NCERT Solutions For Class 11 Maths Chapter 2 Relations And Functions

Class 11 Maths Chapter 2 Relations And Functions

NCERT Solutions For Class 11 Maths Chapter 2 Relations And Functions, (Maths) exam are Students are taught thru NCERT books in some of state board and CBSE Schools. As the chapter involves an end, there is an exercise provided to assist students prepare for evaluation. Students need to clear up those exercises very well because the questions withinside the very last asked from those.

Sometimes, students get stuck withinside the exercises and are not able to clear up all of the questions. To assist students solve all of the questions and maintain their studies with out a doubt, we have provided step by step NCERT Solutions for the students for all classes. These answers will similarly help students in scoring better marks with the assist of properly illustrated solutions as a way to similarly assist the students and answering the questions right.

NCERT Solutions For Class 11 Maths Chapter 2 Relations And Functions

Class 11 Maths Chapter 2 Relations And Functions

Exercise 2.1

1. If find the values of and

Ans. Here

and

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

Ans. Number of elements in set A = 3 and Number of elements in set B = 3

Number of elements in A B = 3 3 = 9

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

Ans. Given: G = {7, 8} and H = {5, 4, 2}

GH = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

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

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

(i) If P = and Q = then P Q =

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

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

Ans. (i) Here P = and Q =

Number of elements in set P = 2 and Number of elements in set Q = 2

Number of elements in P Q = 2 2 = 4

But PQ = and here number of elements in P Q = 2

Therefore, statement is false.

(ii) True

(iii) True

5. If A = find A A A.

Ans. Here A =

A A =

A A A =

6. If A B = find A and B.

Ans. Given: A B =

A = set of first elements = and B = set of second elements =

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

(i)

(ii) A C is a subset of B D.

Ans. Given: A = {1, 2}, B = {1, 2, 3, 4}, C

= {5, 6} and D = {5, 6, 7, 8}

(i) = {1, 2, 3, 4} {5, 6} =

……….(i)

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

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

(AB) (A C) = ……….(ii)

Therefore, from eq. (i) and (ii),

= (A B) (A C)

(ii) A C = {(1, 5), (1, 6), (2, 5), (2, 6)

B D = {(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),

Therefore, it is clear that each element of A C is present in B D.

A C B D

8. Let A = {1, 2} and B = {3, 4}, write A B. How many sub sets will A B have? List them.

Ans. Given: A = {1, 2} and B = {3, 4}

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

Number of elements in A B = 4

Therefore, Number of subsets of AB = = 16

The subsets are:

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

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

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

9. Let A and B be two sets such that and If are in A B.

Ans. Here

A and B

But it is given that and

A = and B = {1, 2}

10. The Cartesian Product A A has 9 elements among which are found and (0, 1). Find the set A and the remaining elements of A A.

Ans. Here

A and A

But it is given that which implies that

A =

And A A =

Therefore, the remaining elements of A A are

and

Exercise 2.2

1. Let A = {1, 2, 3, ……., 14}. Define a relation R from A to A by R = where Write down its domain co-domain and range.

Ans. Given: A = {1, 2, 3, ……….., 14}

The ordered pairs which satisfy are (1, 3), (2, 6), (3, 9) and (4, 12).

R = {(1, 3), (2, 6), (3, 9), (4, 12)}

Domain = {1, 2, 3, 4}

Range = {3, 6, 9, 12}

Co-domain = {1, 2, 3, ……….., 14}

2. Define a relation R on the set N of natural numbers R = is a natural number less than 4: Depict this relationship using roster form. Write down the domain and the range.

Ans. Given: R =

Putting = 1, 2, 3 in we get = 6, 7, 8

R = {(1, 6), (2, 7), (3, 8)}

Domain = {1, 2, 3}

Range = {6, 7, 8}

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

Ans. Given: A = {1, 2, 3, 5} and B = {4, 6, 9}, A, B

= (1 – 4), (1 – 6), (1 – 9), (2 – 4), (2 – 6), (2 – 9), (3 – 4), (3 – 6) (3 – 9),

(5 – 4), (5 – 6), (5 – 9)

R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6) (5, 4), (5, 6)}

4. Figure shows a relationship between the sets P and Q. Write this relation:

(i) in set-builder form

(ii) roster form

What is its domain and range?

