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 $\arg \left( {{z}_{1}} \right)<\arg \left( {{z}_{2}} \right)<\arg \left( {{z}_{3}} \right)$.
(i)
Find ${{z}_{1}}$, ${{z}_{2}}$ and ${{z}_{3}}$ in the exact form $r{{\text{e}}^{\text{i}\,\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.
[6]
The complex number $w$ has modulus $\sqrt{2}$ and argument $\frac{\pi }{24}$.
(ii)
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 $O$.
(iii)
Mark, on an Argand diagram, the points ${{Z}_{2}}$, ${{Z}_{3}}$ and ${{Z}_{4}}$.
[2]
(iv)
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]
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