Question

Let $G$ be the sum of infinite geometric series whose first term is $\sin \theta$ and $\mathrm{common}$ ratio is $\cos \theta$ while $G^{\prime }$ be the sum of different infinite G.P. whose first term is 1 and common ratio is $\frac{1}{2}$. The number of solutions of $G=G^{\prime }$ in $[0,2\pi ]$, is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (B)

$G=\frac{\sin \theta }{1-\cos \theta }$ and $G^{\prime }=\frac{1}{1-(1/2)}=2$
$\therefore G=G^{\prime }\Rightarrow \frac{\sin \theta }{1-\cos \theta }=2\Rightarrow {\sin }^2\theta =4\left(1+{\cos }^2\theta -2\cos \theta \right)$
$\Rightarrow 1-{\cos }^2\theta =4+4{\cos }^2\theta -8\cos \theta$
$\Rightarrow (\cos \theta -1)(5\cos \theta -3)=0$
$\therefore \cos \theta =\frac{3}{5}\text{ and }\cos \theta =1\text{ (rejected) }$
$\Rightarrow \theta \in$ I and IV quadrant but in IV quadrant $\sin \theta$ is negative therefore rejected.
$\therefore$ Number of solution are 1.