(a)
Do not use a calculator in answering this question.
The complex number $w$ is given by $w=\frac{{{\left( \cos \frac{\pi }{3}+\text{i}\sin \frac{\pi }{3} \right)}^{4}}}{{{\left( \cos \frac{\pi }{6}-\text{i}\sin \frac{\pi }{6} \right)}^{5}}}$.
(i) Find $\left| w \right|$ and the exact value of $\arg \left( w \right)$. Hence find the value of ${{w}^{6}}$.
[3]
(ii) Express $\frac{1+{{\text{e}}^{\text{i}\theta }}}{1+{{\text{e}}^{-\text{i}\theta }}}$ in the modulus-argument form. Hence find the value of ${{\left( 1+w \right)}^{6}}+{{\left( 1+{{w}^{*}} \right)}^{6}}$.
[4]
(b)
Given that $z=\sqrt{3}+\text{i}$, by first expressing $z$ in the form $r{{\text{e}}^{\text{i}\theta }}$, find the first three positive integer $n$ such that ${{z}^{n}}$ is purely imaginary.
[4]
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