The number of solutions \(x\) to the equation \[7\sin x +2\cos ^2 x =5,\] in the range \(0\le x<2\pi\), is

\(1\),

\(2\),

\(3\),

\(4\)

So we must have \(\sin x=3\), which has no solutions for real \(x\), or \(2\sin x=1\), that is, \(\sin x=\dfrac{1}{2}\). This has two solutions in the relevant range (namely \(\dfrac{\pi}{6}\) and \(\dfrac{5\pi}{6}\)).

So the answer is (b).