NCERT solutions for class 11 exercise 2.1 of chapter 2-Relations and Functions

NCERT solutions for class 11 exercise 2.1 of chapter 2-Relations and Functions are created here for the students of 11 class in boosting their preparation of the exams, class test, and worksheet. These NCERT solutions are the solutions of all unsolved questions of exercise 2.1 of chapter 2-Relations and Functions of NCERT maths text book for class 11. The NCERT solutions of exercise 2.1 of chapter 2-Relations and Functions are based on the cartesian product of the sets.

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NCERT solutions for class 11 exercise 2.1 of chapter 2-Relations and Functions

PDF of NCERT solutions for class 11 exercise 2.1 of chapter 2-Relations and Functions

Exercise 2.2

Exercise 2.3

NCERT Solutions for class 11 maths

Chapter 1-Sets	Chapter 9-Sequences and Series
Chapter 2- Relations and functions	Chapter 10- Straight Lines
Chapter 3- Trigonometry	Chapter 11-Conic Sections
Chapter 4-Principle of mathematical induction	Chapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbers	Chapter 13- Limits and Derivatives
Chapter 6- Linear Inequalities	Chapter 14-Mathematical Reasoning
Chapter 7- Permutations and Combinations	Chapter 15- Statistics
Chapter 8- Binomial Theorem	Chapter 16- Probability

CBSE Class 11-Question paper of maths 2015

CBSE Class 11-Question paper of maths 2015

CBSE Class 11 – Second unit test of maths 2021 with solutions

Study notes of Maths and Science NCERT and CBSE from class 9 to 12

NCERT solutions for class 11 exercise 2.1 of chapter 2-Relations and Functions

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Q1. If (x/3  + 1,  y – 2/3) = (5/3,  1/3), find the value of x and y.

Ans. We are given the two ordered pair which are equal to each other

(x/3  + 1,  y – 2/3) = (5/3,  1/3)

The corresponding elements of  LHS and RHS should be equal to each other

x/3  + 1 = 5/3

x/3 = 5/3 – 1

x/3 = (5-3)/3

x = 2

y – 2/3 = 1/3

y = 1/3 + 2/3

y = 1

Therefore the value of x = 2 and y = 1

Q2. If set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B)?

Ans. The number of elements in set A = 3

We are given the set B = {3, 4, 5}

The number of elements in set (A × B) = (The number of elements in set A) × (The number of elements in the set B)= 3 × 3 = 9

Therefore the number of elements in (A × B) are = 9

Q3. If G = {7,8} and H = {5,4,2} , find  G × H and  H × G.

Ans. We are given G = {7,8} and H = {5,4,2}

It is known to us that the cartesian product of the two non-empty sets A and B is defined as follows.

A × B = { (a,b) : a ∈ A, b ∈ B}

Therefore,we have

G × H = {7,8} × {5,4,2} = { (7,5), (7,4),(7,2),(8,5),(8,4),(8,2)}

H × G =  {5,4,2}×  {7,8} = { (5,7), (5,8),(4,7),(4,8),(2,7),(2,8)}

Q4. State whether each of the following statement are true or false. If the statement is false, rewrite the given statement correctly.

(i) If P = {m, n} and Q = {n, m}, then P × Q = {(m, n), (n, m)}

(ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.

(iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ Φ) = Φ 

Ans. (i) If P = {m, n} and Q = {n, m}, then

P × Q = {(m, m), (m, n), (n, m), (n, n)}

Therefore the given ststement is false

(ii) The given ststement is true

(iii) The given ststement is true

Q5. If  A = { -1, 1} , find A × A × A.

Ans. It is known to us that the cartesian product of the three non-empty sets A , B  and C is defined as follows.

A × B × C = {(a,b,c), a ∈ A, b ∈ B, c ∈ C}

Here B = A, C = A

Since A = {-1,1}

Therefore

A × A × A =(-1,1) × (-1,1)× (-1,1) = {(-1,-1),(-1,1),(1,-1),(1,1)}×(-1,1)={(-1,-1,-1),(-1,-1,1),(-1,1,-1),(-1,1,1),(1,-1,-1),(1,-1,1),(1,1,-1),(1,1,1)}

Q6. If  A × B = {(a,x),(a,y),(b,x),(b,y)}. Find A and B.

