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Given the complex numbers

\n

\$a=\\var{z1},\\quad b=\\var{z2},\\quad c=\\var{z3},\\quad d=\\var{z4},\$

\n

\n

\$x+iy\$

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

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No brackets are required or allowed.

", "strings": ["(", ")"]}, "prompt": "

$a+b=$

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No brackets are required or allowed.

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$a+c=$

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No brackets are required or allowed.

", "strings": ["(", ")"]}, "prompt": "

$a-b=$

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No brackets are required or allowed.

", "strings": ["(", ")"]}, "prompt": "

$a-c=$

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No brackets are required or allowed.

", "strings": ["(", ")"]}, "prompt": "

$a+d=$

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No brackets are required or allowed.

", "strings": ["(", ")"]}, "prompt": "

$b-d=$

", "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "vsetrange": [0, 1], "checkvariablenames": false, "vsetrangepoints": 5, "checkingtype": "absdiff"}], "variables": {"z1": {"group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "name": "z1", "description": "", "templateType": "anything"}, "z3": {"group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "name": "z3", "description": "", "templateType": "anything"}, "z2": {"group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "name": "z2", "description": "", "templateType": "anything"}, "z4": {"group": "Ungrouped variables", "definition": "random(-10..10)+random(-10..10)i", "name": "z4", "description": "", "templateType": "anything"}}, "type": "question"}, {"name": "Complex number arithmetic: multiply and divide", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "statement": "

Given the complex numbers

\n

\$a=\\var{a},\\quad b=\\var{b},\\quad c=\\var{c},\\quad d=\\var{d},\$

\n

\n

\$x+iy\$

", "preamble": {"js": "", "css": ""}, "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}\$

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

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "ungrouped_variables": ["a", "b", "c", "d", "twocans"], "parts": [{"checkingaccuracy": 0.001, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "stepsPenalty": 0, "prompt": "

$ab=$

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

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

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No brackets are required or allowed.

", "showStrings": false}, "scripts": {}, "showpreview": false, "checkvariablenames": false, "checkingtype": "absdiff", "marks": "2", "answer": "{a*b}"}, {"checkingaccuracy": 0.001, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "checkvariablenames": false, "prompt": "

$ac=$

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No brackets are required or allowed.

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$\\frac{a}{b}=$

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What is the complex conjugate of $b$?

\n

$\\overline{b}=$

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Multiply the numerator ($\\var{a}$) by $\\overline{b}$.

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Multiply the denominator ($\\var{b}$) by $\\overline{b}$.

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\n

\$\\frac{a}{b}=\\frac{a\\overline{b}}{b\\overline{b}}\$

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No brackets are required or allowed.

", "showStrings": false}, "scripts": {}, "showpreview": false, "checkvariablenames": false, "checkingtype": "absdiff", "marks": "2", "answer": "{a/b}"}, {"checkingaccuracy": 0.001, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "checkvariablenames": false, "prompt": "

$\\frac{a}{c}=$

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No brackets are required or allowed.

", "showStrings": false}, "scripts": {}, "showpreview": false, "checkingtype": "absdiff", "marks": "2", "answer": "{a/c}"}, {"checkingaccuracy": 0.001, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "checkvariablenames": false, "prompt": "

$ad=$

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No brackets are required or allowed.

", "showStrings": false}, "scripts": {}, "showpreview": false, "checkingtype": "absdiff", "marks": "2", "answer": "{a*d}"}, {"checkingaccuracy": 0.001, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "checkvariablenames": false, "prompt": "

$\\frac{b}{d}=$

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No brackets are required or allowed.

", "showStrings": false}, "scripts": {}, "showpreview": false, "checkingtype": "absdiff", "marks": "2", "answer": "{b/d}"}], "variables": {"b": {"name": "b", "description": "", "templateType": "anything", "definition": "random(-10..10)+random(-10..10)i", "group": "Ungrouped variables"}, "d": {"name": "d", "description": "", "templateType": "anything", "definition": "random(-10..10)+random(-10..10)i", "group": "Ungrouped variables"}, "twocans": {"name": "twocans", "description": "", "templateType": "anything", "definition": "(a*conj(b))", "group": "Ungrouped variables"}, "a": {"name": "a", "description": "", "templateType": "anything", "definition": "random(-10..10)+random(-10..10)i", "group": "Ungrouped variables"}, "c": {"name": "c", "description": "", "templateType": "anything", "definition": "random(-10..10)+random(-10..10)i", "group": "Ungrouped variables"}}, "tags": [], "rulesets": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "functions": {}, "variable_groups": [], "type": "question"}, {"name": "Solve quadratic with complex roots (simpler)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "advice": "", "functions": {}, "tags": [], "statement": "

Solve the quadratic equations below. You may use the following formula: if $ax^2+bx+c=0$ then

\n

\$x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}\$

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Solve

\n

\$\\simplify{x^2-2*{re(a)}*x+{abs(a)^2}}=0\$

\n

$x_1=$ [[0]] (solution with positive imaginary part)

\n

$x_2=$ [[1]] (solution with negative imaginary part)

\n

", "showCorrectAnswer": true, "scripts": {}, "gaps": [{"marks": "2", "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "answer": "{a}", "showFeedbackIcon": true, "variableReplacements": [], "type": "jme", "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst"}, {"marks": "2", "showpreview": false, "checkingtype": "absdiff", "scripts": {}, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "answer": "{conj(a)}", "showFeedbackIcon": true, "variableReplacements": [], "type": "jme", "showCorrectAnswer": true, "expectedvariablenames": [], "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst"}], "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)+i*random(1..5)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "description": "", "name": "c"}, "b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-5..5)+i*random(-5..-1)", "description": "", "name": "b"}}, "metadata": {"licence": "None specified", "description": ""}, "ungrouped_variables": ["a", "b", "c"], "rulesets": {}, "type": "question"}, {"name": "Convert between polar and rectangular forms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "advice": "", "functions": {}, "tags": [], "statement": "

This question is about converting complex numbers from polar to rectangular form and from rectangular to polar form. Give your answers correct to at least 1 decimal place.

", "variable_groups": [{"name": "P to R", "variables": ["r1", "theta1", "x1", "y1", "theta2", "r2", "x2", "y2"]}, {"name": "R to P", "variables": ["x3", "y3", "r3", "theta3", "x4", "y4", "r4", "theta4"]}], "parts": [{"marks": 1, "showpreview": false, "checkingtype": "dp", "scripts": {}, "vsetrange": [0, 1], "checkingaccuracy": "1", "checkvariablenames": false, "answer": "{x1+i*y1}", "showFeedbackIcon": true, "variableReplacements": [], "type": "jme", "prompt": "

Write the number $\\var{r1}\\angle\\var{theta1}^\\circ$ in rectangular form.

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Write the number $\\var{r2}\\angle\\var{theta2}^\\circ$ in rectangular form.

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Write the number $\\simplify{{x3}+i*{y3}}$ in polar form.

\n

[[0]]$\\angle$ [[1]]$^\\circ$

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Write the number $\\simplify{{x4}+i*{y4}}$ in polar form.

\n

[[0]]$\\angle$ [[1]]$^\\circ$