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)

$3{\sin }^{2}x-7\sin x+2=0$
$\Rightarrow 3\sin 2x-6\sin x-\sin x+2=0$
$\Rightarrow 3\sin x(\sin x-2)-(\sin x-2)=0$
$\Rightarrow (3\sin x-1)(\sin x-2)=0$
$\Rightarrow \sin x=\frac{1}{3}$ or $2$
$\Rightarrow \sin x=\frac{1}{3}$
$(\because \sin x\ne 2)$
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 $=6$