A) \[-3\]
B) \[-2\]
C) 1
D) 0
E) 1
Correct Answer: E
Solution :
\[{{z}_{1}}.{{z}_{2}}.{{z}_{3}}....\infty \] \[=\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right)\left( \cos \frac{\pi }{{{2}^{2}}}+i\sin \frac{\pi }{{{2}^{2}}} \right)\] \[\times \left( \cos \frac{\pi }{{{2}^{3}}}+i\sin \frac{\pi }{{{2}^{3}}} \right)...\] \[=\cos \left( \frac{\pi }{2}+\frac{\pi }{{{2}^{2}}}+\frac{\pi }{{{2}^{3}}}+.... \right)\] \[+i\sin \left( \frac{\pi }{2}+\frac{\pi }{{{2}^{2}}}+\frac{\pi }{{{2}^{3}}}+.... \right)\] \[=\cos \left( \frac{\pi /2}{1-1/2} \right)+i\sin \left( \frac{\pi /2}{1-1/2} \right)\] \[=\cos \pi +i\sin \pi =-1\]
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