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प्रश्न
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
विकल्प
60°
120°
210°
240°
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उत्तर
240°
\[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\]
\[\text { Rationalising the denominator,} \]
\[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \times \frac{1 - i\sqrt{3}}{1 - i\sqrt{3}}\]
\[ = \frac{1 + 3 i^2 - 2\sqrt{3} i}{1 - 3 i^2}\]
\[ = \frac{- 2 - 2\sqrt{3} i}{4} \left( \because i^2 = - 1 \right)\]
\[ = \frac{- 1}{2} - i\frac{\sqrt{3}}{2}\]
\[\tan \alpha = \left| \frac{Im (z)}{Re (z)} \right|\]
\[\text { Then,} \tan \alpha = \left| \frac{\frac{- \sqrt{3}}{2}}{\frac{- 1}{2}} \right|\]
\[ = \sqrt{3} \]
\[ \Rightarrow \alpha = 60°\]
\[\text { Since the points } \left( \frac{- 1}{2}, \frac{- \sqrt{3}}{2} \right) \text { lie in the third quadrant, the argument is given by}: \]
\[\theta = 180° + 60°\]
\[ = 240°\]
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