Compute $$\mathop {\lim }\limits_{x \to 3} {{{x^4} - 81} \over {2{x^2} - 5x - 3}}$$

Two numbers are chosen independently and uniformly at random from the set {1, 2, ...., 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is ______.

Let G be an arbitrary group. Consider the following relations on G :

R1: ∀a,b ∈ G, aR₁b if and only if ∃g ∈ G such that a = g^-1bg

R2: ∀a,b ∈ G, aR₂b if and only if a = b^-1

Which of the above is/are equivalence relation/relations?

The following C program is executed on a Unix/Linux system:

         #include < unistd.h >
            int main ()
            {
                  int i ;
                  for (i=0; i<10; i++)
                         if (i%2 == 0) fork ( ) ;
                  return 0 ;
            }

The total number of child processes created is _____.