Let S be a sample space and two mutually exclusive events A and B be such that $$A\, \cup \,B = \,S$$. If P(.) denotes the probability of the event, the maximum value of P(A) P(B) is ________________.

Suppose you want to move from 0 to 100 on the number line. In each step, you either move right by a unit distance or you take a shortcut. A shortcut is simply a pre specified pair of integers i, j with i < j. Given a shortcut i, j if you are at position i on the number line, you may directly move to j. suppose T(k) denotes the smallest number of steps needed to move from k to 100. Suppose further that there is at most 1 shortcut involving any number, and in particular from 9 there is a shortcut to 15. Let y and z be such that T(9) = 1+ min(T(y),T(z)). Then the value of the product yz is _______.

If $$G$$ is a forest with $$n$$ vertices and $$k$$ connected components, how many edges does $$G$$ have?

The value of the integral given below is $$$\int_0^\pi {{x^2}\,\cos \,x\,dx} $$$