# If Z = − 2 1 + I √ 3 ,Then the Value of Arg (Z) is - Mathematics

MCQ

If $z = \frac{- 2}{1 + i\sqrt{3}}$,then the value of arg (z) is

#### Options

• π

• $\frac{\pi}{3}$

• $\frac{2\pi}{3}$

• $\frac{\pi}{4}$

#### Solution

$\frac{2\pi}{3}$

z =$\frac{- 2}{1 + i\sqrt{3}}$

Rationalising z, we get,

$z = \frac{- 2}{1 + i\sqrt{3}} \times \frac{1 - i\sqrt{3}}{1 - i\sqrt{3}}$

$\Rightarrow z = \frac{- 2 + i2\sqrt{3}}{1 + 3}$

$\Rightarrow z = \frac{- 1 + i\sqrt{3}}{2}$

$\Rightarrow z = \frac{- 1}{2} + \frac{i\sqrt{3}}{2}$

$\tan \alpha = \left| \frac{Im(z)}{Re(z)} \right|$

$= \sqrt{3}$

$\Rightarrow \alpha = \frac{\pi}{3}$

$\text { Since, z lies in the second quadrant } .$

$\text { Therefore,}\arg (z) = \pi - \frac{\pi}{3}$

$= \frac{2\pi}{3}$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 13 Complex Numbers
Q 8 | Page 64