A) 0
B) 5
C) 6
D) 10
Correct Answer: C
Solution :
\[3{{\sin }^{2}}x-7\sin x+2=0\] \[\Rightarrow \] \[3{{\sin }^{2}}x-6\sin x-\sin x+2=0\] \[\Rightarrow \] \[3\sin (\sin x-2)-(\sin x-2)=0\] \[\Rightarrow \] \[(3\sin x-1)\,(\sin x-2)=0\]\[\Rightarrow \] \[\sin x=\frac{1}{3}\text{ or 2}\] \[\Rightarrow \] \[\sin x=\frac{1}{3}\], (\[\because \,\,\sin x\ne 2\]) Let \[{{\sin }^{-1}}\frac{1}{3}=\alpha \], \[0<\alpha <\frac{\pi }{2}\] are the solutions in \[[0,\text{ }5\pi ]\]. 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.
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