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

Column $$I$$
(A) If $$a=1$$ and $$b=0,$$ then $$(x, y)$$
(B) If $$a=1$$ and $$b=1,$$ then $$(x, y)$$
(C) If $$a=1$$ and $$b=2,$$ then $$(x, y)$$
(D) If $$a=2$$ and $$b=2,$$ then $$(x, y)$$

Column $$II$$
(p) lies on the circle $${x^2} + {y^2} = 1$$
(q) lies on $$\left( {{x^2} - 1} \right)\left( {{y^2} - 1} \right) = 0$$
(r) lies on $$y=x$$
(s) lies on $$\left( {4{x^2} - 1} \right)\left( {{y^2} - 1} \right) = 0$$

Match the following

Column $$I$$

(A) $$\sum\limits_{i = 1}^\infty {{{\tan }^{ - 1}}\left( {{1 \over {2{i^2}}}} \right) = t,} $$ then tan $$t=$$

(B) Sides $$a, b, c$$ of a triangle $$ABC$$ are in $$AP$$ and
$$\cos {\theta _1} = {a \over {b + c}},\,\cos {\theta _2} = {b \over {a + c}},\cos {\theta _3} = {c \over {a + b}},$$
then $${\tan ^2}\left( {{{{\theta _1}} \over 2}} \right) + {\tan ^2}\left( {{{{\theta _3}} \over 2}} \right) = $$

(C) A line is perpendicular to $$x + 2y + 2z = 0$$ and
passes through $$(0, 1, 0)$$. The perpendicular distance of this line from the origin is

Column $$II$$

(p) $$1$$

(q) $${{\sqrt 5 } \over 3}$$

(r) $${2 \over 3}$$

Prove that $$\cos \,ta{n^{ - 1}}\sin \,{\cot ^{ - 1}}x = \sqrt {{{{x^2} + 1} \over {{x^2} + 2}}} $$.

Find all the solution of $$4$$ $${\cos ^2}x\sin x - 2{\sin ^2}x = 3\sin x$$