List of top Mathematics Questions on Relations and functions asked in UP Board XII

A function \( f: \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 3x \) for all \( x \in \mathbb{R} \). Then function \( f \) will be:

Suppose that \( A = \{ 1, 2, 3 \} \), \( B = \{ 4, 5, 6, 7 \} \), and \( f = \{ (1, 4), (2, 5), (3, 6) \} \) be a function from \( A \) to \( B \). Then \( f \) is:

Prove that: \[ \begin{vmatrix} 1 + a & 1 & 1 \\ 1 & 1 + b & 1 \\ 1 & 1 & 1 + c \end{vmatrix} = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right). \]

Prove that the  \( f(x) = x^2 \) is continuous at \( x \neq 0 \).

The principal value of the  \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) will be:
 

{(a)Function \( f : \mathbb{R} \to \mathbb{R} \) is defined by \( f(x) = 5x, \forall x \in \mathbb{R} \). Select the correct answer:}
If the matrices \( X + Y = \begin{bmatrix} 5 & 2 \\ 0 & 9 \end{bmatrix} \) and \( X - Y = \begin{bmatrix} 3 & 6 \\ 0 & -1 \end{bmatrix} \), then find the matrices \( X \) and \( Y \).
(b) Find the value of \( \tan^{-1}(\sqrt{3}) - \cot^{-1}(-\sqrt{3}) \):
(a) If \( A = \{1, 2, 3\} \), \( B = \{4, 5\} \), then find the number of relations from \( A \) to \( B \):

Prove that the number of equivalence relations in the set \( \{1, 2, 3\} \) including \( \{(1, 2)\} \) and \( \{(2, 1)\} \) is 2.

A relation \( R = \{(a, b) : a = b - 2, b \geq 6 \} \) is defined on the set \( \mathbb{N} \). Then the correct answer will be:

A relation \( R \) is defined on the set \( \mathbb{Q}^* \) of non-zero rational numbers as follows: \[ a \, R \, b \quad \text{if} \quad a = \frac{1}{b}. \] Then on \( \mathbb{Q}^* \), this relation \( R \) is:
(b) Show that the points \( A(2,3,4) \), \( B(-1,-2,1) \), and \( C(5,8,7) \) are collinear.
(c) The modulus function \( f: \mathbb{R} \to \mathbb{R}^+ \) is given by \( f(x) = |x| \); then it will be:
(a) Prove that the function \( f(x) = |x - 1| \) is continuous at \( x = 1 \):

If \( R \) is the relation "less than" from \( A = \{1,2,3,4,5\} \) to \( B = \{1,4,5\} \), find the set of ordered pairs corresponding to \( R \). Also, define this relation from \( B \) to \( A \).

\( R \) is a relation on a set of natural numbers \( N \) defined by \[ R = \{(a, b): a, b \in N \text{ and } a = b^2 \} \] Is \( (a, b) \in R \), \( (b, c) \in R \Rightarrow (a, c) \in R \) true? Justify it by one example.

In the set of real numbers, the relation \( R \) defined by \( R = \{(a, b) : a \leq b^2 \} \) is:

If \( A = \{1, 2, 3\}, B = \{a, b\} \), then the number of functions from A to B will be:
A relation \( R \) is defined in \( \mathbb{N} \times \mathbb{N} \) as follows: \[ (a, b) \, R \, (c, d) { if and only if } ad = bc. \] Prove that \( R \) is an equivalence relation.
If the numbers of elements of two finite sets \( A \) and \( B \) are \( m \) and \( n \) respectively, then the total number of relations from \( A \) to \( B \) will be
If \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), then the function from A to B will be
Prove that the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x - 1 \), when \( x>2 \), and \( f(1) = f(2) = 1 \), is onto but not one-to-one.

A relation \( R = \{(x, y) : \text{Number of pages in} \, x \text{ and } y \text{ are equal} \} \) is defined on the set \( A \) of all books in a college library. Prove that \( R \) is an equivalence relation.

Let a relation \( R = \{(a, b) : (a - b) \text{ is a multiple of 5} \} \) be defined on the set \( \mathbb{Z} \) (set of integers). Prove that \( R \) is an equivalence relation.