Question

If $\theta_1, \theta_2, \theta_3, \theta_4$ are the roots of the equation $\sin (\theta+\alpha)= k \sin 2 \theta$, no two of which differ by a multiple of $2 \pi$, then $\theta_1+\theta_2+\theta_3+\theta_4$ is equal to -

• A. $2 n \pi, n \in Z$
• B. $(2 n +1) \pi, n \in Z$
• C. $n \pi, n \in Z$
• D. none of these

Answer 1

Answer: (B)

$\sin (\theta+\alpha)=k \sin 2 \theta $
$\sin \theta \cos \alpha+\cos \theta \sin \alpha=2 k \sin \theta \cos \theta$
Change the equation in $\tan \frac{\theta}{2}$ form and let $\tan \frac{\theta}{2}=t$, then obtained equation is $\sin \alpha t^4-(2 \cos \alpha+4 k) t^3+t(4 k-2 \cos \alpha)-\sin \alpha=0$ $S _1=\frac{2 \cos \alpha+4 k }{\sin \alpha}, S _2=0$
$S _3=\frac{4 k -2 \cos \alpha}{\sin \alpha}, S _4=-1$
$\tan \frac{\left(\theta_1+\theta_2+\theta_3+\theta_4\right)}{2}=\frac{ S _1- S _3}{1- S _2+ S _4}=\infty$
$\Rightarrow \frac{\theta_1+\theta_2+\theta_3+\theta_4}{2}=(2 n +1) \frac{\pi}{2}$
$\Rightarrow \theta_1+\theta_2+\theta_3+\theta_4=(2 n +1) \pi, n \in$ integer