Question

Which of the following is true for $z=\frac{(3+2i\sin \theta )}{(1-2i\sin \theta )}$,where $i=\sqrt{-}1$ ?

• A. Z is purely real for $\theta =n\pi \pm \frac{\pi }{3},nϵz$
• B. Z is purely imaginary for $\theta =n\pi \pm \frac{\pi }{2},nϵz$
• C. Z is purely real for $\theta =n\pi ,nϵz$
• D. None of these

Answer 1

The correct option is C Z is purely real for $\theta =n\pi ,nϵz$

$=\frac{3+2i\sin \theta }{1-2i\sin \theta }$

$=\frac{(3+2i\sin \theta )(1+2i\sin \theta )}{(1-2i\sin \theta )(1+2i\sin \theta )}$

$=\frac{3+8i\sin \theta -4{\sin }^2\theta }{1+4{\sin }^2\theta }$

It will be real only if $8i\sin \theta =0$
$\theta =n\pi$