202212231206 Solution to 2002-CE-AMATH-I-18

(a) Let $z=\cos\theta +i\sin\theta$ , where $-\pi<\theta \leqslant \pi$ . Show that

$|z^2+1|^2=2(1+\cos 2\theta )$ .

Hence, or otherwise, find the greatest value of $|z^2+1|$ .
(b) $w$ is a complex number such that $|w|=3$ .
i. Show that the greatest value of $|w^2+9|$ is $18$ .
ii. Explain why the equation

$w^4-81=100i(w^2-9)$

has only two roots.

Roughwork.

(a)

$\begin{aligned} \textrm{LHS} & = |z^2+1|^2 \\ & = |(\cos\theta +i\sin\theta )^2+1|^2 \\ & = |(\cos^2\theta -\sin^2\theta +2i\sin\theta\cos\theta )+1|^2 \\ & = |(2\cos^2\theta )+i(2\sin\theta\cos\theta )|^2 \\ & = \big(\sqrt{(2\cos^2\theta )^2+(2\sin\theta\cos\theta )^2}\big)^2 \\ & = 4\cos^4\theta +4\sin^2\theta\cos^2\theta \\ & = 4\cos^2\theta (\cos^2\theta +\sin^2\theta ) \\ & = 4\cos^2\theta \\ \end{aligned}$