# NCERT solutions for Mathematics Exemplar Class 11 chapter 2 - Relations and Functions [Latest edition]

#### Chapters ## Chapter 2: Relations and Functions

Solved ExamplesExercise
Solved Examples [Pages 22 - 27]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 2 Relations and Functions Solved Examples [Pages 22 - 27]

Solved Examples | Q 1.(i) | Page 22

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine A × B

Solved Examples | Q 1.(ii) | Page 22

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine B × A

Solved Examples | Q 1.(iii) | Page 22

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine is A × B = B × A?

Solved Examples | Q 1.(iv) | Page 22

Let A = {1, 2, 3, 4} and B = {5, 7, 9}. Determine is n (A × B) = n (B × A)?

Solved Examples | Q 2.(i) | Page 23

Find x and y if: (4x + 3, y) = (3x + 5, – 2)

Solved Examples | Q 2.(ii) | Page 23

Find x and y if: (x – y, x + y) = (6, 10)

Solved Examples | Q 3 | Page 23

If A = {2, 4, 6, 9} and B = {4, 6, 18, 27, 54}, a ∈ A, b ∈ B, find the set of ordered pairs such that 'a' is factor of 'b' and a < b.

Solved Examples | Q 4 | Page 23

Find the domain and range of the relation R given by R = {(x, y) : y = x + 6/x; where x, y ∈ N and x < 6}.

Solved Examples | Q 5.(i) | Page 24

R1 = {(2, 3), (1/2, 0), (2, 7), (-4, 6)}

Solved Examples | Q 5.(ii) | Page 24

R2 = {(x, |x |) | x is a real number}

Solved Examples | Q 6 | Page 24

Find the domain for which the functions f(x) = 2x2 – 1 and g(x) = 1 – 3x are equal.

Solved Examples | Q 7.(i) | Page 24

Find the domain of the following function.

f(x) = x/(x^2 + 3x + 2)

Solved Examples | Q 7.(ii) | Page 24

Find the domain of the following function.

f(x) = [x] + x

Solved Examples | Q 8.(i) | Page 25

Find the range of the following functions given by |x - 4|/(x - 4)

Solved Examples | Q 8.(ii) | Page 25

Find the range of the following functions given by sqrt(16 - x^2)

Solved Examples | Q 9 | Page 25

Redefine the function which is given by f(x) = |x - 1| + |1 + x|, -2 ≤ x ≤ 2

Solved Examples | Q 10 | Page 26

Find the domain of the function f given by f(x) = 1/sqrt([x]^2 - [x] - 6)

#### Objective Type Questions any 4 given possible

Solved Examples | Q 11 | Page 26

The domain of the function f defined by f(x) = 1/sqrt(x - |x|) is ______.

• R

• R

• R–

• None of these

Solved Examples | Q 12 | Page 27

If f(x) = x^3 - 1/x^3, then f(x) + f(1/x) is equal to ______.

• 2x3

• 2 1/x^3

• 0

• 1

Solved Examples | Q 13 | Page 27

Let A and B be any two sets such that n(B) = p, n(A) = q then the total number of functions f : A → B is equal to ______.

Solved Examples | Q 14 | Page 27

Let f and g be two functions given by f = {(2, 4), (5, 6), (8, – 1), (10, – 3)} g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, – 5)} then. Domain of f + g is ______.

Exercise [Pages 27 - 33]

### NCERT solutions for Mathematics Exemplar Class 11 Chapter 2 Relations and Functions Exercise [Pages 27 - 33]

Exercise | Q 1.(i) | Page 27

Let A = {–1, 2, 3} and B = {1, 3}. Determine A × B

Exercise | Q 1.(ii) | Page 27

Let A = {–1, 2, 3} and B = {1, 3}. Determine B × A

Exercise | Q 1.(iii) | Page 27

Let A = {–1, 2, 3} and B = {1, 3}. Determine B × B

Exercise | Q 1.(iv) | Page 27

Let A = {–1, 2, 3} and B = {1, 3}. Determine A × A

Exercise | Q 2 | Page 28

If P = {x : x < 3, x ∈ N}, Q = {x : x ≤ 2, x ∈ W}. Find (P ∪ Q) × (P ∩ Q), where W is the set of whole numbers.

