2021 NYJC P1 Q9

The complex numbers \(z\) and \(w\) where \(w\ne 0\) satisfy the relation

\(2z=\left| w \right|+1\).

1. It is given that \(a\) is a real number and that \(a\) and \(z\) satisfy the equation \(2{{z}^{3}}-5{{z}^{2}}+2z+\left( a+3 \right)\mathbf{i}=0\).
   Explain, with justification, why \(a=-3\) and that the only possible value of \(z\)is \(2\).[6]

It is given that \(\arg w=\frac{\pi }{3}\).

2. Express the complex number \(w\) in the form \(p+q\mathbf{i}\) where \(p\) and \(q\)are in non-trigonometric form.[2]
3. Find the least integer \(n\) such that \(\left| {{w}^{n}} \right|>\mathrm{20212021}\).[2]
4. Find the least positive integer \(k\) such that \({{w}^{k}}\) is a positive real number.[2]