NYJC/2021/JC2/Prelim/P1/Q09

The complex numbers $z$ and $w$ where $w \neq 0$ satisfy the relation
$$2 z=|w|+1 .$$
(i) It is given that $a$ is a real number and that $a$ and $z$ satisfy the equation $2 z^3-5 z^2+2 z+(a+3) i =0$. 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 $p+q$ i where $p$ and $q$ are in non-trigonometric form.

[2]

(iii) Find the least integer $n$ such that $\left|w^n\right|>20212021$.

[2]

(iv) Find the least positive integer $k$ such that $w^k$ is a positive real number.

[2]