Relations and Functions

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Class 12 Maths Chapter 1
Relation and Functions
Important Questions

Q: Q1 : What is a Relation in Mathematics?

Ans: A relation in mathematics is a set of ordered pairs, typically defined between two sets. It establishes a relationship between the elements of these sets.

Q: Q2 : How is a Function Different from a Relation?

Ans: A function is a special type of relation where each element in the domain (the first set) is associated with exactly one element in the codomain (the second set). In a relation, an element in the first set can be associated with multiple elements in the second set.

Q: Q3 : What is the Domain of a Function?

Ans: The domain of a function is the complete set of possible input values (usually x-values) for which the function is defined.

Q: Q4 : What is the Range of a Function?

Ans: In a circle, tangents drawn from an external point are equal in length. They are both perpendicular to the radius drawn to the point of contact on the circle.

Q: Q5: What Does it Mean for a Function to be One-to-One (Injective)?

Ans: A function is one-to-one, or injective, if no two different elements in the domain map to the same element in the codomain.

Preparing for Class 12 Maths exams, especially Chapter 1 on Relations and Functions, is crucial for a strong foundation in mathematics. Here are CBSE class 12 maths relations and functions important questions you should be familiar with. Questions are designed to test understanding of the fundamental concepts in this chapter. Interacting with diverse question styles helps students confront uncertainties, ensuring they’re fully prepared for the upcoming exams. Handling these varied questions doesn’t just enhance confidence but also sharpens their problem-solving abilities.

Table of Contents

Introduction

CBSE class 12 maths relations and functions important questions help students in understanding the chapter in more detailed way. Relations and functions chapter encompasses several key topics. Students delve into the basics of relations, understanding the concepts of ordered pairs, Cartesian products, and types of relations like reflexive, symmetric, and transitive. Functions, a vital part of this module, introduce learners to concepts such as domain, codomain, and range. They explore different types of functions like one-to-one, onto, and composite functions. Furthermore, the syllabus typically includes an in-depth study of inverse trigonometric functions, their properties, and graphs. This segment often extends to cover binary operations, including their properties and applications within the realm of relations and functions.

What are Relation and Functions?

“Relation” and “Function” are fundamental concepts in mathematics, particularly in the field of set theory and algebra.
Relation
A relation is a way to describe a relationship between sets. It is defined as a set of ordered pairs. Each ordered pair consists of two elements where the first element is drawn from one set (often called the domain) and the second element from another set (often called the codomain).

Example: Consider two sets, A = {1, 2} and B = {3, 4}. A relation from A to B could be R = {(1, 3), (2, 4)}, where each pair (a, b) indicates that a from A is related to b from B.

Function A function is a special type of relation with a unique characteristic: each element in the domain is associated with exactly one element in the codomain. In simpler terms, a function is a rule that assigns each input exactly one output.

Notation: A function f from a set X to a set Y is written as f: X → Y.
Example: Consider a function f(x) = $x^2$ . Here, for each real number x (input), the function gives a unique output $x^2$ .

Class 12 Relation and Functions Important Questions and Answers

Q1. The relation R in the set of natural numbers N defined as R = { (x, y) : y = x + 5 and x < 4 } is :

Options

(a) reflexive
(b) symmetric
(c) transitive
(d) none of these

Ans. (d) none of these
Explanation:
R = {(1, 6), (2, 7), (3, 8)}
·.· (6, 6) ∉ R
⇒ R is not Reflexive.
For (2, 7) ∈ R
⇒ (2, 7) ∈ R but (7, 2) ∉ R
⇒ R is not symmetric.
Now, since there is no pair in R such that (x, y)and (y, z)∈R, then (x, z)cannot belong to R.
∴ R is transitive.
⇒ R is not equivalence

Q2. For the set A = { 1, 2, 3 }, define a relation R in the set A as follows R = { (1, 1), (2, 2), (3, 3), (1, 3)} Then, the ordered pair to be added to R to make it the smallest equivalence relation is

Options

(a) (1, 3)
(b) (3, 1)
(c) (2, 1)
(d) (1, 2)

Ans. (b) (3, 1)

Explanation:
(1, 1,), (2, 2) and (3, 3)∈ R
⇒ R is Reflexive.
(1, 3) ∈ R
(1, 3) ∈ R but (3, 1) ∉
R ⇒ R is not symmetric.
To make R an equivalence we can add (3, 1).
Clearly R is reflexive and transitive. For R to be symmetric we should add (3, 1) in R.

Q3. Let A = {1, 2, 3} and B = {a, b, c, d }. Which of the following relation from A to B is not a function?

