Let $${H_1},{H_2},....,{H_n}$$ be mutually exclusive and exhaustive events with $$P\left( {{H_1}} \right) > 0,i = 1,2,.....,n.$$ Let $$E$$ be any other event with $$0 < P\left( E \right) < 1.$$
STATEMENT-1:
$$P\left( {{H_1}|E} \right) > P\left( {E|{H_1}} \right).P\left( {{H_1}} \right)$$ for $$i=1,2,....,n$$ because

STATEMENT-2: $$\sum\limits_{i = 1}^n {P\left( {{H_i}} \right)} = 1.$$

Let $$ABCD$$ be a quadrilateral with area $$18$$, with side $$AB$$ parallel to the side $$CD$$ and $$2AB=CD$$. Let $$AD$$ be perpendicular to $$AB$$ and $$CD$$. If a circle is drawn inside the quadrilateral $$ABCD$$ touching all the sides, then its radius is

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 $$\left( {4{x^2} - 1} \right)\left( {{y^2} - 1} \right) = 0$$

The tangent to the curve $$y = {e^x}$$ drawn at the point $$\left( {c,{e^c}} \right)$$ intersects the line joining the points $$\left( {c - 1,{e^{c - 1}}} \right)$$ and $$\left( {c + 1,{e^{c + 1}}} \right)$$