Question

$If\frac{1+i\cos \theta }{1-2i\cos \theta }$ is imaginary then $\theta =$

• A. $2n\pi +\frac{\pi }{2}$
• B. $2n\pi -\frac{\pi }{2}$
• C. $2n\pi \pm \frac{\pi }{2}$
• D. $n\pi \pm \frac{\pi }{4}$

Answer 1

The correct option is D $n\pi \pm \frac{\pi }{4}$

$\frac{1+i\cos \theta }{1-2i\cos \theta }\ast \frac{(1+2i\cos \theta )}{(1+2i\cos \theta )}=0+ki$
$\Rightarrow \frac{1+2i\cos \theta +i\cos \theta -2{\cos }^2\theta }{(1^2+4{\cos }^{2}co{s}^2\theta )}=0+ki$
$\Rightarrow 1-2{\cos }^2\theta =0$
$\Rightarrow \cos \theta =\pm \sqrt{1/2}$
$\Rightarrow \theta =n\pi \pm \pi /4$