#### Online Mock Tests

#### Chapters

Chapter 2: Relations and Functions

Chapter 3: Trigonometric Functions

Chapter 4: Principle of Mathematical Induction

Chapter 5: Complex Numbers and Quadratic Equations

Chapter 6: Linear Inequalities

Chapter 7: Permutations and Combinations

Chapter 8: Binomial Theorem

Chapter 9: Sequences and Series

Chapter 10: Straight Lines

Chapter 11: Conic Sections

Chapter 12: Introduction to Three Dimensional Geometry

Chapter 13: Limits and Derivatives

Chapter 14: Mathematical Reasoning

Chapter 15: Statistics

Chapter 16: Probability

## Chapter 2: Relations and Functions

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

#### Short Answer

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

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

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

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

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

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

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.

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}.

Is the following relation a function? Justify your answer

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

Is the following relation a function? Justify your answer

R_{2} = {(x, |x |) | x is a real number}

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

Find the domain of the following function.

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

Find the domain of the following function.

f(x) = [x] + x

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

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

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

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

#### Objective Type Questions any 4 given possible

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

R

R

^{+ }R

^{– }None of these

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

2x

^{3}`2 1/x^3`

0

1

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 ______.

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 ______.

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

#### Short Answer

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

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

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

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

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.

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

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

In the following find a and b

(2a + b, a – b) = (8, 3)

In the following find a and b

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

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

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

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

Given R = {(x, y) : x, y ∈ W, x^{2} + y^{2} = 25}. Find the domain and Range of R.

If R_{1} = {(x, y) | y = 2x + 7, where x ∈ R and – 5 ≤ x ≤ 5} is a relation. Then find the domain and Range of R_{1}.

If R_{2} = {(x, y) | x and y are integers and x^{2} + y^{2} = 64} is a relation. Then find R_{2}.

If R_{3} = {(x, x) | x is a real number} is a relation. Then find domain and range of R_{3}.

Is the given relation a function? Give reasons for your answer.

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

Is the given relation a function? Give reasons for your answer.

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

Is the given relation a function? Give reasons for your answer.

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

Is the given relation a function? Give reasons for your answer.

s = {(n, n^{2}) | n is a positive integer}

Is the given relation a function? Give reasons for your answer.

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

If f and g are real functions defined by f(x) = x^{2} + 7 and g(x) = 3x + 5, find the following:

f(3) + g(– 5)

If f and g are real functions defined by f(x) = x^{2} + 7 and g(x) = 3x + 5, find the following:

`f(1/2) xx g(14)`

If f and g are real functions defined by f(x) = x^{2} + 7 and g(x) = 3x + 5, find the following:

f(– 2) + g(– 1)

If f and g are real functions defined by f(x) = x^{2} + 7 and g(x) = 3x + 5, find the following:

f(t) – f(– 2)

If f and g are real functions defined by f(x) = x^{2} + 7 and g(x) = 3x + 5, find the following:

`(f(t) - f(5))/(t - 5)`, if t ≠ 5

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)?

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)?

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

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

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

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

Express the following functions as set of ordered pairs and determine their range.

f : X → R, f(x) = x^{3} + 1, where X = {–1, 0, 3, 9, 7}

Find the values of x for which the functions f(x) = 3x^{2} – 1 and g(x) = 3 + x are equal.

#### Long Answer

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?

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

#### Objective Type Questions from 24 to 35

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 ______.

m

^{n}n

^{m}– 1mn – 1

2

^{mn}– 1

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)

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

`[1/3, 1]`

`[-1, 1/3]`

`(-oo, -1] ∪ [1/3, oo)`

`[- 1/3, 1]`

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

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

(– a, a)

[– a, a]

[0, a]

(– a, 0]

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

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)`

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}

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)`

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)

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]`

The domain for which the functions defined by f(x) = 3x^{2} – 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 :

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 ______.

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)} |

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

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

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

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

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

## 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.

Further, we at Shaalaa.com provide 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 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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