Argand diagram transformations

<p><strong>Do not use a calculator in answering this question.</strong></p>
<p>The complex number $z$ is given by $z=1-\sqrt{3}\mathrm{i}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $z$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that $1+z+{{z}^{2}}+{{z}^{3}}+{{z}^{4}}+{{z}^{5}}$ is purely imaginary.</span></div><p style="text-align: right;">[4]</p></li></ol>
<p>The complex number $w$ is given by $w=\frac{\mathrm{i}\sqrt{3}}{{{z}^{n}}}$.</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Show that $w$ cannot be a real number for all integer values of $n$.</span></div><p style="text-align: right;">[4]</p></li></ol>

**Do not use a calculator in answering this question.**

The complex number $z$ is given by $z=1-\sqrt{3}\mathrm{i}$.

Express $z$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, where $r>0$ and $-\pi <\theta \le \pi $.

Show that $1+z+{{z}^{2}}+{{z}^{3}}+{{z}^{4}}+{{z}^{5}}$ is purely imaginary.

The complex number $w$ is given by $w=\frac{\mathrm{i}\sqrt{3}}{{{z}^{n}}}$.

Show that $w$ cannot be a real number for all integer values of $n$.

2022 TJC J2 MYE P1 Q6

2022 TJC J2 MYE P1 Q6

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