A) \[\pm \,\,\left( \cos \frac{5\pi }{48}+i\sin \frac{5\pi }{48} \right)\]
B) \[\pm \,\,\left( \cos \frac{7\pi }{48}+i\sin \frac{7\pi }{48} \right)\]
C) \[\pm \,\,\left( \cos \frac{19\pi }{48}-i\sin \frac{19\pi }{48} \right)\]
D) None of these.
Correct Answer: A
Solution :
\[{{x}^{4}}=\frac{\sqrt{3}-1}{2\sqrt{2}}+i\frac{\sqrt{3}+1}{2\sqrt{2}}=\cos \frac{5\pi }{12}+i\,\sin \frac{5\pi }{12}\] So, \[x=\cos \left\{ \frac{k\pi }{2}+\frac{5\pi }{48} \right\}+i\,\,\sin \left\{ \frac{k\pi }{2}+\frac{5\pi }{48} \right\}\] \[k=0,\,\,1,\,\,2,\,\,3\] \[\therefore \] Roots are \[\cos \frac{5\pi }{48}+i\,sin\,\frac{5\pi }{48} \,for\,\,k=0\] \[\cos \frac{29\pi }{48}+i\,sin\,\frac{29\pi }{48} \,for\,\,k=1\] \[\cos \frac{53\pi }{48}+i\,sin\,\frac{53\pi }{48} =-\left( \cos \frac{5\pi }{48}+i\sin \frac{5\pi }{48} \right)\,\,for\,\,k=2\] \[\cos \frac{77\pi }{48}+i\,sin\,\frac{77\pi }{48} =-\left( \cos \frac{29\pi }{48}+i\sin \frac{29\pi }{48} \right)\] for \[\operatorname{k}=3\]
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