by Tiwari Academy
NCERT Textbook Solutions for Class 12 Mathematics Chapter 1 Relations and Functions form the foundation for understanding advanced mathematical concepts in set theory and algebraic structures. NCERT Detailed Solutions for Class 12 Maths Book Chapter 1 PDF is a comprehensive resource providing step-by-step derivations, proofs, and detailed solutions to all exercises. NCERT Grade 12th Math book solutions not only include clear explanations of mathematical operations, but also address frequently asked questions like domain, codomain, range and types of relations, such as reflexive, symmetric, transitive and equivalence relations. Whether you’re exploring binary operations, solving composite functions or tackling tricky inverse function problems, this guide ensures every student gains a strong conceptual and analytical understanding.
NCERT Textbook Solutions Class 12 Maths Chapter 1 Relations and Functions
You can download NCERT Class 12 Maths Textbook Chapter 1 solutions free in PDF format to access exercise-wise solutions and illustrative diagrams offline. For efficient revisions, NCERT Class 12 Math Chapter 1 notes encapsulate key formulas, theorems and definitions, summarizing the chapter with mathematical precision. NCERT solutions for Class 12 Maths Chapter 1 Relations and Functions in Hindi and English Medium revised and updated for academic year 2024-25. As per new NCERT textbook there are only three exercises (including miscellaneous) in class 12th Maths chapter 1. Earlier there were five, rest two are deleted from the syllabus now.
Detailed NCERT Textbook Solutions for Class 12 Maths Chapter 1 Exercises
Students often face challenges solving Class 12 Mathematics Chapter 1 exercise solutions, as this chapter introduces key concepts like equivalence relations, composition of functions and inverse relations. With our NCERT Class 12th Maths Chapter 1 answers, you’ll find step-by-step solutions to each question. For example, solving problems involving the composition of two functions or verifying a relation’s reflexivity, symmetry or transitivity becomes simpler with video solutions and illustrations.
Our solutions guide also includes MCQ solutions that help you prepare for competitive exams. Whether you’re looking for revision notes or practice questions, our resource ensures you’re fully equipped for exams. Class 12 NCERT Mathematics Book Chapter 1 PDF download is ideal for students who want accessible learning material anytime. Covering all aspects of the chapter, our solutions ensure a clear understanding of theoretical and applied mathematics.
Class 12 Maths Chapter 1 Solutions in Hindi
Key Features of NCERT Class 12 Mathematics Chapter 1 Solutions
NCERT Class 12 Maths Chapter 1 textbook solutions by experts are crafted to simplify complex topics, making it easier for students to grasp essential concepts. With NCERT Class 12th Mathematics Chapter 1 problem-solving techniques, you can master questions from the basic to advanced level. The guide includes important questions and step-by-step solutions to help you ace board exams. By focusing on problem-solving strategies, this resource allows students to approach topics such as the inverse of functions and equivalence relations confidently.
NCERT Class 12 Math Chapter 1 exercise-wise solutions help build accuracy in solving questions. Class 12 Maths NCERT Book Chapter 1 solution guide is a must-have for anyone aiming to improve mathematical skills and score better in exams. For easy access, the NCERT Class 12 Maths Chapter 1 solutions free download offers flexibility, letting you revise on the go.
| Class: 12 | Mathematics |
| Chapter 1: | Relations and Functions |
| Content: | Textbook Exercises and Extra Questions |
| Content Type: | PDF, Images, Text and Videos |
| Session: | 2024-25 |
| Medium: | English and Hindi Medium |
Class 12 Maths Chapter 1 Solutions in English Medium
Important Points for Class 12 Maths Chapter 1 Relations and Functions
Key topics include types of relations (reflexive, symmetric, transitive and equivalence), domain and range of functions, composition of functions and inverse of functions. Understanding solved examples, exercise-wise solutions and important questions from NCERT Solutions for Class 12 Maths Chapter 1 is crucial for scoring well in exams.
| Day | Topics to Cover | Resources to Use | Practice Activity |
|---|---|---|---|
| Day 1 | Introduction to Relations and Functions | NCERT textbook, Notes | Revise definitions and solve 5 examples |
| Day 2 | Types of Relations (Reflexive, Symmetric, Transitive) | NCERT Solutions PDF, Video tutorials | Solve Exercise 1.1 |
| Day 3 | Equivalence Relations | NCERT examples, Revision notes | Practice important questions |
| Day 4 | Functions and their Composition | Solved examples, Video explanations | Solve Exercise 1.2 |
| Day 5 | Inverse of Functions | NCERT Solutions PDF, Practice guide | Solve Exercise 1.3 |
| Day 6 | Revision and Problem Solving | NCERT Revision notes | Attempt previous year’s questions |
NCERT solutions for Class 12 Maths Chapter 1
Class XII Mathematics all exercises including miscellaneous are in PDF Hindi & English Medium along with NCERT Solutions Apps free download. Download assignments based on Relations and functions and Previous Years Questions asked in CBSE board, important questions for practice as per latest CBSE Curriculum.
