Question

$\frac{3+2isin\theta }{1-2isin\theta }$ will be purely imaginary, if $\theta$ =

[Where n is an integer]

[IIT 1976; Pb. CET 2003]

• A. $2n\pi \pm \frac{\pi }{3}$
• B. $n\pi +\frac{\pi }{3}$
• C. $n\pi \pm \frac{\pi }{3}$
• D. None of these

Answer 1

The correct option is C $n\pi \pm \frac{\pi }{3}$

$\frac{3+2isin\theta }{1-2isin\theta }$ will be purely imaginary, if the real part vanishes, i.e., $\frac{3-4si{n}^2\theta }{1+4si{n}^2\theta }=0$

$\Rightarrow 3-4si{n}^2\theta =0$ (only if $\theta$ be real)
$\Rightarrow sin\theta \pm \frac{\sqrt{3}}{2}=sin(\pm \frac{\pi }{3})$
$\Rightarrow \theta =n\pi +(-1{)}^n(\pm \frac{\pi }{3})=n\pi \pm \frac{\pi }{3}$