The number of solutions of the pair of equations $$$\,2{\sin ^2}\theta - \cos 2\theta = 0$$$ $$$2co{s^2}\theta - 3\sin \theta = 0$$$

in the interval $$\left[ {0,2\pi } \right]$$

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
(P) The number of elements in the set $$\left\{x \in [-\pi,\pi] : \sin^6 x + \cos^4 x = 1 \right\}$$	(1) is 1
(Q) The number of elements in the set $$\left\{x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] : \sin^2 x + \cos^6 x = 1 \right\}$$	(2) is 2
(R) The number of elements in the set $$\left\{x \in [-\pi,\pi] : \cos^2\left(\frac{x}{2}\right) - \sin^2 x = \frac{1}{2} \right\}$$	(3) is 3
(S) The number of elements in the set $$\left\{x \in [-2\pi,2\pi] : 6\sin^2\left(\frac{x}{2}\right) - \cos 3x = 3 \right\}$$	(4) is 4 (5) is 5

Let $\frac{\pi}{2} < x < \pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then

$$ \left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x) $$

is equal to :

Consider the following lists :

List-I	List-II
(I) $$\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$$	(P) has two elements
(II) $$\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\}$$	(Q) has three elements
(III) $$\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}$$	(R) has four elements
(IV) $$\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$$	(S) has five elements
	(T) has six elements

The correct option is: