2022 TJC J2 MYE P1 Q6
Timothy Gan
2022 TJC J2 MYE P1 Q6
Do not use a calculator in answering this question.
The complex number $z$ is given by $z=1-\sqrt{3}\text{i}$.
(i)
Express $z$ in the form $r{{\text{e}}^{\text{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.
[2]
(ii)
Show that $1+z+{{z}^{2}}+{{z}^{3}}+{{z}^{4}}+{{z}^{5}}$ is purely imaginary.
[4]
The complex number $w$ is given by $w=\frac{\text{i}\sqrt{3}}{{{z}^{n}}}$.
(iii)
Show that $w$ cannot be a real number for all integer values of $n$.
[4]
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