Question 1420a

Complex Numbers

2020 Paper 1 Question 4 Variant

Question

Three complex numbers are

\begin{align*} z_1 &= 1 + \sqrt{3} \mathrm{i}, \\ z_2 &= \sqrt{3} - \mathrm{i} \; \textrm{ and } \\ z_3 &= {\textstyle 2 \left( \cos \left( - \frac{2}{3} \pi \right) + \mathrm{i} \sin \left( - \frac{2}{3} \pi \right) \right)}. \end{align*}

(a)

Find

\frac{z_1}{z_2z_3}

in the form

{r (\cos \theta + \mathrm{i} \sin \theta),}

where

{r>0}

and

{-\pi < \theta \leq \pi.}

[4]

A fourth complex number,

{z_4,}

is such that

\frac{z_1z_4}{z_2z_3}

is purely imaginary and

\left| \frac{z_1z_4}{z_2z_3} \right| = 1.

(b)

Find the possible values of

{z_4}

in the form

{r (\cos \theta + \mathrm{i} \sin \theta),}

where

{r>0}

and

{-\pi < \theta \leq \pi.}

[3]

Answer

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