Argand diagram transformations

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"><strong>Do not use a calculator in answering this question.</strong></span></div><div class="list-item-paragraph">The complex number $w$ is given by $w=\frac{{{\left( \cos \frac{\pi }{3}+\mathrm{i}\sin \frac{\pi }{3} \right)}^{4}}}{{{\left( \cos \frac{\pi }{6}-\mathrm{i}\sin \frac{\pi }{6} \right)}^{5}}}$.</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find $\left| w \right|$ and the exact value of $\arg \left( w \right)$. Hence find the value of ${{w}^{6}}$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">Express $\frac{1+{{\mathrm{e}}^{\mathrm{i}\theta }}}{1+{{\mathrm{e}}^{-\mathrm{i}\theta }}}$ in the modulus-argument form. Hence find the value of ${{\left( 1+w \right)}^{6}}+{{\left( 1+{{w}^{*}} \right)}^{6}}$.</span></div><p style="text-align: right;">[4]</p></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Given that $z=\sqrt{3}+\mathrm{i}$, by first expressing $z$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, find the first three positive integer $n$ such that ${{z}^{n}}$ is purely imaginary.</span></div><p style="text-align: right;">[4]</p></li></ol>

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

The complex number $w$ is given by $w=\frac{{{\left( \cos \frac{\pi }{3}+\mathrm{i}\sin \frac{\pi }{3} \right)}^{4}}}{{{\left( \cos \frac{\pi }{6}-\mathrm{i}\sin \frac{\pi }{6} \right)}^{5}}}$.

Find $\left| w \right|$ and the exact value of $\arg \left( w \right)$. Hence find the value of ${{w}^{6}}$.

Express $\frac{1+{{\mathrm{e}}^{\mathrm{i}\theta }}}{1+{{\mathrm{e}}^{-\mathrm{i}\theta }}}$ in the modulus-argument form. Hence find the value of ${{\left( 1+w \right)}^{6}}+{{\left( 1+{{w}^{*}} \right)}^{6}}$.

Given that $z=\sqrt{3}+\mathrm{i}$, by first expressing $z$ in the form $r{{\mathrm{e}}^{\mathrm{i}\theta }}$, find the first three positive integer $n$ such that ${{z}^{n}}$ is purely imaginary.

2022 ACJC J2 MYE P1 Q6

2022 ACJC J2 MYE P1 Q6

Video Solution:

Solution:

Solution locked