12th Maths Chapter 1 Solutions
NCERT solutions for Class 12 Maths Chapter 1 all exercises are given to free download in PDF. NCERT Books as well as Solutions are available in English Hindi Medium. Ask your doubts in Maths, Physics and other subjects, NIOS Board admission query and CBSE Board information through Discussion Forum. NCERT Solutions and Offline apps are based on latest CBSE Syllabus.
Class 12 Maths Chapter 1 Study Material
- Class 12 Maths NCERT Book Chapter 1
- Class 12 Maths Revision Book Chapter 1
- Class 12 Maths Revision Book Answers
- Download Class 12 Maths Assignment 1
- Download Class 12 Maths Assignment 2
- Download Class 12 Maths Assignment 2 Answers
- Download Class 12 Maths Assignment 3
- Download Class 12 Maths Assignment 4
- Class 12 Maths NCERT Solutions
- Class 12 all Subjects NCERT Solutions
Important Questions on 12th Maths Chapter 1
Determine whether each of the following relation are reflexive, symmetric and transitive: Relation R in the set A = {1, 2, 3… 13, 14} defined as R = {(x, y): 3x – y = 0}
A = {1, 2, 3 … 13, 14}
R = {(x, y): 3x − y = 0}
∴ R = {(1, 3), (2, 6), (3, 9), (4, 12)}
R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R.
Also, R is not symmetric as (1, 3) ∈ R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0]
Also, R is not transitive as (1, 3), (3, 9) ∈ R, but (1, 9) ∉ R. [3(1) − 9 ≠ 0] Hence, R is neither reflexive, nor symmetric, nor transitive.
Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b²} is neither reflexive nor symmetric nor transitive.
R = {(a, b): a ≤ b²}
It can be observed that (1/2,1/2)∉R, since, 1/2>(1/2)²
∴ R is not reflexive.
Now, (1, 4) ∈ R as 1 < 42
But, 4 is not less than 12.
∴ (4, 1) ∉ R ∴ R is not symmetric.
Now, (3, 2), (2, 1.5) ∈ R [as 3 < 2² = 4 and 2 < (1.5)² = 2.25]
But, 3 > (1.5)² = 2.25
∴ (3, 1.5) ∉ R
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
Let A = {1, 2, 3, 4, 5, 6}.
A relation R is defined on set A as: R = {(a, b): b = a + 1}
∴ R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}
We can find (a, a) ∉ R, where a ∈ A.
For instance, (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R
∴ R is not reflexive.
It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.
∴ R is not symmetric.
Now, (1, 2), (2, 3) ∈ R but, (1, 3) ∉ R
∴ R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.
Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Let A = {1, 2, 3}. A relation R on A is defined as R = {(1, 2), (2, 1)}.
It is clear that (1, 1), (2, 2), (3, 3) ∉ R,
∴ R is not reflexive.
Now, as (1, 2) ∈ R and (2, 1) ∈ R, then R is symmetric.
Now, (1, 2) and (2, 1) ∈ R, however, (1, 1) ∉ R,
∴ R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.
Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.
Set A is the set of all books in the library of a college.
R = {x, y): x and y have the same number of pages}
Now, R is reflexive since (x, x) ∈ R as x and x has the same number of pages.
Let (x, y) ∈ R
⇒ x and y have the same number of pages.
⇒ y and x have the same number of pages.
⇒ (y, x) ∈ R
∴ R is symmetric.
Now, let (x, y) ∈R and (y, z) ∈ R.
⇒ x and y and have the same number of pages and y and z have the same number of pages.
⇒ x and z have the same number of pages.
⇒ (x, z) ∈ R ∴ R is transitive.
Hence, R is an equivalence relation.
R = {(P, Q): Distance of point P from the origin is the same as the distance of point Q from the origin}
Clearly, (P, P) ∈ R since the distance of point P from the origin is always the same as the distance of the same point P from the origin.
∴ R is reflexive.
Now, Let (P, Q) ∈ R.
⇒ The distance of point P from the origin is the same as the distance of point Q from the origin.
⇒ The distance of point Q from the origin is the same as the distance of point P from the origin.
⇒ (Q, P) ∈ R, ∴ R is symmetric.
Now, Let (P, Q), (Q, S) ∈ R.
⇒ The distance of points P and Q from the origin is the same and also, the distance of points Q and S from the origin is the same.
⇒ The distance of points P and S from the origin is the same.
⇒ (P, S) ∈ R ∴ R is transitive.
Therefore, R is an equivalence relation.
The set of all points related to P ≠ (0, 0) will be those points whose distance from the origin is the same as the distance of point P from the origin.
