Let $$F(x)$$ be an indefinite integral of $$si{n^2}x.$$

STATEMENT-1: The function $$F(x)$$ satisfies $$F\left( {x + \pi } \right) = F\left( x \right)$$
for all real $$x$$. because

STATEMENT-2: $${\sin ^2}\left( {x + \pi } \right) = {\sin ^2}x$$ for all real $$x$$.

Match the integrals in Column $$I$$ with the values in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS$$.

Column $$I$$
(A) $$\int\limits_{ - 1}^1 {{{dx} \over {1 + {x^2}}}} $$
(B) $$\int\limits_0^1 {{{dx} \over {\sqrt {1 - {x^2}} }}} $$
(C) $$\int\limits_2^3 {{{dx} \over {1 - {x^2}}}} $$
(D) $$\int\limits_1^2 {{{dx} \over {x\sqrt {{x^2} - 1} }}} $$

Column $$II$$
(p) $${1 \over 2}\log \left( {{2 \over 3}} \right)$$
(q) $$2\log \left( {{2 \over 3}} \right)$$
(r) $${{\pi \over 3}}$$
(s) $${{\pi \over 2}}$$

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

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