Ans. (i) Relation R in set-builder form is R =

(ii) Relation R in roster form is R = {(5 3), (6, 4), (7, 5)

Domain = {5, 6, 7}

Range = {3, 4, 5}

5. Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by is exactly divisible by

(i) Write R in roster form.

(ii) Find the domain of R.

(iii) Find the range of R.

Ans. Given: A = {1, 2, 3, 4, 6}

A set of ordered pairs where is exactly divisible by

(i) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (4, 6)}

(ii) Domain of R = {1, 2, 3, 4, 6}

(iii) Range of R = {1, 2, 3, 4, 6}

6. Determine the domain and range of the relation R defined by

R =

Ans. Given: R = =

and

Putting we get

Domain of R = {0, 1, 2, 3, 4 5}

Range of R = {0, 1, 2, 3, 4 5}

7. Write the relation R = is a prime number less than in roster form.

Ans. Given: R =

Putting = 2, 3, 5, 7

R = {(2, 8), (3, 27), 5, 125), (7, 343)}

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

Ans. Given: A = and B = {1, 2}

Number of elements in set A = 3 and Number of elements in set B = 2

Number of subsets of

Number of relations from A to

9. Let R be the relation on Z defined by R = is an integer}. Find the domain and range of R.

Ans. Given: R =

Domain of R = Z

Range of R = Z

Exercise 2.3

1. Which of the following 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)}

Ans. (i) Given: Relation is {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

All values of are distinct. Each value of has a unique value of

Therefore, the relation is a function.

Domain of function = {2, 5, 8, 11, 14, 17}

Range of function = {1}

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

All values of are distinct. Each value of has a unique value of

Therefore, the relation is a function.

Domain of function = {2, 4, 6, 8, 10, 12, 14}

Range of function = {1, 2, 3, 4, 5, 6, 7}

(iii) Given: Relation is {(1, 3), (1, 5), (2, 5)}

This relation is not a function because there is an element 1 which is associated to two elements 3 and 5.

2. Find the domain and range of the following real functions:

(i)

(ii)

Ans. (i) Given: . The function is defined for all real values of

Domain of the function = R

Now, when then

When

Therefore, for all real values of

Range of function =

(ii) Given: .

The function is not defined when

Domain of function =

= =

Range of function =

3. A function is defined by Write down the values of:

(i)

(ii)

(iii)

Ans. Given:

(i) Putting

(ii) Putting

(iii) Putting

4. The function which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by Find:

(i)

(ii)

(iii)

(iv) The value of C when

Ans. Given:

(i) Putting C = 0,

(ii) Putting C = 28,

(iii) Putting C = –10,

(iv) Putting = 212,

5. Find the range of each of the following functions:

(i)

(ii) is a real number.

(iii) is a real number.

Ans. (i) Given:, R and

Range of function

= =

(ii) Given: R

for R

Range of function

= =

(iii) Given:, R

Range of function = R

Miscellaneous Exercise

1. The relation is defined by . The relation is defined by. Show that is a function and is not a function.

Ans. Given: and

At and

It is observed that takes unique value at each point in its domain [0, 10]. Therefore, is a function.

Now, and

At and

Therefore, does not have unique value at

Hence, is not a function.

2. If find

Ans. Given:

and

3. Find the domain of the function

Ans. Given:

is a rational function of

assumes real values of all

NCERT Solutions For Class 11 Maths Chapter 2 Relations And Functions

Class 11 Maths Chapter 2 Relations And Functions

Exercise 2.1

1. If find the values of and

Ans. Here

and

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

Ans. Number of elements in set A = 3 and Number of elements in set B = 3

Number of elements in A B = 3 3 = 9

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

Ans. Given: G = {7, 8} and H = {5, 4, 2}

GH = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}

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

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

(i) If P = and Q = then P Q =

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

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

Ans. (i) Here P = and Q =

Number of elements in set P = 2 and Number of elements in set Q = 2

Number of elements in P Q = 2 2 = 4

But PQ = and here number of elements in P Q = 2

Therefore, statement is false.

(ii) True

(iii) True

5. If A = find A A A.

Ans. Here A =

A A =

A A A =

6. If A B = find A and B.

Ans. Given: A B =

A = set of first elements = and B = set of second elements =

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

(i)

(ii) A C is a subset of B D.

