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 $z_2=\sqrt{3}\left[\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right]=\sqrt{3}\left[\frac{1}{2}+i\frac{\sqrt{3}}{2}\right]$
$\left|z_2\right|=\sqrt{\frac{3}{4}+\frac{9}{4}}=\sqrt{3}$
$\left|z_1{z}_2\right|=\left|z_1\right|\left|z_2\right|=\sqrt{2}\cdot \sqrt{3}=\sqrt{6}$