// Numbas version: exam_results_page_options {"name": "50% 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.
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", "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}\\]
\ncalculate the following. Enter your answers in the form
\n\\[x+yi\\]
\nNote, 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}\\]
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", "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.
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", "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.
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", "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},\\]
\ncalculate the following. Enter your answers in the form
\n\\[x+yi\\]
\nNote, 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, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$
\nSecondly, 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
im
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\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, $|z| = \\sqrt{(\\var{a1})^2+(\\var{b1})^2} = \\sqrt{\\var{a1*a1+b1*b1}} = \\var{dpformat(modz,2)}$
\nSecondly, 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
im
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Use de Moivre's theorem to write the following complex numbers in the form $a+bi$.
\nNote that for these questions, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nImportant: When calculating the final answer in part (iii) of each question, you must use non-rounded 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.
", "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 $z^n$.
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.
\nAlso remember that for this question, arguments of complex numbers lie in the range $-\\pi \\lt \\theta \\le \\pi$.
\nWith the above in mind we can now answer the questions:
\na)
\n\\[ \\begin{eqnarray*} |\\var{z1}|&=&\\sqrt{(\\var{a1})^2+(\\var{b1})^2}\\\\ &=& \\var{abs(z1)}\\\\ &=&\\var{ans1} \\end{eqnarray*} \\] to 3 decimal places.
\nNote 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{m1}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z1}) &=& \\var{arg(z1)}\\\\ &=& \\var{arg1}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z1})^{\\var{n2}})=\\var{n2}\\times \\var{arg(z1)} = \\var{n2*arg(z1)}$ radians.
\nHence 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*} \\]
\nb)
\n\\[ \\begin{eqnarray*} |\\var{z2}|&=&\\sqrt{(\\var{a2})^2+(\\var{b2})^2}\\\\ &=& \\var{abs(z2)}\\\\ &=&\\var{ans2} \\end{eqnarray*} \\] to 3 decimal places.
\nNote 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{m2}.
Hence we see that:
\\[\\begin{eqnarray*} \\arg(\\var{z2}) &=& \\var{arg(z2)}\\\\ &=& \\var{arg2}\\; \\mbox{radians} \\end{eqnarray*} \\] to 3 decimal places.
We have $\\arg((\\var{z2})^{\\var{n4}})=\\var{n4}\\times \\var{arg(z2)} = \\var{n4*arg(z2)}$ radians.
\nHence 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*} \\]
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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.
\nHence find:
\n(iii) $(\\var{z1})^{\\var{n2}}\\;=\\;$[[2]]
\nNote, for part (iii) input a complex number, with real and imaginary parts given correct to the nearest whole number.
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\n(i) $|\\var{z2}|\\;=\\;$ [[0]], to 3 decimal places.
\n(ii) $\\arg(\\var{z2})\\;=\\;$[[1]] radians, to 3 decimal places.
\nHence find:
\n(iii) $(\\var{z2})^{\\var{n4}}\\;=\\;$[[2]]
\nNote, for part (iii) input a complex number. with real and imaginary parts given correct to the nearest whole number.
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"}, "preventleave": true, "startpassword": "y1ai"}, "timing": {"allowPause": true, "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 38 minutes to complete this complex numbers assessment.
\nThis assessment is worth 20% of your overall marks for the WM104 module.
\nContent 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": []}