Question

$27i{z}^3+18z^2-8i=0,$ $\left|z\right|=$

• A. $\frac{2}{3}$
• B. $\frac{4}{9}$
• C. $1$
• D. none

Answer 1

The correct option is A $\frac{2}{3}$

$9i{z}^2(3z-2i)-4(3z-2i)=0$
$\therefore (3z-2i)(9i{z}^2-4)=0$
$\therefore z=\frac{2}{3}i$ or $z^2=\frac{4}{9i}=-\frac{4}{9}i$
$\therefore \left|z\right|=\frac{2}{3}or\left|z^2\right|={\left|z\right|}^2=\frac{4}{9}$
$\therefore \left|z\right|=\frac{2}{3}.$
Hence in either case $\left|z\right|=\frac{2}{3}$ being a + ive number.