2022 TMJC P2 Q2 [Modified]

It is given that \(z=-\sqrt{6}-\mathrm{i}\sqrt{2}\) and \(w=3\left( \cos \frac{5\pi }{7}-\mathrm{i}\sin \frac{5\pi }{7} \right)\). Without the use of a calculator, find the modulus and argument of \(\frac{\mathrm{i}z{{w}^{2}}}{{{w}^{*}}}\) in exact form.[6]

Hence, find the three smallest positive values of \(n\) for which \({{\left( \frac{\mathrm{i}z{{w}^{2}}}{{{w}^{*}}} \right)}^{n}}\) is purely imaginary.[2]