2021 HCI P1 Q2

The complex number \(z\) has modulus \(2\) and argument \(\theta \) and the complex number \(w\) has modulus \(r\) and argument \(-\frac{\pi }{2}\), where \(r\) is a positive constant and \(-\frac{\pi }{2}<\theta <0\). Given that \(\frac{z}{w}=\frac{1}{32}\left( \sin \frac{\pi }{6}+\mathrm{i}\cos \frac{\pi }{6} \right)\), find the values of \(r\) and \(\theta \).[3]

Hence find the \(3\) smallest positive integers \(n\) such that \({{\left( {{z}^{n}} \right)}^{*}}\) is purely imaginary.[3]