For three events A, B and C,

P(Exactly one of A or B occurs)
= P(Exactly one of B or C occurs)
= P (Exactly one of C or A occurs) = $${1 \over 4}$$
and P(All the three events occur simultaneously) = $${1 \over {16}}$$.

Then the probability that at least one of the events occurs, is :

The area (in sq. units) of the region

$$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$$ is

Let $$\omega $$ be a complex number such that 2$$\omega $$ + 1 = z where z = $$\sqrt {-3} $$. If

$$\left| {\matrix{ 1 & 1 & 1 \cr 1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr 1 & {{\omega ^2}} & {{\omega ^7}} \cr } } \right| = 3k$$,

then k is equal to :

C_P and C_v are specific heats at constant pressure and constant volume respectively. It is observed that
C_P – C_v = a for hydrogen gas
C_P – C_v = b for nitrogen gas
The correct relation between a and b is