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

$z = e^{i \theta }$
let $y = \left|z^{3} - z + \alpha \right|$
$y = \left|\right. \left(cos 3 \theta + i sin 3 \theta \right) - \left(cos \theta + i s \text{i} n \theta \right) + 2$
$y^{2} = \left(cos 3 \theta - cos \theta + 2\right)^{2} + \left(sin 3 \theta - sin \theta \right)^{2}$
$y^{2} = 4 cos 3 \theta - 2 cos 2 \theta - 4 cos \theta + 6$
Let $cos \theta = x$
$y^{2} = 16 x^{3} - 4 x^{2} - 16 x + 8$
So $x \in \left[- 1 , 1\right]$
$y_{m a x^{2}} = 13$
$y_{m a x} = \sqrt{13}$