The number of values of $$x$$ in $$\left[ {0,5\pi } \right]$$  satisfying the equation $$3\cos 2x - 10\cos x + 7 = 0$$      is

Solution :
Here, $$6{\cos ^2}x - 10\cos x + 4 = 0\,\,\,\,{\text{or, }}\left( {3\cos x - 2} \right)\left( {\cos x - 1} \right) = 0$$
$$\eqalign{ & \therefore \,\,\cos x = \frac{2}{3},1 \cr & \Rightarrow \,\,x = 2n\pi \pm {\cos ^{ - 1}}\frac{2}{3},2n\pi . \cr} $$
From $$x = 2n\pi $$   we get one solution each for $$n = 0,1,2.$$
From $$x = 2n\pi \pm {\cos ^{ - 1}}\frac{2}{3}$$    we get the following solutions:
when $$n = 0,$$  one solution; when $$n = 1,$$  two solutions; when $$n = 2,$$  two solutions.