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

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?

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 X be a square matrix. Consider the following two statements on X.

I. X is invertible.

II. Determinant of X is non-zero.

Which one of the following is TRUE?