Ans. We are given that

A × B = {(a,x),(a,y),(b,x),(b,y)}

It is known to us that the cartesian product of the two non-empty sets A and B is defined as follows.

A × B = { (a,b) : a ∈ A, b ∈ B}

The first element of A× B shows the element of A and the second element shows the element of B.

Therefore

A = {a,b} and B = {x,y}

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Q7. Let A = {1,2}, B = {1,2,3,4}, C = {5,6} and D ={5,6,7,8}. Verify that.

(i) A × (B ∩C) = (A × B) ∩ (A × C)

(ii) A × C is a subset of  B × D

Ans.

(i) We have to verify that

A × (B ∩C) = (A × B) ∩ (A × C)

Taking LHS

A × (B∩C), in which we are given A = {1,2}, B = {1,2,3,4}, C = {5,6}

Here, (B∩C)= {1,2,3,4}∩ {5,6} = Φ,where Φ is empty set

Now, A × (B∩C) = {1,2} ×Φ = Φ

LHS = Φ

RHS is

(A × B) ∩ (A × C)

A × B = {1,2} × {1,2,3,4} = {(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)}

A × C = {1,2}×{5,6} = {(1,5),(1,6),(2,5),(2,6)}

(A × B) ∩ (A × C)

{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)}∩{(1,5),(1,6),(2,5),(2,6)}

=Φ

RHS is also Φ

Hence A × (B ∩C) = (A × B) ∩ (A × C) , verified

(ii) A × C is a subset of  B × D

Computing A × C as following

A × C = {1,2} × {5,6}= {(1,5),(1,6),(2,5),(2,6)}

Computing B × D as following

B × D ={1,2,3,4}× {5,6,7,8}= {(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8)(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8)}

Observing A × C and B × D ,we see that all the elements of A × C are the elements of  B × D

Hence A × C ⊂ B × D, verified

Q8. Let A = {1,2} and B = {3,4},write A × B.How many subsets will A × B have ? List them.

Ans. A = {1,2} and B ={3,4}

A × B = {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)}

The number of elements in A × B = 4

n(A × B) = 4

Therefore number of subsets of A × B = 2ⁿ = 2⁴= 2× 2× 2×2= 16

16 subsets of A× B = {(1,3),(1,4),(2,3),(2,4)} are computed as following

Φ(empty set)  compulsurily is subset of every set

Taking single element

{(1,3)},{(1,4)},{(2,3)},{(2,4)}

Taking two elements

{(1,3),(1,4)},{(1,3),(2,3)},{(1,3),(2,4)},{(1,4),(2,3)},{(1,4),(2,4)},{(2,3),(2,4)

Taking three elements

{(1,3),(1,4),(2,3)},{(1,3),(2,3),(2,4)},{(1,3),(1,4),(2,4)},{(1,4),{(2,3),(2,4)}

Taking four elements

{(1,3),(1,4),(2,3),(2,4)}

Therefore,the total number of subsets of A × B are 16 listed above

Q9. Let A and B be two sets such that n(A) = 3 and n(B) = 2. If  (x,1), (y,2),(z,1)  are in A × B ,find A and B,where x,y and z are distinct elements.

Ans.We are given that (x,1), (y,2),(z,1)  are in A × B

First elements of the ordered pair of A×B shows the element of A and second elements shows the elements of B.

Therefore x,y,z are the elements of A and 1,2,1 are the elements of B

Since n(A) = 3 and n(B) = 2

Hence the set A = {x,y,z} and set B ={1,2}

Q10. The cartesian product A × A  has 9 elements among which are found (-1,0) and (0,1). Find the set A and the remaining elements of A × A.

