Question

If $z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right]$ and $z_{2}=\sqrt{3}\left[\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right]$ then $\left|z_{1} z_{2}\right|$ is equal to $\sqrt{m}$. Value of $m$ is

• A. 6
• B. 3
• C. 2
• D. 5

Answer 1

Answer: (A)

$z_{1}=\sqrt{2}\left[\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right]=\sqrt{2}\left[\frac{1}{\sqrt{2}}+i \frac{1}{\sqrt{2}}\right]=1+i$
$\left|z_{1}\right|=\sqrt{2}$
and $z_{2}=\sqrt{3}\left[\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right]=\sqrt{3}\left[\frac{1}{2}+i \frac{\sqrt{3}}{2}\right]$
$\left|z_{2}\right|=\sqrt{\frac{3}{4}+\frac{9}{4}}=\sqrt{3}$
$\left|z_{1} z_{2}\right|=\left|z_{1}\right|\left|z_{2}\right|=\sqrt{2} \cdot \sqrt{3}=\sqrt{6}$