Question

For the complex number $z=\sqrt{3}i-i-1-\sqrt{3},$ the correct option(s) is/are

• A. $|z|=2\sqrt{2}$
• B. $\arg \left(z\right)=\frac{11\pi }{12}$
• C. $|z|=2$
• D. $\arg \left(z\right)=\frac{13\pi }{12}$

Answer 1

Answer: (A), (B)

$\arg \left(z\right)=\frac{11\pi }{12}$

We have,
$z=\sqrt{3}i-i-1-\sqrt{3}$
$\Rightarrow |z|=\sqrt{(-\sqrt{3}-1{)}^2+(\sqrt{3}-1{)}^2}=2\sqrt{2}$
and $\tan \alpha =\left|\frac{\sqrt{3}-1}{\sqrt{3}+1}\right|\Rightarrow \alpha =\frac{\pi }{12}$
Clearly, $z$ lies in second quadrant.
$\therefore \arg \left(z\right)=\pi -\alpha =\pi -\frac{\pi }{12}=\frac{11\pi }{12}$