Question 1414b

Complex Numbers

2014 Paper 2 Question 2 Variant

Question

It is given that

{w=- 1 - \sqrt{3} \mathrm{i}.}

(a)

Without using a calculator, find an exact expression for

{w^4.}

Give your answer in the form

{r \mathrm{e}^{\mathrm{i}\theta},}

where

{r > 0}

and

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

[3]

(b)

Without using a calculator, find the three smallest positive whole number values of

{n}

for which

\frac{w^n}{w^*}

is a real number.

[4]