Question

The principal mod-amp form of $-\sqrt{7}+i\sqrt{21}$ is

• A. $2\cos \frac{2\pi }{3}$
• B. $\sqrt{7}cos\frac{2\pi }{3}$
• C. $2\sqrt{7}cos\frac{2\pi }{3}$
• D. none of these

Answer 1

Answer: (C)

$2\sqrt{7}cos\frac{2\pi }{3}$

Let $z=a+ib=-\sqrt{7}+i\sqrt{21}$
$\therefore \left|z\right|=\sqrt{a^2+b^2}=\sqrt{7+21}$$=\sqrt{28}=2\sqrt{7}$
amp $z=\pi -{\tan }^{-1}\left(\left|\frac{b}{a}\right|\right)$ $\because$ $a$ is -ve and $b$ is +ve
$=\pi -{\tan }^{-1}\left(\frac{\sqrt{21}}{\sqrt{7}}\right)$
$=\pi -{\tan }^{-1}\sqrt{3}$
$=\pi -\frac{\pi }{3}$
$=\frac{2\pi }{3}$

Hence, option C is correct