In other words, if O (0, 0) is the origin and OP = k, then the set of all points related to P is at a distance of k from the origin.
Hence, this set of points forms a circle with the centre as the origin and this circle passes through point P.
R = {(P1, P2): P1 and P2 have same the number of sides}
R is reflexive, Since (P1, P1) ∈ R, as the same polygon has the same number of sides with itself.
Let (P1, P2) ∈ R.
⇒ P1 and P2 have the same number of sides.
⇒ P2 and P1 have the same number of sides.
⇒ (P2, P1) ∈ R,
∴ R is symmetric.
Now, Let (P1, P2), (P2, P3) ∈ R.
⇒ P1 and P2 have the same number of sides.
Also, P2 and P3 have the same number of sides.
⇒ P1 and P3 have the same number of sides.
⇒ (P1, P3) ∈ R ∴ R is transitive.
Hence, R is an equivalence relation.
The elements in A related to the right-angled triangle (T) with sides 3, 4, and 5 are those polygons which have 3 sides (Since T is a polygon with 3 sides).
Hence, the set of all elements in A related to triangle T is the set of all triangles.
R = {(L1, L2): L1 is parallel to L2}
R is reflexive as any line L1 is parallel to itself
i.e., (L1, L1) ∈ R.
Now, let (L1, L2) ∈ R.
⇒ L1 is parallel to L2
⇒ L2 is parallel to L1.
⇒ (L2, L1) ∈ R
∴ R is symmetric.
Now, let (L1, L2), (L2, L3) ∈ R.
⇒ L1 is parallel to L2. Also, L2 is parallel to L3.
⇒ L1 is parallel to L3.
∴ R is transitive.
Hence, R is an equivalence relation.
The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x + 4.
Slope of line y = 2x + 4 is m = 2
It is known that parallel lines have the same slopes.
The line parallel to the given line is of the form y = 2x + c, where c ∈ R. Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R.
Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
The functions f: {1, 3, 4} → {1, 2, 5} and
g: {1, 2, 5} → {1, 3} are defined as
f = {(1, 2), (3, 5), (4, 1)} and
g = {(1, 3), (2, 3), (5, 1)}.
gof(1) = g[f(1)] = g(2) = 3 [as f(1) = 2 and g(2) = 3]
gof(3) = g[f(3)] = g(5) = 1 [as f(3) = 5 and g(5) = 1]
gof(4) = g[f(4)] = g(1) = 3 [as f(4) = 1 and g(1) = 3]
∴ gof = {(1, 3), (3, 1), (4, 3)}
Show that the function f: R → R given by f(x) = x³ is injective.
f: R → R is given as f(x) = x³.
For one – one Suppose f(x) = f(y), where x, y ∈ R.
⇒ x³ = y³ … (1)
Now, we need to show that x = y.
Suppose x ≠ y, their cubes will also not be equal.
⇒ x³ ≠ y³
However, this will be a contradiction to (1).
∴ x = y Hence, f is injective.
Before studying this lesson, you should know:
- Concept of set, types of sets, operations on sets
- Concept of ordered pair and cartesian product of set.
- Domain, co-domain and range of a relation and a function
Relation
Let A and B be two sets. Then a relation R from Set A into Set B is a subset of A × B.
Types of Relations
- Reflexive Relation
- Symmetric Relation
- Transitive Relation
Equivalence Relation
A relation R on a set A is said to be an equivalence relation on A iff
- it is reflexive
- it is symmetric
- it is transitive
CLASSIFICATION OF FUNCTIONS
Let f be a function from A to B. If every element of the set B is the image of at least one element of the set A i.e. if there is no unpaired element in the set B then we say that the function f maps the set A onto the set B. Otherwise we say that the function maps the set A into the set B.
Functions for which each element of the set A is mapped to a different element of the set B are said to be one-to-one.
A function can map more than one element of the set A to the same element of the set B. Such a type of function is said to be many-to-one. A function which is both one-to-one and onto is said to be a bijective function.
What does Class 12 Maths Chapter 1 cover?
Class 12 Maths Chapter 1, Relations and Functions, focuses on foundational concepts that are critical for understanding advanced algebra. The chapter discusses topics like domain, range, types of relations, equivalence relations and inverse functions. With NCERT Textbook Solutions for Class 12 Maths Chapter 1 PDF, students can grasp these topics easily with step-by-step solutions to each exercise. This chapter also helps in understanding the composition of functions and practical applications of relations. You can refer to Class 12 Maths Chapter 1 notes and solved examples for better clarity, ensuring a solid understanding of the subject.
What is the core motive of chapter 1 Relations and Functions of 12th standard Maths?
The core motive of chapter 1 (Relations and Functions) of 12th standard Maths is to teach students the following things:
- Empty relation is the relation R in X given by R = φ ⊂ X × X.
