Q83 / 125 · BITSAT 2021 · Mathematics

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

  • Correct answer
    A
    6
  • B
    3
  • C
    2
  • D
    5

Solution

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

Hence, the correct answer is option: 'A'.

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