KVPY Sample Paper KVPY Stream-SX Model Paper-3

Let \[\text{A}=\left\{ \theta \,\in \left( \frac{\pi }{2},\pi \right):\frac{3+2i\text{sin}\theta }{1-2i\text{sin}\theta } \right\}\] purely imaginary. Then the sum of the element in A is:

A) \[\frac{5\pi }{6}\]

B) \[\frac{2\pi }{3}\]

C) \[\frac{3\pi }{4}\]

D) \[\pi \]

Correct Answer: B

Solution :

\[\theta \in \left( -\frac{\pi }{2},\pi \right)\]

\[\frac{3+2i\sin \theta }{1-2i\sin \theta }\] is purely imaginary

\[\Rightarrow \] \[3-4{{\sin }^{2}}\theta =0\]

\[\Rightarrow \] \[{{\sin }^{2}}\theta =\frac{3}{4}\]

\[\theta =n\pi \pm \frac{\pi }{3}\]\[\Rightarrow \] \[\theta =\frac{-\pi }{3},\]\[\frac{\pi }{3},\]\[\frac{2\pi }{3}.\]