A) \[-1\]
B) 1
C) \[\frac{3}{2}\]
D) \[-\frac{1}{2}\]
Correct Answer: B
Solution :
Given that\[{{\left[ \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right]}^{3/4}}\] \[={{[\cos \pi +i\sin \pi ]}^{1/4}}\]. Since the expression has only 4 different roots, therefore on putting \[n=0,\,1,\,2,\,3\] in \[\cos \left[ \frac{2n\pi +\pi }{4} \right]+i\sin \left[ \frac{2n\pi +\pi }{4} \right]\] and multiplying them, we get \[=\left[ \cos \frac{\pi }{4}+i\sin \frac{\pi }{4} \right]\,\left[ \cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4} \right]\,\]\[\left[ \cos \frac{5\pi }{4}+i\sin \frac{5\pi }{4} \right]\,\left[ \cos \frac{7\pi }{4}+i\sin \frac{7\pi }{4} \right]\] \[=\left[ \frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right]\,\left[ -\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}} \right]\,\left[ -\frac{1}{\sqrt{2}}+i\frac{-1}{\sqrt{2}} \right]\,\left[ \frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}} \right]\] \[=\left( -\frac{1}{2}-\frac{1}{2} \right)\,\left( -\frac{1}{2}-\frac{1}{2} \right)=(-1)(-1)=1\].
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