Question

If $z$ is a complex number such that $\left|z\right|=1$ . If maximum value of $\left|z^3-z+2\right|$ is $\text{’}a\text{’}$ , then the sum of digit of $a^2$ is

Answer 1

Answer: 4

$z=e^{i\theta }$
let $y=\left|z^3-z+\alpha \right|$
$y=|\left(cos3\theta +isin3\theta \right)-\left(cos\theta +is\text{i}n\theta \right)+2$
$y^2={\left(cos3\theta -cos\theta +2\right)}^2+{\left(sin3\theta -sin\theta \right)}^2$
$y^2=4cos3\theta -2cos2\theta -4cos\theta +6$
Let $cos\theta =x$
$y^2=16x^3-4x^2-16x+8$
So $x\in \left[-1,1\right]$
$y_{ma{x}^2}=13$
$y_{max}=\sqrt{13}$