2025 RI Complex Numbers Assignment Q4

The complex numbers \(w\) and \(z\) are given by \(w=1+\mathbf{i}\) and \(z=-\frac{\sqrt{6}}{2}+\frac{\sqrt{2}}{2}\mathbf{i}\).

1. Find the modulus and argument of \(w\) and \(z\).[2]
2. On the same Argand diagram, sketch the points \(P\), \(Q\), \(R\) representing the complex numbers \(z\), \(w\) and \(z+w\) respectively. State the geometrical shape of the quadrilateral \(OPRQ\).[3]
3. Referring to the Argand diagram in part (ii), find \(\arg \left( w+z \right)\) and show that \(\tan \left( \frac{11}{24}\pi \right)=\frac{a+\sqrt{2}}{\sqrt{6}+b}\) where \(a\) and \(b\) are constants to be determined.[2]