N2022 P2 Q3

The complex numbers \({{z}_{1}}\), \({{z}_{2}}\) and \({{z}_{3}}\) are such that \({{z}_{1}}=3-\mathbf{i}\sqrt{3}\), \({{z}_{2}}=\frac{1}{2}{{\mathrm{e}}^{\mathbf{i}\frac{2\pi }{5}}}\) and \({{z}_{3}}={{z}_{1}}\times {{z}_{2}}\).

1. Find exactly the modulus and argument of \({{z}_{3}}\).[3]
2. Sketch an Argand diagram showing \(z_1\), \(z_2\) and \(z_3\).[2]
3. Find the smallest positive integer value of \(n\) for which \({{z}_{3}}^{n}\) is purely imaginary. State the modulus and argument of \({{z}_{3}}^{n}\) in this case, giving the modulus in the form \(k\sqrt{3}\), where \(k\) is an integer.[4]