The complex numbers $a$ and $b$ are given by $a=-\left( 1+\sqrt{3}\mathbf{i} \right)$ and $b=\frac{1

Question

The complex numbers $a$ and $b$ are given by $a=-\left( 1+\sqrt{3}\mathbf{i} \right)$ and $b=\frac{1}{2}\left( 1-\mathbf{i} \right)$. Without using a calculator, find the value of ${{a}^{2}}b$ in the form $x+\mathbf{i}y$. [2] By using the moduli and arguments of $a$ and $b$, find the modulus and argument of ${{a}^{2}}b$. [3] Use your answers to parts (i) and (ii) to show that $\sin \frac{5\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}$. [2] The diagram below shows an isosceles right triangle $ABC$, where the points $A$, $B$ and $C$ represent the complex numbers $a$, $b$ and $c$ respectively. Find the exact value of $c$. [2]

Accepted Answer

(i) $\left( \sqrt{3}-1 \right)+\left( \sqrt{3}+1 \right)\mathbf{i}$ (ii) $\arg \left( {{a}^{2}}b \right)=\frac{5\pi }{12}$ (iv) $c=\left( 1-\sqrt{3} \right)+\mathbf{i}$

2016 JJC P2 Q4

2016 JJC P2 Q4

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