2021 ASRJC P2 Q1

The complex numbers \(z\) and \(w\) have moduli \(k\) and \(3{{k}^{2}}\) respectively and arguments \(\alpha \) and \(4\alpha \) respectively, where \(k\) is a positive constant and \(-\frac{\pi }{7}<\alpha \le \frac{\pi }{7}\).

1. Express \(\frac{{{z}^{3}}}{{{w}^{*}}}\) in the form \(r{{\mathrm{e}}^{\mathbf{i}\theta }}\) where \(r>0\) and \(-\pi <\theta \le \pi \).[2]
2. It is given that \(\alpha =\frac{\pi }{21}\). Find the integer values of \(n\) such that \({{\left( \frac{{{z}^{3}}}{{{w}^{*}}} \right)}^{n}}\) is real.[2]