Exercise | Q 3.(i) | Page 28

A = {x : x ∈ W, x < 2} B = {x : x ∈ N, 1 < x < 5} C = {3, 5} find A × (B ∩ C)

Exercise | Q 3.(ii) | Page 28

If A = {x : x ∈ W, x < 2} B = {x : x ∈ N, 1 < x < 5} C = {3, 5} find A × (B ∪ C)

Exercise | Q 4.(i) | Page 28

In the following find a and b

(2a + b, a – b) = (8, 3)

Exercise | Q 4.(ii) | Page 28

In the following find a and b

(a/4, a - 2b) = (0, 6 + b)

Exercise | Q 5.(i) | Page 28

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y = 5

Exercise | Q 5.(ii) | Page 28

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y < 5

Exercise | Q 5.(iii) | Page 28

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y > 8

Exercise | Q 6 | Page 28

Given R = {(x, y) : x, y ∈ W, x2 + y2 = 25}. Find the domain and Range of R.

Exercise | Q 7 | Page 28

If R1 = {(x, y) | y = 2x + 7, where x ∈ R and – 5 ≤ x ≤ 5} is a relation. Then find the domain and Range of R1.

Exercise | Q 8 | Page 28

If R2 = {(x, y) | x and y are integers and x2 + y2 = 64} is a relation. Then find R2.

Exercise | Q 9 | Page 28

If R3 = {(x, x) | x is a real number} is a relation. Then find domain and range of R3.

Exercise | Q 10.(i) | Page 28

h = {(4, 6), (3, 9), (– 11, 6), (3, 11)}

Exercise | Q 10.(ii) | Page 28

f = {(x, x) | x is a real number}

Exercise | Q 10.(iii) | Page 28

g = "n", 1/"n" |"n" is a positive integer

Exercise | Q 10.(iv) | Page 28

s = {(n, n2) | n is a positive integer}

Exercise | Q 10.(v) | Page 28

t = {(x, 3) | x is a real number

Exercise | Q 11.(a) | Page 28

If f and g are real functions defined by f(x) = x2 + 7 and g(x) = 3x + 5, find the following:

f(3) + g(– 5)

Exercise | Q 11.(b) | Page 28

If f and g are real functions defined by f(x) = x2 + 7 and g(x) = 3x + 5, find the following:

f(1/2) xx g(14)

Exercise | Q 11.(c) | Page 28

If f and g are real functions defined by f(x) = x2 + 7 and g(x) = 3x + 5, find the following:

f(– 2) + g(– 1)

Exercise | Q 11.(d) | Page 28

If f and g are real functions defined by f(x) = x2 + 7 and g(x) = 3x + 5, find the following:

f(t) – f(– 2)

Exercise | Q 11.(e) | Page 28

If f and g are real functions defined by f(x) = x2 + 7 and g(x) = 3x + 5, find the following:

(f(t) - f(5))/(t - 5), if t ≠ 5

Exercise | Q 12.(a) | Page 29

Let f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x – 7. For what real numbers x, f(x) = g(x)?

Exercise | Q 12.(b) | Page 29

Let f and g be real functions defined by f(x) = 2x + 1 and g(x) = 4x – 7. For what real numbers x, f(x) < g(x)?

Exercise | Q 13.(i) | Page 29

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x2 + 1, then find f + g

Exercise | Q 13.(ii) | Page 29

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x2 + 1, then find f – g

Exercise | Q 13.(iii) | Page 29

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x2 + 1, then find fg

Exercise | Q 13.(iv) | Page 29

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x2 + 1, then find f/g

Exercise | Q 14 | Page 29

Express the following functions as set of ordered pairs and determine their range.
f : X → R, f(x) = x3 + 1, where X = {–1, 0, 3, 9, 7}

Exercise | Q 15 | Page 29

Find the values of x for which the functions f(x) = 3x2 – 1 and g(x) = 3 + x are equal.

Exercise | Q 16 | Page 29

Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? Justify. If this is described by the relation, g(x) = αx + β, then what values should be assigned to α and β?

