2022 NYJC J2 CT2 P1 Q3
Timothy Gan
2022 NYJC J2 CT2 P1 Q3
The complex number $z$ is given by $z=\frac{{{\left( \cos \left( \frac{1}{12}\pi \right)-\text{i}\sin \left( \frac{1}{12}\pi \right) \right)}^{3}}}{{{\left( a+\text{i} \right)}^{2}}}$ where $a$ is a positive real number.
Given $\left| z \right|=\frac{1}{4}$, find the exact value of $a$.
[2]
Hence, find
(i)
$\arg z$, where $-\pi <\arg z\le \pi $,
[2]
(ii)
the exact real values of $x$ and $y$ which satisfy the equation ${{z}^{4}}={{e}^{x+\text{i}y}}$, where $-\pi <y\le \pi $, giving your answers in simplest form.
[3]
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