Pearson Edexcel IAL Mathematics FP1.1.2 Sum, product & quotient of complex numbers

Practise adding, multiplying and dividing complex numbers, then using modulus or argument conditions to find unknown constants.

Exam points

rationalise a complex quotient and give the result in x + yi form
calculate |z^2 - c| or |z_1z_2| after carrying out complex arithmetic
link real and imaginary parts to an argument condition to solve for a parameter

[Maximum number: 5]

The complex numbers $z_{1}$ and $z_{2}$ are given by

z_{1}=3+5 \mathrm{i} \text { and } z_{2}=-2+6 \mathrm{i}

(a)

Without using your calculator and showing all stages of your working,

[ 3 ]

(i)

express $\frac{z_{1}}{z_{2}}$ in the form a+b i, where a and b are fully simplified fractions.

[ 3 ]

$\frac{z_{1}}{z_{2}}=\frac{3+5 \mathrm{i}}{-2+6 \mathrm{i}} \times \frac{-2-6 \mathrm{i}}{-2-6 \mathrm{i}}=\ldots$

M1

$=\frac{-6-18 \mathrm{i}-10 \mathrm{i}+30}{40}$

A1

$=\frac{3}{5}-\frac{7}{10} \mathrm{i}$

A1

(3)

(b)

Hence determine the value of $\arg\left(\frac{z_1}{z_2}\right)$

Give your answer in radians to 2 decimal places.

[ 2 ]

$\arg\left(\frac{z_1}{z_2}\right)=\tan^{-1}\left(\frac{-7/10}{3/5}\right) =\tan^{-1}\left(-\frac76\right)$

M1

$=-0.86\ \text{radians}\quad\text{or}\quad 5.42\ \text{radians}$

A1

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FP1.1.2 - Sum, product and quotient of complex numbers question 1

Question (a)

Question (i)

Question (b)