Exercise | Q 17.(i) | Page 29

Find the domain of the following functions given by f(x) = 1/sqrt(1 - cos x)

Exercise | Q 17.(ii) | Page 29

Find the domain of the following functions given by f(x) = 1/sqrt(x + |x|)

Exercise | Q 17.(iii) | Page 29

Find the domain of the following functions given by f(x) = x|x|

Exercise | Q 17.(iv) | Page 29

Find the domain of the following functions given by f(x) = (x^3 - x + 3)/(x^2 - 1)

Exercise | Q 17.(v) | Page 29

Find the domain of the following functions given by f(x) = (3x)/(2x - 8)

Exercise | Q 18.(i) | Page 29

Find the range of the following functions given by f(x) = 3/(2 - x^2)

Exercise | Q 18.(ii) | Page 29

Find the range of the following functions given by f(x) = 1 – |x – 2|

Exercise | Q 18.(iii) | Page 29

Find the range of the following functions given by f(x) = |x − 3|

Exercise | Q 18.(iv) | Page 29

Find the range of the following functions given by f(x) = 1 + 3 cos2x

(Hint: –1 ≤ cos 2x ≤ 1 ⇒ –3 ≤ 3 cos 2x ≤ 3 ⇒ –2 ≤ 1 + 3cos 2x ≤ 4)

Exercise | Q 19 | Page 29

Redefine the function f(x) = x − 2 + 2 + x , – 3 ≤ x ≤ 3

Exercise | Q 20.(i) | Page 30

If f(x) = (x - 1)/(x + 1), then show that f(1/x) = – f(x)

Exercise | Q 20.(ii) | Page 30

If f(x) = (x - 1)/(x + 1), then show that f(- 1/x) = (-1)/(f(x))

Exercise | Q 21.(i) | Page 30

Let f(x) = sqrt(x) and g(x) = x be two functions defined in the domain R+ ∪ {0}. Find (f + g)(x)

Exercise | Q 21.(ii) | Page 30

Let f(x) = sqrt(x) and g(x) = x be two functions defined in the domain R+ ∪ {0}. Find (f – g)(x)

Exercise | Q 21.(iii) | Page 30

Let f(x) = sqrt(x) and g(x) = x be two functions defined in the domain R+ ∪ {0}. Find (fg)(x)

Exercise | Q 21.(iv) | Page 30

Let f(x) = sqrt(x) and g(x) = x be two functions defined in the domain R+ ∪ {0}. Find (f/g)(x)

Exercise | Q 22 | Page 30

Find the domain and range of the function f(x) = 1/sqrt(x - 5)

Exercise | Q 23 | Page 30

If f(x) = y = (ax - b)/(cx - a), then prove that f(y) = x.

#### Objective Type Questions from 24 to 35

Exercise | Q 24 | Page 30

Let n(A) = m, and n(B) = n. Then the total number of non-empty relations that can be defined from A to B is ______.

• mn

• nm – 1

• mn – 1

• 2mn – 1

Exercise | Q 25 | Page 30

If [x]2 – 5[x] + 6 = 0, where [ . ] denote the greatest integer function, then ______.

• x ∈ [3, 4]

• x ∈ (2, 3]

• x ∈ [2, 3]

• x ∈ [2, 4)

Exercise | Q 26 | Page 30

Range of f(x) = 1/(1 - 2 cosx) is ______.

• [1/3, 1]

• [-1, 1/3]

• (-oo, -1] ∪ [1/3, oo)

• [- 1/3, 1]

Exercise | Q 27 | Page 31

Let f(x) = sqrt(1 + x^2), then ______.

• f(xy) = f(x) . f(y)

• f(xy) ≥ f(x) . f(y)

• f(xy) ≤ f(x) . f(y)

• None of these

Exercise | Q 28 | Page 31

Domain of sqrt(a^2 - x^2)  (a > 0) is ______.

• (– a, a)

• [– a, a]

• [0, a]

• (– a, 0]

Exercise | Q 29 | Page 31

If f(x) = ax + b, where a and b are integers, f(–1) = – 5 and f(3) = 3, then a and b are equal to ______.

