N2013 P1 Q8

The complex number \(z\) is given by \(z=r{{e}^{\mathrm{i}\theta }}\), where \(r>0\) and \(0\le \theta \le \frac{1}{2}\pi \).

1. Given that \(w=\left( 1-\mathrm{i}\sqrt{3} \right)z\), find \(\left| w \right|\) in terms of \(r\) and \(\arg w\) in terms of \(\theta \).[2]

2. Given that \(r\) has a fixed value, draw an Argand diagram to show the locus of \(z\) as \(\theta\) varies. On the same diagram, show the corresponding locus of \(w\). You should identify the modulus and argument of the end-points of each locus.[4]

3. Given that \(\arg \left( \frac{{{z}^{10}}}{{{w}^{2}}} \right)=\pi \), find \(\theta \).[3]