What is the chromatic number of an $$n$$-vertex simple connected graph which does not contain any odd length cycle? Assume $$n \ge 2$$.

Which one of the following is TRUE for any simple connected undirected graph with more than $$2$$ vertices?

$$\int\limits_0^{\pi /4} {\left( {1 - \tan x} \right)/\left( {1 + \tan x} \right)dx} $$ $$\,\,\,\,\,\,$$ evaluates to

An unbalanced dice (with 6 faces, numbered from 1 to 6) is thrown. The probability that the face value is odd is 90% of the probability that the face valueis even. The probability of getting any even bnumbered face is the same.

If the probability that the face is even given that it is greater than 3 is 0.75, which one of the following options is closest to the probability that the face value exceeds 3?