Question

If $z=\frac{(1+i\sqrt{3}{)}^2}{4i(1-i\sqrt{3})}$ , then $|z|$ and arg(z) are

• A. $1,\pi$
• B. $\frac{1}{2},\frac{\pi }{2}$
• C. $\frac{1}{8},\frac{\pi }{2}$
• D. $\frac{1}{4},\frac{\pi }{2}$

Answer 1

Answer: (B)

$\frac{1}{2},\frac{\pi }{2}$

We have $1+i\sqrt{3}=-2{\omega }^2$
and $1-i\sqrt{3}=-2\omega$
Therefore, $z=\frac{(-2{\omega }^2{)}^2}{4i(-2\omega )}$
$=\frac{4\omega }{4i(-2\omega )}$
$=\frac{-{\omega }^3}{2i}$

$=\frac{i}{2}$
$\therefore |z|=\frac{1}{2}$
$\therefore arg|z|=\frac{\pi }{2}$