• a = – 3, b = –1

• a = 2, b = – 3

• a = 0, b = 2

• a = 2, b = 3

Exercise | Q 30 | Page 31

The domain of the function f defined by f(x) = sqrt(4 - x) + 1/sqrt(x^2 - 1) is equal to ______.

• (– oo, – 1) ∪ (1, 4]

• (– oo, – 1] ∪ (1, 4]

• (– oo, – 1) ∪ [1, 4]

• (– oo, – 1) ∪ [1, 4)

Exercise | Q 31 | Page 31

The domain and range of the real function f defined by f(x) = (4 - x)/(x - 4) is given by ______.

• Domain = R, Range = {–1, 1}

• Domain = R – {1}, Range = R

• Domain = R – {4}, Range = {– 1}

• Domain = R – {– 4}, Range = {–1, 1}

Exercise | Q 32 | Page 31

The domain and range of real function f defined by f(x) = sqrt(x - 1) is given by ______.

• Domain = (1, oo), Range = (0, oo)

• Domain = [1, oo), Range = (0, oo)

• Domain = [1, oo), Range = [0, oo)

• Domain = [1, oo), Range = [0, oo)

Exercise | Q 33 | Page 32

The domain of the function f given by f(x) = (x^2 + 2x + 1)/(x^2 - x - 6) is ______.

• R – {3, – 2}

• R – {–3, 2}

• R – [3, – 2]

• R – (3, – 2)

Exercise | Q 34 | Page 32

The domain and range of the function f given by f(x) = 2 – |x – 5| is ______.

• Domain = R+, Range = (– oo, 1]

• Domain = R, Range = (– oo, 2]

• Domain = R, Range = (– oo, 2)

• Domain = R+, Range = (– oo, 2]

Exercise | Q 35 | Page 32

The domain for which the functions defined by f(x) = 3x2 – 1 and g(x) = 3 + x are equal is ______.

• {- 1, 4/3}

• [-1, 4/3]

• (-1, 4/3)

• [-1, 4/3)

#### Fill in the blanks :

Exercise | Q 36 | Page 32

Let f and g be two real functions given by
f = {(0, 1), (2, 0), (3, – 4), (4, 2), (5, 1)}
g = {(1, 0), (2, 2), (3, – 1), (4, 4), (5, 3)}
then the domain of f . g is given by ______.

Exercise | Q 37 | Page 32

Let f = {(2, 4), (5, 6), (8, – 1), (10, – 3)}
g = {(2, 5), (7, 1), (8, 4), (10, 13), (11, 5)}
be two real functions. Then Match the following :

 Column A Column B f – g {(2, 4/5), (8, (-1)/4), (10, (-3)/13)} f + g {(2, 20), (8, −4), (10, −39)} f . g {(2, −1), (8, −5), (10, −16)} f/g {(2, 9), (8, 3), (10, 10)}
Exercise | Q 38 | Page 33

State True or False for the following statement.

The ordered pair (5, 2) belongs to the relation R = {(x, y) : y = x – 5, x, y ∈ Z}.

• True

• False

Exercise | Q 39 | Page 33

State True or False for the following statement.

If P = {1, 2}, then P × P × P = {(1, 1, 1), (2, 2, 2), (1, 2, 2), (2, 1, 1)}

• True

• False

Exercise | Q 40 | Page 33

State True or False for the following statement.

If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, then (A × B) ∪ (A × C) = {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}.

• True

• False

Exercise | Q 41 | Page 33

State True or False for the following statement.

If (x – 2, y + 5) = (-2, 1/3) are two equal ordered pairs, then x = 4, y = (-14)/3

• True

• False

Exercise | Q 42 | Page 33

State True or False for the following statement.

If A × B = {(a, x), (a, y), (b, x), (b, y)}, then A = {a, b}, B = {x, y}

• True

• False

## Chapter 2: Relations and Functions

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

NCERT solutions for Mathematics Exemplar Class 11 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 Exemplar Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Mathematics Exemplar Class 11 chapter 2 Relations and Functions are Cartesian Product of Sets, Brief Review of Cartesian System of Rectanglar Co-ordinates, Concept of Relation, Concept of 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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