2023 EJC P1 Q10

Do not use a calculator in answering this question.

The complex numbers \(z\) and \(w\) have the same modulus \(r\), and have arguments \(\alpha \) and \(\beta \) respectively.

1. Show that \(z+w=2r\cos \frac{\alpha -\beta }{2}\left( \cos \frac{\alpha +\beta }{2}+\mathbf{i}\,\sin \frac{\alpha +\beta }{2} \right)\) and hence write down expressions for \(\left| z+w \right|\) and for \(\arg \left( z+w \right)\).[4]

It is now given that \(z=2{{\mathrm{e}}^{\mathbf{i}\,\frac{\pi }{12}}}\) and \(w=2{{\mathrm{e}}^{\mathbf{i}\,\frac{5\pi }{12}}}\).

2. Find the values of \(\left| z+w \right|\) and \(\arg \left( z+w \right)\).[2]

Let \(v\) be the complex number such that \(v=\frac{{{z}^{2}}}{w}\).

3. Find \(\left| v \right|\) and \(\arg \left( v \right)\).[3]
4. Let \(O\) be the origin and \(P\) and \(Q\) be points representing \(z+w\) and \(v\) respectively on an Argand diagram. Find the area of triangle \(OPQ\).[2]