2021 NYJC P1 Q9
The complex numbers $z$ and $w$ where $w\ne 0$ satisfy the relation
$2z=\left| w \right|+1$.
(i)
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 $\arg w=\frac{\pi }{3}$.
(ii)
Express the complex number $w$ in the form $p+q\mathbf{i}$ where $p$ and $q$are in non-trigonometric form.
[2]
(iii)
Find the least integer $n$ such that $\left| {{w}^{n}} \right|>\text{20212021}$.
[2]
(iv)
Find the least positive integer $k$ such that ${{w}^{k}}$ is a positive real number.
[2]
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