Class 12 Maths Exercise 1.1 Solutions

Class 12 Maths Exercise 1.2 Solutions

12th Maths Chapter 1 Miscellaneous Exercise

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.

### Class 12 Maths Chapter 1 Solutions in Hindi

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

### 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

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.

### 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.

### 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.

### 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.

### 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.

### 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 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.