Question

The number of values of $x$ in the interval $[0,5 \pi]$ satisfying the equation $3 \sin ^{2} x-7 \sin x+2=0$ is

• A. 0
• B. 5
• C. 6
• D. 10

Answer 1

Answer: (C)

$3 \sin ^{2} x-7 \sin x+2=0$
$\Rightarrow 3 \sin 2 x-6 \sin x-\sin x+2=0$
$\Rightarrow 3 \sin x(\sin x-2)-(\sin x-2)=0$
$\Rightarrow(3 \sin x-1)(\sin x-2)=0$
$\Rightarrow \sin x=\frac{1}{3}$ or $2$
$\Rightarrow \sin x=\frac{1}{3}$
$(\because \sin x \neq 2)$
Let $\sin ^{-1} \frac{1}{3}=\alpha, 0< \alpha< \frac{\pi}{2}$ are the solutions in $[0,5 \pi]$.
Then $\alpha, \pi-\alpha, 2 \pi+\alpha, 3 \pi-\alpha, 4 \pi+\alpha, 5 \pi-\alpha$ are the solutions in $[0,5 \pi]$.
$\therefore$ Required number of solutions $=6$