2021 YIJC P2 Q3

Do not use a calculator in answering this question.

1. The complex number \(z\) is given by \(z=\frac{{{\left( 1-\mathbf{i} \right)}^{3}}}{\sqrt{2}{{\left( a+\mathbf{i} \right)}^{2}}}\), where \(a<0\).
   1. Given that \(\left| z \right|=\frac{1}{2}\), show that \(\arg z=-\frac{5\pi }{12}\).[5]
   2. Hence find the smallest positive integer \(n\) for which \({{z}^{n}}\) has equal real and imaginary parts.[2]
2. The complex number \(q\) is given by \(\frac{{{\mathrm{e}}^{-\mathbf{i}\theta }}}{{{\mathrm{e}}^{\mathbf{i}\theta }}-\mathbf{i}}\), where \(0<\theta <\frac{\pi }{2}\). Show that
   \(\operatorname{Re}\left( q \right)=\frac{1}{2}\left( 1+2\sin \theta \right)\).[4]