Ans. Let the number of elements in set A = n(A)

We are given the number of elements in A × A = 9

The number of elements in the cartesian product of two sets A and B is given as following. If n(A)  is number of elements in set A ,  n(B)  is the number of elements in set B and n(A × B) is the number of elements in (A × B)

n(A) × n(B) = n(A × B)

n(A) × n(A) = n(A × A)

n(A) × n(A) = 9

n(A) = √9 = 3

Two of the ordered pair of A×A among 9  given to us (-1,0) and (0,1)

The first element of the ordered pair shows the element of A and second element also show the elements of A

Therefore set A = {-1,0,1}

A × A = {-1,0,1} × {-1,0,1}

= {(-1,-1),(-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)}

Therefore remaining elements of A × A are

{(-1,-1),(-1,1),(0,-1),(0,0),(1,-1),(1,0),(1,1)}

NCERT Solutions of Science and Maths for Class 9,10,11 and 12

NCERT Solutions for class 9 maths

Chapter 1- Number System	Chapter 9-Areas of parallelogram and triangles
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3- Coordinate Geometry	Chapter 11-Construction
Chapter 4- Linear equations in two variables	Chapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s Geometry	Chapter 13-Surface Areas and Volumes
Chapter 6-Lines and Angles	Chapter 14-Statistics
Chapter 7-Triangles	Chapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science 

Chapter 1-Matter in our surroundings	Chapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?	Chapter 10- Gravitation
Chapter3- Atoms and Molecules	Chapter 11- Work and Energy
Chapter 4-Structure of the Atom	Chapter 12- Sound
Chapter 5-Fundamental unit of life	Chapter 13-Why do we fall ill ?
Chapter 6- Tissues	Chapter 14- Natural Resources
Chapter 7- Diversity in living organism	Chapter 15-Improvement in food resources
Chapter 8- Motion	Last years question papers & sample papers

NCERT Solutions for class 10 maths

Chapter 1-Real number	Chapter 9-Some application of Trigonometry
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3-Linear equations	Chapter 11- Construction
Chapter 4- Quadratic equations	Chapter 12-Area related to circle
Chapter 5-Arithmetic Progression	Chapter 13-Surface areas and Volume
Chapter 6-Triangle	Chapter 14-Statistics
Chapter 7- Co-ordinate geometry	Chapter 15-Probability
Chapter 8-Trigonometry

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NCERT Solutions for Class 10 Science

Chapter 1- Chemical reactions and equations	Chapter 9- Heredity and Evolution
Chapter 2- Acid, Base and Salt	Chapter 10- Light reflection and refraction
Chapter 3- Metals and Non-Metals	Chapter 11- Human eye and colorful world
Chapter 4- Carbon and its Compounds	Chapter 12- Electricity
Chapter 5-Periodic classification of elements	Chapter 13-Magnetic effect of electric current
Chapter 6- Life Process	Chapter 14-Sources of Energy
Chapter 7-Control and Coordination	Chapter 15-Environment
Chapter 8- How do organisms reproduce?	Chapter 16-Management of Natural Resources

NCERT Solutions for class 11 maths

Chapter 1-Sets	Chapter 9-Sequences and Series
Chapter 2- Relations and functions	Chapter 10- Straight Lines
Chapter 3- Trigonometry	Chapter 11-Conic Sections
Chapter 4-Principle of mathematical induction	Chapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbers	Chapter 13- Limits and Derivatives
Chapter 6- Linear Inequalities	Chapter 14-Mathematical Reasoning
Chapter 7- Permutations and Combinations	Chapter 15- Statistics
Chapter 8- Binomial Theorem	Chapter 16- Probability

CBSE Class 11-Question paper of maths 2015

CBSE Class 11 – Second unit test of maths 2021 with solutions

NCERT solutions for class 12 maths

Chapter 1-Relations and Functions	Chapter 9-Differential Equations
Chapter 2-Inverse Trigonometric Functions	Chapter 10-Vector Algebra
Chapter 3-Matrices	Chapter 11 – Three Dimensional Geometry
Chapter 4-Determinants	Chapter 12-Linear Programming
Chapter 5- Continuity and Differentiability	Chapter 13-Probability
Chapter 6- Application of Derivation	CBSE Class 12- Question paper of maths 2021 with solutions
Chapter 7- Integrals
Chapter 8-Application of Integrals

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