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Question
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
Options
π
\[\frac{\pi}{3}\]
\[\frac{2\pi}{3}\]
\[\frac{\pi}{4}\]
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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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