2018 EJC P1 Q6 (b)

Do not use a calculator in answering this question.

It is given that \(q=-\sqrt{3}-\mathrm{i}\).

1. Find an exact expression for \({{q}^{6}}\), giving your answer in the form \(r{{\mathrm{e}}^{\mathrm{i}\theta }}\), where \(r>0\) and \(0\le \theta <2\pi \).[3]
2. Find the three smallest positive whole number values of \(n\) for which \(\frac{{{q}^{n}}}{{{q}^{*}}}\) is purely imaginary.[4]