<p>The complex numbers $a$ and $b$ are given by $a=-\left( 1+\sqrt{3}\mathbf{i} \right)$ and $b=\frac{1}{2}\left( 1-\mathbf{i} \right)$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Without using a calculator, find the value of ${{a}^{2}}b$ in the form $x+\mathbf{i}y$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">By using the moduli and arguments of $a$ and $b$, find the modulus and argument of ${{a}^{2}}b$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Use your answers to parts (i) and (ii) to show that $\sin \frac{5\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}$.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">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 $c$.</span></div><p style="text-align: right;">[2]</p></li></ol>
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1759222096991-8gm7y3hzzp6.png" width="400" class="aligncenter" /></figure>

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 $x+\mathbf{i}y$.

By using the moduli and arguments of $a$ and $b$, find the modulus and argument of ${{a}^{2}}b$.

Use your answers to parts (i) and (ii) to show that $\sin \frac{5\pi }{12}=\frac{\sqrt{3}+1}{2\sqrt{2}}$.

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 $c$.

2016 JJC P2 Q4

2016 JJC P2 Q4

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