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# NCERT solutions for Class 11 Mathematics chapter 2 - Relations and Functions

## Chapter 2: Relations and Functions

Ex. 2.10Ex. 2.20Ex. 2.30Ex. 2.40

#### Chapter 2: Relations and Functions Exercise 2.10 solutions [Pages 33 - 34]

Ex. 2.10 | Q 1 | Page 33

If (x/3+1, y-2/3)=(5/3,1/3),find the values of x and y.

Ex. 2.10 | Q 2 | Page 33

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

Ex. 2.10 | Q 3 | Page 33

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

Ex. 2.10 | Q 4.1 | Page 33

State whether of  the statement is true or false. If the statement is false, re-write the given statement correctly:

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

• True

• False

Ex. 2.10 | Q 4.2 | Page 33

State whether of  the statement is true or false. If the statement is false, re-write the given statement correctly:

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

• True

• False

Ex. 2.10 | Q 4.3 | Page 33

State whether of  the statement is true or false. If the statement is false, re-write the given statement correctly:

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

• True

• False

Ex. 2.10 | Q 5 | Page 33

If A = {–1, 1}, find A × A × A.

Ex. 2.10 | Q 6 | Page 33

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

Ex. 2.10 | Q 7.1 | Page 33

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

Ex. 2.10 | Q 7.2 | Page 33

Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that   A × C is a subset of B × D

Ex. 2.10 | Q 8 | Page 33

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

Ex. 2.10 | Q 9 | Page 33

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.

Ex. 2.10 | Q 10 | Page 34

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.

#### Chapter 2: Relations and Functions Exercise 2.20 solutions [Pages 35 - 36]

Ex. 2.20 | Q 1 | Page 35

Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

Ex. 2.20 | Q 2 | Page 36

Define a relation R on the set N of natural numbers by R = {(xy): y = x + 5, x is a natural number less than 4; xy ∈ N}. Depict this relationship using roster form. Write down the domain and the range.

Ex. 2.20 | Q 3 | Page 36

A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(xy): the difference between x and y is odd; x ∈ A, ∈ B}. Write R in roster form.

Ex. 2.20 | Q 4 | Page 36

The given figure shows a relationship between the sets P and Q. write this relation

(i) in set-builder form (ii) in roster form.

What is its domain and range?

Ex. 2.20 | Q 5 | Page 36

Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(ab): ab ∈ A, b is exactly divisible by a}.

(i) Write R in roster form

(ii) Find the domain of R

(iii) Find the range of R.

Ex. 2.20 | Q 6 | Page 36

Determine the domain and range of the relation R defined by R = {(xx + 5): x ∈ {0, 1, 2, 3, 4, 5}}.

Ex. 2.20 | Q 7 | Page 36

Write the relation R = {(xx3): is a prime number less than 10} in roster form.

Ex. 2.20 | Q 8 | Page 36

Let A = {xy, z} and B = {1, 2}. Find the number of relations from A to B.

Ex. 2.20 | Q 9 | Page 36

Let R be the relation on Z defined by R = {(ab): ab ∈ Z– b is an integer}. Find the domain and range of R.

#### Chapter 2: Relations and Functions Exercise 2.30 solutions [Page 44]

Ex. 2.30 | Q 1 | Page 44

Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.

(i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

(ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)}

(iii) {(1, 3), (1, 5), (2, 5)}

Ex. 2.30 | Q 2.1 | Page 44

Find the domain and range of the given real function:

f(x) = –|x|

Ex. 2.30 | Q 2.2 | Page 44

Find the domain and range of the following real function:

f(x) = sqrt(9 - x^2)

Ex. 2.30 | Q 3 | Page 44

A function f is defined by f(x) = 2x – 5. Write down the values of

(i) f(0), (ii) f(7), (iii) f(–3)

Ex. 2.30 | Q 4 | Page 44

The function ‘t’ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = "9C"/5 + 32

Find

(i) t(0)

(ii) t(28)

(iii) t(–10)

(iv) The value of C, when t(C) = 212.

Ex. 2.30 | Q 5.1 | Page 44

Find the range of each of the following functions.

f(x) = 2 – 3xx ∈ Rx > 0.

Ex. 2.30 | Q 5.2 | Page 44

Find the range of each of the following functions.

f(x) = xx is a real number

Ex. 2.30 | Q 5.2 | Page 44

Find the range of each of the following functions.

f(x) = x2 + 2, x, is a real number.

#### Chapter 2: Relations and Functions Exercise 2.40 solutions [Pages 46 - 47]

Ex. 2.40 | Q 1 | Page 46

The relation f is defined by f(x) = {(x^2,0<=x<=3),(3x,3<=x<=10):}

The relation g is defined by  g(x) = {(x^2, 0 <= x <= 2),(3x,2<= x <= 10):}

Show that f is a function and g is not a function.

Ex. 2.40 | Q 2 | Page 46

If f(x) = x2, find (f(1.1) - f(1))/((1.1 - 1))

Ex. 2.40 | Q 3 | Page 46

Find the domain of the function  f(x) = (x^2 + 2x + 1)/(x^2 - 8x + 12)

Ex. 2.40 | Q 4 | Page 46

Find the domain and the range of the real function f defined by f(x)=sqrt((x-1))

Ex. 2.40 | Q 5 | Page 46

Find the domain and the range of the real function f defined by f (x) = |x – 1|.

Ex. 2.40 | Q 6 | Page 46

Let f = {(x, x^2/(1+x^2)):x ∈ R} be a function from R into R. Determine the range of f.

Ex. 2.40 | Q 7 | Page 46

Let fgR → R be defined, respectively by f(x) = + 1, g(x) = 2x – 3. Find f + gf – g and f/g

Ex. 2.40 | Q 8 | Page 46

Let = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers ab. Determine ab.

Ex. 2.40 | Q 9 | Page 46

Let R be a relation from N to N defined by R = {(ab): ab ∈ N and a = b2}. Are the following true?

(i) (aa) ∈ R, for all a ∈ N

(ii) (ab) ∈ R, implies (ba) ∈ R

(iii) (ab) ∈ R, (bc) ∈ R implies (ac) ∈ R.

Ex. 2.40 | Q 10 | Page 46

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) f is a relation from A to B (ii) f is a function from A to B.

Ex. 2.40 | Q 11 | Page 47

Let f be the subset of Z × Z defined by = {(aba + b): ab ∈ Z}. Is f a function from Z to Z: justify your answer.

Ex. 2.40 | Q 12 | Page 47

Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.

## Chapter 2: Relations and Functions

Ex. 2.10Ex. 2.20Ex. 2.30Ex. 2.40

## NCERT solutions for Class 11 Mathematics chapter 2 - Relations and Functions

NCERT solutions for Class 11 Maths chapter 2 (Relations and Functions) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Textbook for Class 11 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics chapter 2 Relations and Functions are Cartesian Product of Sets, Brief Review of Cartesian System of Rectanglar Co-ordinates, Relation, Functions, Some Functions and Their Graphs, Algebra of Real Functions, Ordered Pairs, Equality of Ordered Pairs, Pictorial Diagrams, Graph of Function, Pictorial Representation of a Function, Exponential Function, Logarithmic Functions.

Using NCERT Class 11 solutions Relations and Functions exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer NCERT Textbook Solutions to score more in exam.

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