Are these practice questions for Relations and Functions based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Relations and Functions free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Relations and Functions mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Relations and Functions for full coverage.

How many practice questions are there for Relations and Functions?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Relations and Functions Class 12 Mathematics Practice Questions with Answers

easy

Consider the set

S = \{1, 2, 3, 4\}

. Create a relation R on S that is reflexive and symmetric but fails to be transitive. Justify your construction.

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easy

Given two functions

f(x) = x^2

and

g(x) = x + 3

, calculate the value of

(gof)(2)

.

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easy

Justify whether the function

f: \mathbf{Z} \to \mathbf{Z}

defined by

f(x) = |x| + x

is injective.

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easy

Analyze if the function

f: \mathbf{Z} \rightarrow \mathbf{Z}

defined by

f(x) = |x|

is one-one.

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easy

Identify if the relation R = {(1,1), (2,2), (3,3)} on set A = {1, 2, 3} is reflexive.

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easy

Explain the difference between a relation and a function with a simple example.

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easy

What is a bijective function?

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easy

Define a universal relation.

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easy

Let the set be

A = \{a, b, c\}

. Analyze if the relation

R = \{(a, a), (b, b), (a, c)\}

defined on set A is reflexive. Justify your answer.

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easy

Let

S

be the set of all straight lines in the Cartesian plane. A relation

R

is defined on

S

as

L_1 R L_2

if

L_1

is parallel to

L_2

. Justify that

R

is an equivalence relation. (Note: A line is considered parallel to itself).

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easy

Describe the conditions for a relation R on a set A to be an equivalence relation.

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medium

Let R be a relation on the set of integers

\mathbf{Z}

defined by

aRb

if

a^2 - b^2

is divisible by 3. Prove that R is an equivalence relation.

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medium

Explain why the function f: N → N given by f(x) = x + 1 is one-one but not onto.

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medium

Create a relation R on the set A = {1, 2, 3} that is symmetric and transitive, but not reflexive. Justify your answer.

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medium

Let

f: A \to B

and

g: B \to C

be two surjective functions. Formulate a proof to show that the composition

g \circ f: A \to C

is also surjective.

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medium

If

f: \mathbf{R} \rightarrow \mathbf{R}

and

g: \mathbf{R} \rightarrow \mathbf{R}

are given by

f(x) = |x|

and

g(x) = 5x - 2

, calculate

fog(x)

and

gof(x)

. Demonstrate that

fog \neq gof

.

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medium

Let R be a relation on the set of integers

\mathbf{Z}

defined by

R = \{(a, b) : a + b

is an even integer

\}.

Analyze if R is symmetric.

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medium

Analyze if the function

f: \mathbf{N} \rightarrow \mathbf{N}

defined by

f(x) = x + 1

is onto.

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medium

Let

A = \mathbf{R} - \{2\}

and

B = \mathbf{R} - \{1\}

. The function

f: A \to B

is defined by

f(x) = \frac{x-1}{x-2}

. Formulate the inverse function

g: B \to A

and justify that it is indeed the inverse by showing

g \circ f = I_A

and

f \circ g = I_B

.

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medium

Let A = {1, 2, 3}. List the elements of the relation R = {(a, b) : |a - b| = 1}. Also, state its domain and range.

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medium

Demonstrate that the relation R in the set

\{1, 2, 3\}

given by

R = \{(1, 2), (2, 1)\}

is symmetric but neither reflexive nor transitive.

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medium

Explain the concepts of one-one (injective) and onto (surjective) functions. Give an example of a function that is one-one but not onto, and another example that is onto but not one-one, using finite sets.

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medium

Explain what is meant by the domain of a relation R from a set A to a set B.

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medium

Given a function f: {1, 2} → {a, b} defined by f(1) = a, f(2) = b, and a function g: {a, b} → {x, y} defined by g(a) = x, g(b) = y. Describe the composition function gof.

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medium

Summarize the properties of a symmetric relation. Given an example of a relation that is symmetric but not reflexive.

