A) \[2n\pi ,\], n is an integer
B) \[2n\pi +\frac{\pi }{2},\] n is an integer
C) \[2n\pi -\frac{\pi }{2},\] n is an integer
D) \[n\pi +\frac{\pi }{2},\] n is an integer
Correct Answer: A
Solution :
Let \[z=\frac{1+i\sin \theta }{1-2i\,\sin \theta }\times \frac{1+2i\,\sin \theta }{1+2i\,\sin \theta }\] \[=\frac{1-2{{\sin }^{2}}\theta +3i\,(\sin \theta )}{1+4{{\sin }^{2}}\theta }\] For z to be real, \[3\,\sin \theta =0\] \[\Rightarrow \] \[\theta =n\pi \]
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