2022 NYJC J2 CT2 P1 Q3

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 \(a\).[2]

Hence, find

1. \(\arg z\), where \(-\pi <\arg z\le \pi \),[2]
2. 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.[3]