Which of the following is true for `z=(3+2isintheta)(1-2sintheta)w h e r ei=sqrt(-1)` ? z is purely real for `theta=npi+-pi/3,n in Z` z is purely imaginary for `theta=npi+-pi/2,n in Z` z is purely real for `theta=npi,n in Z` none of these

Options

A. z is purely real for `theta=n pi pm pi//3, n in Z`
B. z is purely imaginary for `theta=npi pm pi//2, n in Z`
C. z is purely real for `theta=n pi, n in Z`
D. none of these

Answer

C

Explanation

Let `Z=(3+2i sin theta)/(1-2 I sin theta)=((3+2i sin theta)(1+2i sin theta))/((1-2i sin theta)(1+2i sin theta))`
`=((3-4 sin^(2) theta)+8i sin theta)/(1+4 sin^(2) theta)`
Therefore, the real part of `Z=(3-4 sin^(2) theta)/(1+4 sin^(2) theta)`
and the imaginary part of `Z=(8 sin theta)/(1+4 sin^(2) theta)`
Z is real, if imaginary part`=(8 sin theta)/(1+4 sin^(2) theta)=0`
or `sin theta=0 or theta = n pi, AA n in I`
Z is purely imaginary, if real part
`(3-4 sin^(2) theta)//(1+4 sin^(2) theta)=0`
or `sin^(2)theta =3//4=sin^(2) (pi//3)`
or `theta=n pi pm pi//3, AA n in I`