Q78 / 150 · MHT CET 2024 (16 May Shift 1) · Mathematics
The number of values of $x$ in the interval $(0,5 \pi)$ satisfying the equation $3 \sin ^2 x-7 \sin x+2=0$
- A0
- B5
- Correct answerC6
- D10
Solution
$\begin{aligned}
& 3 \sin ^2 x-7 \sin x+2=0 \\
& \Rightarrow 3 \sin ^2 x-6 \sin x-\sin x+2=0 \\
& \Rightarrow 3 \sin x(\sin x-2)-(\sin x-2)=0 \\
& \Rightarrow(3 \sin x-1)(\sin x-2)=0 \\
& \Rightarrow \sin x=\frac{1}{3} \text { or } 2 \\
& \Rightarrow \sin x=\frac{1}{3}
\end{aligned} \ldots .[\because \sin x \neq 2]$
Let $\sin ^{-1} \frac{1}{3}=\alpha, 0 \lt \alpha \lt \frac{\pi}{2}$ 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 $[0,5 \pi]$. $\therefore \quad$ number of solutions $=6$
Let $\sin ^{-1} \frac{1}{3}=\alpha, 0 \lt \alpha \lt \frac{\pi}{2}$ 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 $[0,5 \pi]$. $\therefore \quad$ number of solutions $=6$
Hence, the correct answer is option: 'C'.