Do not use a calculator in answering this question. The complex number $z$ has modulus $1$ and a

Question

Do not use a calculator in answering this question. The complex number $z$ has modulus $1$ and argument $\theta$, where $\frac{\pi}{2} < \theta < \pi$, and the complex number $w$ is given by $w = \mathrm{i}\sqrt{3}z$. The point $P$ on the Argand diagram represents $z$. On the copy of the Argand diagram with origin $O$ in the Printed Answer Booklet, plot the points $Q$ and $R$ to represent $w$ and $z - w$ respectively. Show clearly the geometrical relationship between the points $P$, $Q$ and $R$.[3] Find the area of the quadrilateral $ORPQ$.[1] It is given that $\theta = \frac{3\pi}{4}$. Find $z$ in the form $x + y\mathrm{i}$, where $x$ and $y$ are real numbers.[2] Show that $z - w = k\left[ (\sqrt{3} - 1) + (\sqrt{3} + 1)\mathrm{i} \right]$, where $k$ is a constant to be determined.[2] Hence show that $\tan \frac{5\pi}{12} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$.[1]

Accepted Answer

(b)$\sqrt 3$ (c)$z=-\frac{1}{\sqrt 2}+\frac{1}{\sqrt 2}i$ (d)$k=\frac{1}{\sqrt 2}$

2025 VJC P2 Q4

2025 VJC P2 Q4