2017 SRJC P1 Q3

For \(\alpha ,\,\beta \in \mathbb{R}\) such that \(2\alpha <\beta \), the complex numbers \({{z}_{1}}={{e}^{ia}}\)and \({{z}_{2}}=2{{e}^{\mathrm{i}\beta }}\) are represented by the points \(P\) and \(Q\) respectively in the Argand diagram below.

Find the modulus and argument of the complex numbers given by \(\frac{\mathrm{i}}{\mathrm{2}}{{z}_{2}}\) and \(\frac{{{z}_{1}}^{2}}{{{z}_{2}}}\).

Copy the given Argand diagram onto your answer script and indicate clearly the following points representing the corresponding complex numbers on your diagram.

1. \(A:\frac{\mathrm{i}}{2}{{z}_{2}}\) ;
2. \(B:\frac{{{z}_{1}}^{2}}{{{z}_{2}}}\)

You are expected to indicate clearly the relevant moduli and arguments for parts (i) and (ii) on your Argand diagram. If \(\beta =\frac{11}{12}\pi \), find the smallest positive integer \(n\) such that the point representing the complex number \({{({{z}_{2}})}^{n}}\) lies on the negative real axis.