2019 DHS P2 Q4

The complex number \(w\) has modulus \(\sqrt{2}\) and argument \(\frac{1}{4}\pi \) and the complex number \(z\) has modulus \(\sqrt{2}\) and argument \(\frac{5}{6}\pi \).

1. By first expressing \(w\) and \(z\) in the form \(x+\mathrm{i}y\), find the exact real and imaginary parts of \(w+z\).[3]
2. On the first 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 \(\arg (w+z)\) 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.

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