Find the values of n such that ${\left( {1 + \sqrt 3 {\text{i}}} \right)^n}$ is a real number.

EITHER

changing to modulus-argument form

r = 2

$\theta  = \arctan \sqrt 3  = \frac{\pi }{3}$     (M1)A1

$\Rightarrow 1 + {\sqrt 3 ^n} = {2^n}\left( {\cos \frac{{n\pi }}{3} + {\text{i}}\sin \frac{{n\pi }}{3}} \right)$     M1

if $\sin \frac{{n\pi }}{3} = 0 \Rightarrow n = \{ 0,{\text{ }} \pm 3,{\text{ }} \pm 6,{\text{ }} \ldots \} {\text{ }}$     (M1)A1     N2

OR

$\theta = \arctan \sqrt 3 = \frac{\pi }{3}$     (M1)(A1)

    M1

$n \in \mathbb{R} \Rightarrow \frac{{n\pi }}{3} = k\pi ,{\text{ }}k \in \mathbb{Z}$     M1

$\Rightarrow n = 3k,{\text{ }}k \in \mathbb{Z}$     A1     N2

[5 marks]

Some candidates did not consider changing the number to modulus-argument form. Among those that did this successfully, many considered individual values of n, or only positive values. Very few candidates considered negative multiples of 3.