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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

  • \[\frac{\pi}{3}\]

  • \[\frac{2\pi}{3}\]

  • \[\frac{\pi}{4}\]

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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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APPEARS IN

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