Question 1414b

Complex Numbers
2014 Paper 2 Question 2 Variant

Question

It is given that w=3+i.{w=\sqrt{3} + \mathrm{i}.}
(a)
Without using a calculator, find an exact expression for w8.{w^8.} Give your answer in the form reiθ,{r \mathrm{e}^{\mathrm{i}\theta},} where r>0{r > 0} and π<θπ.{-\pi < \theta \leq \pi.}
[3]
(b)
Without using a calculator, find the three smallest positive whole number values of n{n} for which wnw{\displaystyle \frac{w^n}{w^*}} is a real number.
[4]

Answer