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medium

Define reflexive, symmetric, and transitive relations. Provide one example for each type of relation on the set A = {a, b, c}.

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medium

Let L be the set of all lines in the XY plane. A relation R is defined on L as

R = \{(L_1, L_2) : L_1

is parallel to

L_2\}

. Analyze if R is an equivalence relation.

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medium

Formulate a concise argument to prove that a function

f: A \to B

from a finite set A to a smaller finite set B (where

|A| > |B|

) cannot be injective.

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medium

Define an empty relation in a set A.

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medium

Create a function

f: \mathbf{N} \to \mathbf{N}

that is surjective but not injective, and justify both properties.

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medium

Formulate a proof to show that if

R_1

and

R_2

are two equivalence relations on a set

A

, then their intersection

R_1 \cap R_2

is also an equivalence relation on

A

.

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medium

Analyze the injectivity and surjectivity of the function

f: \mathbf{R} \rightarrow \mathbf{R}

given by

f(x) = 3x + 5

.

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medium

Let

A = \{1, 2, 3\}

. A relation

R

is defined on

A

as

R = \{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3)\}

. Examine if

R

is reflexive, symmetric, or transitive.

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medium

Let

f: \mathbf{R} \to \mathbf{R}

be defined by

f(x) = \sin(x)

and

g: \mathbf{R} \to \mathbf{R}

be defined by

g(x) = x^2

. Formulate the composite functions

g \circ f

and

f \circ g

. Evaluate whether each composite function is injective or surjective, justifying your conclusions.

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hard

Let

A = \mathbf{R} - \{3\}

and

B = \mathbf{R} - \{1\}

. Analyze if the function

f: A \rightarrow B

defined by

f(x) = \frac{x - 2}{x - 3}

is a bijective function.

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hard

Consider the function

f: \mathbf{R}^+ \rightarrow [4, \infty)

given by

f(x) = x^2 + 4

, where

\mathbf{R}^+

is the set of all non-negative real numbers. Demonstrate that

f

is invertible and calculate its inverse.

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hard

Describe what an equivalence class is. Consider the relation R on the set of integers Z defined by R = {(a, b) : a ≡ b (mod 3)}. Find the equivalence classes [0], [1], and [2].

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hard

Identify the type of relation R in the set Z of integers defined as R = {(x, y): x - y is an even integer}. Explain your choice.

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hard

Let

f: \mathbf{N} \rightarrow \mathbf{N}

be defined by

f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \ \frac{n+1}{2}, & \text{if } n \text{ is odd} \end{cases}

. Analyze whether the function

f

is bijective.

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hard

Design a function

f: \mathbf{N} \to \mathbf{Z}

(from the set of natural numbers to the set of integers) that is bijective. Formulate a proof to justify that your designed function is both one-one and onto.

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hard

Let

\mathbf{N}

be the set of natural numbers. A relation

R

is defined on the set

\mathbf{N} \times \mathbf{N}

by

(a, b) R (c, d)

if and only if

ad = bc

. Prove that

R

is an equivalence relation. Furthermore, describe the equivalence class of

(2, 3)

.

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hard

Let

f: \mathbf{N} \rightarrow S

be a function defined as

f(x) = 4x^2 + 12x + 15

, where S is the range of

f

. Demonstrate that

f

is invertible and calculate the inverse of

f

.

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hard

Demonstrate that the relation R on the set

\mathbf{N} \times \mathbf{N}

defined by

(a, b) R (c, d)

if and only if

a + d = b + c

is an equivalence relation.

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hard

Evaluate if the function

f: \mathbf{R} \to \mathbf{R}

defined by

f(x) = x|x|

is bijective. Justify your conclusion by checking for injectivity and surjectivity.

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hard

Evaluate the statement: "If the composition

g \circ f

is one-one, then the function

g

must be one-one." Justify your conclusion with a counterexample.

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Relations and FunctionsClass 12 Mathematics Practice Questions