$R_1$ = {(1, a), (2, b), (3, c)}
$R_2$ = {(1, a), (2, b), (3, d )}
$R_3$ = {(1, a), (1, b), (2, c), (3, d )}

Explanation:
$R_3$ is not a function since 1 ∈ A has two images a, b ∈ B. $R_1\text{ and }R_2$ are functions since in these relations, every element of A has a unique image in B.

Q4. If f(x) = $\tan^{-1}$ x, then find the relation in f(x), f(y) and f(x + y).

Explanation:
We have $f(x) = tan^{–1} x\\$ x = tan f(x) ...(1)
From equation (1),
y = tan f(y)
and x + y = tan f(x + y)
Hence, tan f (x) + tan f (y) = tan f (x + y) is the required relation.

Q5. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

Explanation:
Here, f : A → B is defined as {(1, 4), (2, 5), (3, 6)}.
Since the images of distinct elements of A under f are distinct as :
f(1) = 4, f(2) = 5 and f(3) = 6
From above it is evident that $\\ x_1 ≠ x_2 \text{ and }f (x_1) ≠ f(x_2)\\$ .
f : A → B is one-one.

CBSE Class 12 Maths Chapter wise Important Questions

Chapter No.	Chapter Name
Chapter 1	Relations and Functions
Chapter 2	Inverse Trigonometric Functions
Chapter 3	Matrices
Chapter 4	Determinants
Chapter 5	Continuity and Differentiability
Chapter 6	Applications of Derivatives
Chapter 7	Integrals
Chapter 8	Applications of the Integrals
Chapter 9	Differential Equations
Chapter 10	Vectors
Chapter 11	Three - dimensional Geometry
Chapter 12	Linear Programming
Chapter 13	Probability

Conclusion

The concepts of relations and functions are integral to the understanding of mathematical principles and are widely applied in various fields of study. Exploring CBSE class 12 maths relations and functions important questions, becomes essential for a comprehensive grasp of this fundamental aspect of mathematics. oswal.io provide class 12 Relation and Function important questions and answers. Utilising these resources ensures students acquire the knowledge necessary to excel in their studies and examinations, inviting them to delve deeper into the captivating realm of relations and functions.

Frequently Asked Questions

Q1 : What is a Relation in Mathematics?

Ans: A relation in mathematics is a set of ordered pairs, typically defined between two sets. It establishes a relationship between the elements of these sets.

Q2 : How is a Function Different from a Relation?

Ans: A function is a special type of relation where each element in the domain (the first set) is associated with exactly one element in the codomain (the second set). In a relation, an element in the first set can be associated with multiple elements in the second set.

Chapter Wise Important Questions for CBSE Board Class 12 Maths
Relations and Functions
Inverse Trigonometric Functions
Matrices
Determinants
Continuity and Differentiability
Applications of Derivatives
Integrals
Applications of the Integrals
Differential Equations
Vectors
Three - dimensional Geometry
Linear Programming
Probability

Class 12 Maths Chapter 1 Relation and Functions Important Questions

Introduction

What are Relation and Functions?

Class 12 Relation and Functions Important Questions and Answers

Q1. The relation R in the set of natural numbers N defined as R = { (x, y) : y = x + 5 and x < 4 } is :

Options

(a) reflexive (b) symmetric (c) transitive (d) none of these

Q2. For the set A = { 1, 2, 3 }, define a relation R in the set A as follows R = { (1, 1), (2, 2), (3, 3), (1, 3)} Then, the ordered pair to be added to R to make it the smallest equivalence relation is

Options

(a) (1, 3) (b) (3, 1) (c) (2, 1) (d) (1, 2)

Q3. Let A = {1, 2, 3} and B = {a, b, c, d }. Which of the following relation from A to B is not a function?

R_1 = {(1, a), (2, b), (3, c)} R_2 = {(1, a), (2, b), (3, d )} R_3 = {(1, a), (1, b), (2, c), (3, d )}

Q4. If f(x) = \tan^{-1} x, then find the relation in f(x), f(y) and f(x + y).

Q5. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.

CBSE Class 12 Maths Chapter wise Important Questions

Conclusion

Frequently Asked Questions

Q1 : What is a Relation in Mathematics?

Q2 : How is a Function Different from a Relation?

Q3 : What is the Domain of a Function?

Q4 : What is the Range of a Function?

Q5: What Does it Mean for a Function to be One-to-One (Injective)?

Class 12 Maths Chapter 1
Relation and Functions
Important Questions

(a) reflexive
(b) symmetric
(c) transitive
(d) none of these

(a) (1, 3)
(b) (3, 1)
(c) (2, 1)
(d) (1, 2)

$R_1$ = {(1, a), (2, b), (3, c)}
$R_2$ = {(1, a), (2, b), (3, d )}
$R_3$ = {(1, a), (1, b), (2, c), (3, d )}

Q4. If f(x) = $\tan^{-1}$ x, then find the relation in f(x), f(y) and f(x + y).