2019 TMJC BT Q4

The complex numbers \({{z}_{1}}\) and \({{z}_{2}}\) are given by \(\sqrt{3}+\mathbf{i}\) and \(-1+\sqrt{3}\mathbf{i}\) respectively.

1. Find the modulus and argument of \({{z}_{1}}\) and \({{z}_{2}}\).[3]
2. In an Argand diagram with origin \(O\), the complex numbers \({{z}_{1}}\) and \({{z}_{2}}\) are represented by the points \(P\) and \(Q\) respectively. With the aid of an Argand diagram, show that \(\angle POQ=\frac{\pi }{2}\).[2]
3. Find \({{z}_{1}}+{{z}_{2}}\) in the form \(a+b\,\mathbf{i}\), where \(a\),\(b\in \mathbb{R}\). The complex number \({{z}_{1}}+{{z}_{2}}\) is represented by the point \(R\) in the Argand diagram. Hence use the Argand diagram to find the exact argument of \({{z}_{1}}+{{z}_{2}}\) and show that \(\tan \left( \frac{5\pi }{12} \right)=\frac{\sqrt{3}+1}{\sqrt{3}-1}\).[3]