Explanation

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