2018 CJC P1 Q11

It is given that \(z=-\frac{1}{2}\) is a root of the equation

\(8{{z}^{3}}+\left( 4-4\sqrt{2} \right){{z}^{2}}+\left( 2-2\sqrt{2} \right)z+1=0\)

The roots of the equation are denoted by \({{z}_{1}}\), \({{z}_{2}}\), \({{z}_{3}}\), where \(\arg \left( {{z}_{1}} \right)<\arg \left( {{z}_{2}} \right)<\arg \left( {{z}_{3}} \right)\).

1. Find \({{z}_{1}}\), \({{z}_{2}}\) and \({{z}_{3}}\) in the exact form \(r{{\mathrm{e}}^{\mathrm{i}\,\theta }}\), where \(r>0\) and \(-\pi <\theta \le \pi \).[6]

The complex number \(w\) has modulus \(\sqrt{2}\) and argument \(\frac{\pi }{24}\).

2. Find the modulus and argument of \({{z}_{4}}\), where \({{z}_{4}}=\frac{{{w}^{2}}}{{{z}_{1}}}\).[3]

The complex numbers \({{z}_{2}}\) , \({{z}_{3}}\) and \({{z}_{4}}\) are represented by the points \({{Z}_{2}}\), \({{Z}_{3}}\) and \({{Z}_{4}}\) respectively in an Argand diagram with origin \(O\).

3. Mark, on an Argand diagram, the points \({{Z}_{2}}\), \({{Z}_{3}}\) and \({{Z}_{4}}\).[2]
4. By considering \(\sin \left( A-B \right)\) with suitable values of \(A\) and \(B\), show that \(\sin \left( \frac{\pi }{12} \right)=\frac{\sqrt{2}}{4}\left( \sqrt{3}-1 \right)\).
   Hence or otherwise, find the exact area of the quadrilateral \(O{{Z}_{3}}{{Z}_{4}}{{Z}_{2}}\).[2]