A) 2
B) 6
C) 5
D) 3
Correct Answer: D
Solution :
First rationalize the number \[\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\times \left( \frac{1+i\sqrt{3}}{1+i\sqrt{3}} \right)=\left( \frac{-2+i2\sqrt{3}}{4} \right)=\left( \frac{1-i\sqrt{3}}{-2} \right)\] ?(1) \[\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\times \left( \frac{1-i\sqrt{3}}{1-i\sqrt{3}} \right)=\left( \frac{4}{-2-i2\sqrt{3}} \right)=\left( \frac{-2}{1+i\sqrt{3}} \right)\] ?.(2) Using (1)and (2) \[{{\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)}^{3}}=\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\times \left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\times \left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\] \[=\left( \frac{1+i\sqrt{3}}{1-i\sqrt{3}} \right)\times \left( \frac{-2}{1+i\sqrt{3}} \right)\times \left( \frac{1-i\sqrt{3}}{-2} \right)=1\] Therefore correct Answer is 3 so correct option is D
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