A) \[i\]
B) \[1\]
C) \[-1\]
D) \[-i\]
Correct Answer: C
Solution :
\[{{x}_{1}}{{x}_{2}}{{x}_{3}}=\left[ \cos \left( \frac{\pi }{2} \right)+i\sin \left( \frac{\pi }{2} \right) \right]\] \[\left[ \cos \left( \frac{\pi }{{{2}^{2}}} \right)+i\sin \left( \frac{\pi }{{{2}^{2}}} \right) \right].....\] \[=\cos \left( \frac{\pi }{2}+\frac{\pi }{{{2}^{2}}}+\frac{\pi }{{{2}^{3}}}+.... \right)\] \[+i\sin \left( \frac{\pi }{2}+\frac{\pi }{{{2}^{2}}}+\frac{\pi }{{{2}^{3}}}+.... \right)\] \[=\cos \left( \frac{\pi /2}{1-\frac{1}{2}} \right)+i\sin \left( \frac{\pi /2}{1-\frac{1}{2}} \right)\] \[=\cos (\pi )+\sin (\pi )=-1\]
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