A) \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{6}\]
D) \[{{\sin }^{-1}}\left( \frac{\sqrt{3}}{4} \right)\]
Correct Answer: A
Solution :
\[|Z=\frac{2+3i\sin \theta }{1-2\sin \theta }\] \[\Rightarrow \]\[|Z=\frac{(2+3i\sin \theta )(1+2i\sin \theta )}{1+4{{\sin }^{2}}\theta }\] \[=\frac{\left( 2-6i{{\sin }^{2}}\theta \right)+7i\sin \theta )}{1+4{{\sin }^{2}}\theta }\] for purely imaginary Z, Re(Z) = 0 \[\Rightarrow \] \[2-6{{\sin }^{2}}\theta =0\Rightarrow \sin \theta =\pm \frac{1}{\sqrt{3}}\] \[\Rightarrow \]\[\theta =\pm {{\sin }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]
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