<p>The complex number $z$ is given by $z=\frac{{{\left( \cos \left( \frac{1}{12}\pi \right)-\mathrm{i}\sin \left( \frac{1}{12}\pi \right) \right)}^{3}}}{{{\left( a+\mathrm{i} \right)}^{2}}}$ where $a$ is a positive real number.</p>
<p>Given $\left| z \right|=\frac{1}{4}$, find the exact value of $a$.</p><p style="text-align: right;">[2]</p>
<p>Hence, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">$\arg z$, where $-\pi <\arg z\le \pi $,</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">the exact real values of $x$ and $y$ which satisfy the equation ${{z}^{4}}={{e}^{x+\mathrm{i}y}}$, where $-\pi < y\le \pi $, giving your answers in simplest form.</span></div><p style="text-align: right;">[3]</p></li></ol>

The complex number $z$ is given by $z=\frac{{{\left( \cos \left( \frac{1}{12}\pi \right)-\mathrm{i}\sin \left( \frac{1}{12}\pi \right) \right)}^{3}}}{{{\left( a+\mathrm{i} \right)}^{2}}}$ where $a$ is a positive real number.

Given $\left| z \right|=\frac{1}{4}$, find the exact value of $a$.

the exact real values of $x$ and $y$ which satisfy the equation ${{z}^{4}}={{e}^{x+\mathrm{i}y}}$, where $-\pi < y\le \pi $, giving your answers in simplest form.

2022 NYJC J2 CT2 P1 Q3

2022 NYJC J2 CT2 P1 Q3

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