- Universal relation is the relation R in X given by R = X × X.
- Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.
- Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.
- Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.
- Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.
- A function f: X → Y is one-one (or injective) if f(x₁) = f(x₂) ⇒x₁ = x₂ ∀ x₁, x₂∈ X.
- A function f: X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
- A function f: X → Y is one-one and onto (or bijective), if f is both one-one and onto.
- The composition of functions f: A → B and g: B → C is the function
gof: A → C given by gof(x) = g(f(x)) ∀ x ∈ A. - A function f: X → Y is invertible if and only if f is one-one and onto.
- Given a finite set X, a function f: X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set.
- A binary operation ∗ on a set A is a function ∗ from A × A to A.
- An element e ∈ X is the identity element for binary operation ∗: X × X → X, if a ∗ e = a = e ∗ a ∀ a ∈ X.
- An element a ∈ X is invertible for binary operation ∗: X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where, e is the identity for the binary operation ∗. The element b is called inverse of a and is denoted by a^(–1).
- An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀ a, b in X.
- An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) ∀ a, b, c in X.
Where can I find NCERT Solutions for Class 12 Maths Chapter 1?
You can access NCERT Class 12 Mathematics Chapter 1 solutions free from various online educational platforms like Tiwari Academy. These solutions provide exercise-wise solutions for all problems, including MCQ solutions for quick practice. NCERT Class 12 Maths Chapter 1 PDF download option allows offline access to study material. These solutions include diagrams, step-by-step explanations and detailed approaches to solving questions. The video solutions provided in some guides are particularly useful for visual learners. For quick reference and revisions, you can also refer to NCERT Class 12 Maths Chapter 1 notes that summarize the chapter effectively.
What should be revised before starting chapter 1 of 12th standard Maths?
Before starting chapter 1 (Relations and Functions) of 12th standard Maths, students should revise chapter 2 (Relations and Functions) of grade 11th Maths. Chapter 2 of class 11th Maths works as a base for chapter 1 of class 12th Maths.
How do NCERT Solutions for Class 12 Maths Chapter 1 help in exams?
Class 12 Maths NCERT Chapter 1 solutions by experts are tailored to help students excel in their exams by offering detailed and step-by-step solutions for all exercises. These resources simplify complex problems involving inverse functions and relations, ensuring a better understanding of core concepts. With important questions and solved examples, students can practice and improve their problem-solving skills. NCERT Class 12 Maths Chapter 1 PDF download provides convenient access to all exercises for revision. These solutions also include problem-solving techniques for board exams and competitive exams, making them a valuable resource.
Does chapter 1 of grade 12th Maths has any miscellaneous exercise?
Yes, chapter 1 of grade 12th Maths has a miscellaneous exercise. There are five exercises in chapter 1 of class 12th Maths, and the last exercise is the miscellaneous exercise of chapter 1 of grade 12th Maths.
How are exercise-wise solutions structured in NCERT Class 12 Maths Chapter 1?
The exercise-wise solutions in NCERT Class 12 Mathematics textbook Chapter 1 answers are designed to address each question comprehensively. Each exercise starts with an explanation of concepts like equivalence relations and the composition of functions. This is followed by solved examples and practice questions. The Class 12 NCERT Math Chapter 1 exercise solutions are structured to build understanding gradually, starting from basic problems and advancing to more complex ones. The revision notes and video solutions available alongside these solutions help clarify doubts effectively. Students can rely on these resources for a systematic approach to mastering the chapter.
Is NCERT chapter 1 of 12th Standard Maths short or lengthy?
Chapter 1 of 12th standard Maths is lengthy. There are five exercises in this chapter.
In the first exercise, there are 6 examples (examples 1 to 6) and 16 questions.
The second exercise contains eight examples (examples 7 to 14) and 12 questions.
The last (Miscellaneous) exercise has 11 examples (examples 41 to 51) and 19 questions.
So, there are good examples and just a little questions in chapter 1 of class 12th Maths.
Are Class 12 Maths Chapter 1 resources available for free?
Yes, you can access NCERT Class 12 Mathematics Book Chapter 1 solutions free on several educational websites. These resources include step-by-step solutions, MCQ solutions and Class 12 Maths Chapter 1 PDF download options for offline study. Some platforms also offer video solutions to explain concepts visually, which is helpful for quick understanding. In addition, free revision notes summarize key topics, making them ideal for last-minute preparation. These resources are created by experts to provide exercise-wise solutions and ensure students have all the tools they need to master Relations and Functions in Class 12 Maths.
Are there any theorems in chapter 1 of class 12th Maths?
Yes, there are two theorems in chapter 1 of class 12th Maths. Both the theorems are nice and easy. Proofs of these theorems are short and easy. Students can easily understand the proofs of these theorems.
