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. $10$
• B. $5$
• C. $6$
• D. $0$

Answer 1

The correct option is C $6$

$3{\sin }^{2}x-7\sin x+2=0$

$\Rightarrow 3{\sin }^{2}x-6\sin x-\sin x+2=0$

$\Rightarrow (3\sin x-1)(\sin x-2)=0$

$\Rightarrow (3\sin x-1)=0\text{or}(\sin x-2)=0$

$\Rightarrow \sin x=\frac{1}{3}\text{or}\sin x=2$

As $\sin x\in [-1,1]$, so $\sin x=\frac{1}{2}$

Since $\sin x$ is positive in first and second quadrant.

So, $\sin x=\frac{1}{3}$ has $2$ solutions in $[0,\pi ]$

Similarly, $2$ solutions in $[2\pi ,3\pi ]$

Similarly, $2$ solution in $[4\pi ,5\pi ]$

Therefore, the total number of solution are $6$.
Hence, the option (C) i s correct.