Let
`A = {0 in (-(pi)/(2), pi):(3 + 2i sin theta)/(1 - 2i sin theta)" is purely imaginary"}`
Then the sum of the elements in A is

Options

A. `(3pi)/(4)`
B. `(5 pi)/(6)`
C. `pi`
D. `(2 pi )/(3)`

Answer

A

Explanation

Let `z=((3+2i sin theta)/(1-2i sin theta))xx((1+2 I sin theta)/(1+2 I sin theta))`
(reationaliisng the denomunator)
`=(3-4 sin ^2 theta + 8 I sin theta)/(1+ 4 sin ^2 theta)i`
As z is purely imaginary , so real part of z=0
`therefore (3- 4 sin ^2 theta )/(1+ 4 sin ^2 theta )=0 rArr 3 - 4 sin^2 thet=0 `
`rArr sin^2 theta = 3/4 rArr sin theta = pm (sqrt(3))/(2)`

`rArr theta ne {-(pi)/3,(pi)/3,(2 pi)/3}`
Sum of values of `theta = (2 pi) /3`