Let R be a relation defined on the set Z of all integers and xRy, when x + 2y is divisible by 3, then

• A is not transitive

• R is symmetric only

• R is an equivalence relation

• R is not an equivalence relation

Let f : N $\to$ R be such that f(1) = 1 and f(1) + 2f(2) +  3f(3) + ... + nf(n) = n(n+ 1) f(n), for all n  $\in$ N, n $\ge$ 2, where N is the set of natural numbers and R is the set of real numbers. Then, the value of f(500) is

• 1000

• 500

• 1/500

• 1/1000

13.

For the function f(x) = $\left[\frac{1}{\left[\mathrm{x}\right]}\right]$. where [x] denotes the greatest integer less than or equal to x, which of the following statements are true?

• The domain is $\left(- \infty , \infty \right)$

• The range is $\left\{0\right\} \cup \left\{- 1\right\} \cup \left\{1\right\}$

• The domain is $\left(- \infty , 0\right) \cup [1, \infty )$

• The range is $\left\{0\right\} \cup \left\{1\right\}$

If R be the set of all real numbers and f : R ➔ R is given by f(x) = 3x² + 1. Then, the set f^-1([1, 6]) is

• $\left\{- \sqrt{\frac{5}{3}}, 0, \sqrt{\frac{5}{3}}\right\}$

• $\left[- \sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right]$

• $\left[- \sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right]$

• $\left(- \sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right)$

Let X_n = $\left\{\mathrm{z} = \mathrm{x} + \mathrm{iy} : {\left|\mathrm{z}\right|}^2 \le \frac{1}{\mathrm{n}}\right\}$ for all integers $\mathrm{n} \ge 1. \mathrm{Then}, \underset{\mathrm{n} = 1}{\overset{\infty }{\cap }} {\mathrm{X}}_{\mathrm{n}}$ is

• a singleton set

• not a finite set

• an empty set

• a finite set with more than one element

The number of onto functions from the set {1, 2, ... , 11} to the set {1, 2, ... , 10} is

• $5 \times 11!$

• 10!

• $\frac{11!}{2}$

• $10 \times 11!$

If f(x) = 2¹⁰⁰x + 1, g(x) = 3¹⁰⁰x + 1,  then the set ofreal numbers x such that f{g(x)} = x is 

• empty

• a singleton

• a finite set with more than one element

• infinite

For any two sets A and B, A - (A - B) equals

• B

• A - B

• $\mathrm{A} \cap \mathrm{B}$

• ${\mathrm{A}}^{\mathrm{C}} \cap {\mathrm{B}}^{\mathrm{C}}$

Three sets A, B, C are such that A = B ∩ C and B = C ∩ A, then

• A ⊂ B

• A ⊃ B

• A ≡ B

• A ⊂ B'

If A = {x : x² - 5x + 6 = 0}, B={2, 4}, C = {4, 5}, then A x (B ∩ C) is

• {(2, 4), (3, 4)}

• 
  {(4, 2), (4, 3)}

• {(2, 4), (3, 4), (4, 4)}

• {(2, 2), (3, 3), (4, 4), (5, 5)}