[cot x]dx, where [.] denotes the greatest integer function, is equal to

• π/2

• 1

• -1

• – π/2 

D.

– π/2 

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

• Interval	Function
  (-∞, ∞)	x3 – 3x2 + 3x + 3

• Interval	Function
  [2, ∞)	2x3 – 3x2 – 12x + 6

• Interval	Function
  (-∞, 1/3]	3x2 – 2x + 1

• Interval	Function
  (- ∞, -4]	x3 + 6x2 + 6

A real valued function f(x) satisfies the functional equation f(x – y) = f(x) f(y) – f(a – x) f(a + y) where a is a given constant and f(0) = 1, f(2a – x) is equal to

• –f(x)

• f(x)

• f(a) + f(a – x)

• f(-x)

Consider the following relations:
R = {(x, y)| x, y are real numbers and x = wy for some rational number w}; S = {(m/p, p/q)| m, n, p and q are integers such that n, q ≠ 0 and qm = pn}. Then

• R is an equivalence relation but S is not an equivalence relation

• neither R nor S is an equivalence relation

• S is an equivalence relation but R is not an equivalence relation

• R and S both are equivalence relations

For real x, let f(x) = x³+ 5x + 1, then

• f is one–one but not onto R

• f is onto R but not one–one

• f is one–one and onto R

• f is neither one–one nor onto R