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Quiz covering basic arithmetic with complex numbers, solving a quadratic with complex solutions and converting to/from polar and rectangular forms.

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Given the complex numbers

\n

\\[a=\\var{z1}\\\\b=\\var{z2}\\\\c=\\var{z3}\\\\d=\\var{z4}\\]

\n

calculate the following. Enter your answers in the form

\n

\\[x+yi\\]

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$a+b=$

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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.

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$a-b=$

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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.

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$a+d=$

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No brackets are required or allowed.

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$\\simplify{{n}*b+{m}*d}=$

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Given the complex numbers

\n

\\[a=\\var{a}\\\\b=\\var{b}\\\\c=\\var{c}\\]

\n

calculate the following. Enter your answers in the form

\n

\\[x+yi\\]

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$ab=$

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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.

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$ac=$

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No brackets are required or allowed.

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$cb=$

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No brackets are required or allowed.

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Solve the quadratic equation:

\n

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

\n

You may use the following formula: if $ax^2+bx+c=0$ then

\n

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

\n

Enter your answers in the form $x+yi$.

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

\n

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

\n

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This question is about converting complex numbers from polar to rectangular form and from rectangular to polar form. The argument $\\theta$ should be in the range

\n

\\[-180<\\theta\\leq180\\]

\n

Give your answers correct to at least 1 decimal place.

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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]]$^\\circ$

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Let $a$, $b$ and $c$ be the complex numbers below.

\n

\\[a=\\var{a_m}\\angle\\var{a_a}\\qquad b=\\var{b_m}\\angle\\var{b_a}\\qquad c=\\var{c_m}\\angle\\var{c_a}\\]

\n

Calculate the following in polar form, simplifying where needed.

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$bc=\\;$[[0]]$\\angle\\;$[[1]]$^\\circ$

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Write the argument in the range $180^\\circ<\\theta\\leq180^\\circ$.

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$\\frac{c}{a}=\\;$[[0]]$\\angle\\;$[[1]]$^\\circ$

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Write the argument in the range $180^\\circ<\\theta\\leq180^\\circ$.

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$a^\\var{n}=\\;$[[0]]$\\angle\\;$[[1]]$^\\circ$

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Write the argument in the range $180^\\circ<\\theta\\leq180^\\circ$.

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