Match List $$I$$ with List $$II$$ and select the correct answer using the code given below the lists:

List $$I$$
$$P.$$$$\,\,\,\,\,$$ $${\left( {{1 \over {{y^2}}}{{\left( {{{\cos \left( {{{\tan }^{ - 1}}y} \right) + y\sin \left( {{{\tan }^{ - 1}}y} \right)} \over {\cot \left( {{{\sin }^{ - 1}}y} \right) + \tan \left( {{{\sin }^{ - 1}}y} \right)}}} \right)}^2} + {y^4}} \right)^{1/2}}$$ takes value

$$Q.$$ $$\,\,\,\,$$ If $$\cos x + \cos y + \cos z = 0 = \sin x + \sin y + \sin z$$ then
possible value of $$\cos {{x - y} \over 2}$$ is

$$R.$$ $$\,\,\,\,\,$$ If $$\cos \left( {{\pi \over 4} - x} \right)\cos 2x + \sin x\sin 2\sec x = \cos x\sin 2x\sec x + $$
$$\cos \left( {{\pi \over 4} + x} \right)\cos 2x$$ then possible value of $$\sec x$$ is

$$S.$$ $$\,\,\,\,\,$$ If $$\cot \left( {{{\sin }^{ - 1}}\sqrt {1 - {x^2}} } \right) = \sin \left( {{{\tan }^{ - 1}}\left( {x\sqrt 6 } \right)} \right),\,\,x \ne 0,$$
Then possible value of $$x$$ is

List $$II$$
$$1.$$ $$\,\,\,\,\,$$ $${1 \over 2}\sqrt {{5 \over 3}} $$

$$2.$$ $$\,\,\,\,\,$$ $$\sqrt 2 $$

$$3.$$ $$\,\,\,\,\,$$ $${1 \over 2}$$

$$1.$$ $$\,\,\,\,$$ $$1$$

The value of $$\cot \left( {\sum\limits_{n = 1}^{23} {{{\cot }^{ - 1}}} \left( {1 + \sum\limits_{k = 1}^n {2k} } \right)} \right)$$ is

If $$0 < x < 1$$, then

$$\sqrt {1 + {x^2}} {\left[ {{{\left\{ {x\cos \left( {{{\cot }^{ - 1}}x} \right) + \sin \left( {{{\cot }^{ - 1}}x} \right)} \right\}}^2} - 1} \right]^{1/2}} = $$

Let $$(x,y)$$ be such that $${\sin ^{ - 1}}(ax) + {\cos ^{ - 1}}(y) + {\cos ^{ - 1}}(bxy) = {\pi \over 2}$$.

Match the statements in Column I with the statements in Column II.

	Column I		Column II
(A)	If $$a=1$$ and $$b=0$$, then $$(x,y)$$	(P)	lies on the circle $$x^2+y^2=1$$
(B)	If $$a=1$$ and $$b=1$$, then $$(x,y)$$	(Q)	lies on $$(x^2-1)(y^2-1)=0$$
(C)	If $$a=1$$ and $$b=2$$, then $$(x,y)$$	(R)	lies on $$y=x$$
(D)	If $$a=2$$ and $$b=2$$, then $$(x,y)$$	(S)	lies on $$(4x^2-1)(y^2-1)=0$$