}\\\\ &=& \\var{abs(z3)}\\\\ &=&\\var{ans3} \\end{eqnarray*} \\] to 3 decimal places.

\n

Argument.

\n

{m3}

\n

Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z3}) &=& \\var{arg(z3)}\\\\ &=& \\var{arg3}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

\n

d)Modulus.

\n

\\[ \\begin{eqnarray*} |\\var{z4}|&=&\\sqrt{(\\var{a3})^2+(\\var{b3})^2}\\\\ &=& \\var{abs(z4)}\\\\ &=&\\var{ans4} \\end{eqnarray*} \\] to 3 decimal places.

\n

Argument.

\n

{m4}

\n

Hence we see that: \\[\\begin{eqnarray*} \\arg(\\var{z4}) &=& \\var{arg(z4)}\\\\ &=& \\var{arg4}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.

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"description": "", "templateType": "anything"}, "ans1": {"name": "ans1", "group": "Ungrouped variables", "definition": "precround(abs(z1),3)", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "s5*random(1..9)", "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"}, "q1": {"name": "q1", "group": "Ungrouped variables", "definition": "'The complex number is in the first quadrant.'", "description": "", "templateType": "anything"}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "s1": {"name": "s1", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=4,1,-1)", "description": "", "templateType": "anything"}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.001", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "s1*random(2..9)", "description": "", "templateType": "anything"}, "c4": {"name": "c4", "group": "Ungrouped variables", "definition": "if(a4=f,f+1,f)", "description": "", "templateType": "anything"}, "s2": {"name": "s2", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=3,-1,1)", "description": "", "templateType": "anything"}, "q2": {"name": "q2", "group": "Ungrouped variables", "definition": "'The complex number is in the second quadrant.'", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "Ungrouped variables", "definition": "switch(t=1,q3,t=2,q4,t=3,q1,q3)", "description": "", "templateType": "anything"}, "d2": {"name": "d2", "group": "Ungrouped variables", "definition": "s7*random(1..9)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..5)", "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(3..9)", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Ungrouped variables", "definition": "switch(t=1,q4,t=2,q2,t=3,q3,q1)", "description": "", "templateType": "anything"}, "q3": {"name": "q3", "group": "Ungrouped variables", "definition": "'The complex number is in the third quadrant.'", "description": "", "templateType": "anything"}, "s4": {"name": "s4", "group": "Ungrouped variables", "definition": "switch(t=1,-1,t=4,-1,1)", "description": "", "templateType": "anything"}, "ans2": {"name": "ans2", "group": "Ungrouped variables", "definition": "precround(abs(z2),3)", "description": "", "templateType": "anything"}, "q4": {"name": "q4", "group": "Ungrouped variables", "definition": "'The complex number is in the fourth quadrant.'", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "s3*random(1..9)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "s8*random(1..9)", "description": "", "templateType": "anything"}, "s3": {"name": "s3", "group": "Ungrouped variables", "definition": "switch(t=1,1,t=2,-1,t=3,-1,1)", "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": ["ans1", "ans2", "ans3", "ans4", "b4", "b1", "b2", "b3", "d4", "d2", "q1", "q3", "q2", "q4", "s3", "s2", "s1", "s7", "s6", "s5", "s4", "m4", "m1", "z2", "m3", "arg1", "z3", "arg2", "arg3", "tol", "arg4", "m2", "a1", "a3", "s8", "a4", "z4", "z5", "z6", "z1", "c2", "c4", "f", "n", "a2", "t"], "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": "

$|\\var{z1}|=\\;\\;$[[0]], $\\arg(\\var{z1})=\\;\\;$[[1]] radians

\n

Input both answers to 3 decimal places.

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$|\\var{z2}|=\\;\\;$[[0]], $\\arg(\\var{z2})=\\;\\;$[[1]] radians

\n

Input both answers to 3 decimal places.

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$|\\var{z3}|=\\;\\;$[[0]], $\\arg(\\var{z3})=\\;\\;$[[1]] radians

\n

Input both answers to 3 decimal places.

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$|\\var{z4}|=\\;\\;$[[0]], $\\arg(\\var{z4})=\\;\\;$[[1]] radians

\n

Input both answers to 3 decimal places.

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Given are the complex numbers $z_1=\\var{rad1}cis(\\var{arg1}^\\circ)$ and $z_2=\\var{rad2}cis(\\var{arg2}^\\circ)$, where the argument is in degrees.

\n

Find the following in polar form with $0^\\circ \\leq \\theta <360^\\circ$. You might input fractions (/) and exponents (^).

\n

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$z_1 \\cdot z_2=$ [[0]] $cis$ ([[1]])

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$z_1/z_2=$ [[0]] $cis$ ([[1]])

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$z_1^\\var{power}=$ [[0]] $cis$ ([[1]])

", "gaps": [{"type": "jme", "useCustomName": true, "customName": "radius", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{rad1}^{power}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": []}, {"type": "numberentry", "useCustomName": true, "customName": "Arg", "marks": 1, "showCorrectAnswer": false, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "mod({power}*{arg1},360) ", "maxValue": "mod({power}*{arg1},360) ", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}]}, {"name": "Polar to Rectangular form", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Kariane Ouellet", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4505/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

The following are complex numbers in polar form. Write them in rectangular form using the exact values (input division / and exponents ^if needed).

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a) $\\simplify{z={{rada}}(cos({{arga}}pi/6)+i*sin({{arga}}pi/6))}$

\n

The rectangular form is: [[0]]

\n

\n

b) $\\simplify{z={{radb}}(cos({{argb}}pi/4)+i*sin({{argb}}pi/4))}$

\n

The rectangular form is: [[1]]

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Find all possible solutions to the equation $z^5=\\var{rad}cis(\\var{arg}^\\circ)$.

\n

Give your answers in polar form with $0^\\circ \\leq \\theta <360^\\circ$, write the radii and angles in decimal form with 2 decimal places.

\n

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The solutions are:

\n

a) $z_0=$[[5]] $cis($[[0]]$^\\circ)$

\n

b) $z_1=$[[5]] $cis($[[1]]$^\\circ)$

\n

c) $z_2=$[[5]] $cis($[[2]]$^\\circ)$

\n

d) $z_3=$[[5]]$ cis($[[3]]$^\\circ)$

\n

e) $z_4=$[[5]] $cis($[[4]]$^\\circ)$

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Select the graph that represents the locus of points $|z-1+2i|<2$.

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Select the graph that represents the locus of points $|z-1+2i|>2$.

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Select the graph that represents the locus of points $|z+1-2i|<2$.

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Select the graph that represents the locus of points $|z+1-2i|>2$.

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Select the graph that represents the locus of points $|z-2+i|<2$.

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Select the graph that represents the locus of points $|z-2+i|>2$.

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Select the graph that represents the locus of points $|z+2-i|>2$.

", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": false, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "2", "showCellAnswerState": false, "choices": ["", "", "", ""], "matrix": ["0", "0", "0", "1"], "distractors": ["", "", "", ""]}]}, {"name": "Locus of Points 8", "extensions": [], "custom_part_types": [], "resources": [["question-resources/a1.png", "/srv/numbas/media/question-resources/a1.png"], ["question-resources/a2.png", "/srv/numbas/media/question-resources/a2.png"], ["question-resources/a3.png", "/srv/numbas/media/question-resources/a3.png"], ["question-resources/a4.png", "/srv/numbas/media/question-resources/a4.png"], ["question-resources/b1_1YkxNrM.png", "/srv/numbas/media/question-resources/b1_1YkxNrM.png"], ["question-resources/b2_8sKC3IF.png", "/srv/numbas/media/question-resources/b2_8sKC3IF.png"], ["question-resources/b3.png", "/srv/numbas/media/question-resources/b3.png"], ["question-resources/b1_DnHgge4.png", "/srv/numbas/media/question-resources/b1_DnHgge4.png"], ["question-resources/b4.png", "/srv/numbas/media/question-resources/b4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Kariane Ouellet", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4505/"}], "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "

Select the graph that represents the locus of points $|z+2-i|<2$.

", "advice": "", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": false, "showFeedbackIcon": false, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "2", "showCellAnswerState": false, "choices": ["", "", "", ""], "matrix": ["0", "0", "1", "0"], "distractors": ["", "", "", ""]}]}]}], "navigation": {"allowregen": false, "reverse": false, "browse": false, "allowsteps": true, "showfrontpage": true, "showresultspage": "never", "onleave": {"action": "warnifunattempted", "message": "

Do not leave the question blank, as you won't be able to come back to it.

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

You are out of time!

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

You have 5 minutes left.

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

This is your Complex Quiz!

\n

It will count as academic result, and it's based on your score, not completion. Moreover, to get your score, you have to submit pictures of written solutions on google classroom. Note that I will cross check hand writting to comfirm it was written by you. If you don't submit written solutions, you automatically get 0.

\n\n

Don't hesitate to contact me if you have questions throughout.

\n

Good luck!

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