Question

The real value of $\theta$ for which the expression $\frac{1+i\cos \theta }{1-2i\cos \theta }$ is a real number is:

• A. $n\pi +(-1{)}^n\frac{\pi }{4}$
• B. $2n\pi \pm \frac{\pi }{2}$
• C. none of these
• D. $n\pi +\frac{\pi }{4}$

Answer 1

The correct option is B $2n\pi \pm \frac{\pi }{2}$

Let $z=\frac{1+i\cos \theta }{1-2i\cos \theta }$

Now,

$\Rightarrow z=\frac{1+i\cos \theta }{1-2i\cos \theta }\times \frac{1+2i\cos \theta }{1+2i\cos \theta }$

$\Rightarrow z=\frac{1+2i\cos \theta +i\cos \theta +2i^2{\cos }^2\theta }{1^2-(2i\cos \theta {)}^2}$

$\Rightarrow z=\frac{1-2{\cos }^2\theta +3i\cos \theta }{1+4{\cos }^2\theta }$ $[\because i^2=-1]$

As $z$ is a real number, so

$\Rightarrow Im\left(z\right)=0$

$\Rightarrow \frac{3\cos \theta }{1+4{\cos }^2\theta }=0$

$\Rightarrow \cos \theta =0$ $[\because 1+4{\cos }^2\theta >0]$

$\Rightarrow \cos \theta =\cos \frac{\pi }{2}$

$\Rightarrow \theta =2n\pi \pm \frac{\pi }{2},n\in I$

$[\because \cos \alpha =\cos \theta \Rightarrow \alpha =2n\pi \pm \theta ,n\in I]$

Hence, option $\left(C\right)$ is correct.