Question

Let $G$ be the sum of infinite geometric series whose first term is $\sin \theta$ and $\operatorname{common}$ 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.