A) \[\frac{1+\sqrt{3}i}{2}\]
B) \[\frac{1-\sqrt{3}i}{2}\]
C) \[\frac{-\sqrt{3}-i}{2}\]
D) None of these
Correct Answer: C
Solution :
(a) \[{{\left( \frac{1+\sqrt{3}i}{2} \right)}^{3}}={{\left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)}^{3}}\] \[=\cos \pi +i\sin \pi \] \[=-1\] (b) \[{{\left( \frac{1-\sqrt{3}i}{2} \right)}^{3}}={{\left( \cos \frac{\pi }{3}-i\sin \frac{\pi }{3} \right)}^{3}}\] \[=\cos \pi -i\sin \pi \] \[=-1\] (c) \[{{\left( \frac{-\sqrt{3}-i}{2} \right)}^{3}}=-{{\left( \frac{\sqrt{3}+1}{2} \right)}^{3}}\] \[=-{{\left( \cos \frac{\pi }{6}+i\sin \frac{\pi }{6} \right)}^{3}}\] \[=-\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right)\] \[=-i\] \[\therefore \]Option (c) is correct.
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