Ans. Given: A = {1, 2}, B = {1, 2, 3, 4}, C

= {5, 6} and D = {5, 6, 7, 8}

(i) = {1, 2, 3, 4} {5, 6} =

……….(i)

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

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

(AB) (A C) = ……….(ii)

Therefore, from eq. (i) and (ii),

= (A B) (A C)

(ii) A C = {(1, 5), (1, 6), (2, 5), (2, 6)

B D = {(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),

Therefore, it is clear that each element of A C is present in B D.

A C B D

8. Let A = {1, 2} and B = {3, 4}, write A B. How many sub sets will A B have? List them.

Ans. Given: A = {1, 2} and B = {3, 4}

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

Number of elements in A B = 4

Therefore, Number of subsets of AB = = 16

The subsets are:

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

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

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

9. Let A and B be two sets such that and If are in A B.

Ans. Here

A and B

But it is given that and

A = and B = {1, 2}

10. The Cartesian Product A A has 9 elements among which are found and (0, 1). Find the set A and the remaining elements of A A.

Ans. Here

A and A

But it is given that which implies that

A =

And A A =

Therefore, the remaining elements of A A are

and

Exercise 2.2

1. Let A = {1, 2, 3, ……., 14}. Define a relation R from A to A by R = where Write down its domain co-domain and range.

Ans. Given: A = {1, 2, 3, ……….., 14}

The ordered pairs which satisfy are (1, 3), (2, 6), (3, 9) and (4, 12).

R = {(1, 3), (2, 6), (3, 9), (4, 12)}

Domain = {1, 2, 3, 4}

Range = {3, 6, 9, 12}

Co-domain = {1, 2, 3, ……….., 14}

2. Define a relation R on the set N of natural numbers R = is a natural number less than 4: Depict this relationship using roster form. Write down the domain and the range.

Ans. Given: R =

Putting = 1, 2, 3 in we get = 6, 7, 8

R = {(1, 6), (2, 7), (3, 8)}

Domain = {1, 2, 3}

Range = {6, 7, 8}

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

Ans. Given: A = {1, 2, 3, 5} and B = {4, 6, 9}, A, B

= (1 – 4), (1 – 6), (1 – 9), (2 – 4), (2 – 6), (2 – 9), (3 – 4), (3 – 6) (3 – 9),

(5 – 4), (5 – 6), (5 – 9)

R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6) (5, 4), (5, 6)}

4. Figure shows a relationship between the sets P and Q. Write this relation:

(i) in set-builder form

(ii) roster form

What is its domain and range?

Ans. (i) Relation R in set-builder form is R =

(ii) Relation R in roster form is R = {(5 3), (6, 4), (7, 5)

Domain = {5, 6, 7}

Range = {3, 4, 5}

5. Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by is exactly divisible by

(i) Write R in roster form.

(ii) Find the domain of R.

(iii) Find the range of R.

Ans. Given: A = {1, 2, 3, 4, 6}

A set of ordered pairs where is exactly divisible by

(i) R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (4, 6)}

(ii) Domain of R = {1, 2, 3, 4, 6}

(iii) Range of R = {1, 2, 3, 4, 6}

6. Determine the domain and range of the relation R defined by

R =

Ans. Given: R = =

and

Putting we get

Domain of R = {0, 1, 2, 3, 4 5}

Range of R = {0, 1, 2, 3, 4 5}

7. Write the relation R = is a prime number less than in roster form.

Ans. Given: R =

Putting = 2, 3, 5, 7

R = {(2, 8), (3, 27), 5, 125), (7, 343)}

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

Ans. Given: A = and B = {1, 2}

Number of elements in set A = 3 and Number of elements in set B = 2

Number of subsets of

Number of relations from A to

9. Let R be the relation on Z defined by R = is an integer}. Find the domain and range of R.

Ans. Given: R =

Domain of R = Z

Range of R = Z

Exercise 2.3

1. Which of the following 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)}

Ans. (i) Given: Relation is {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)}

All values of are distinct. Each value of has a unique value of

Therefore, the relation is a function.

Domain of function = {2, 5, 8, 11, 14, 17}

Range of function = {1}

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

All values of are distinct. Each value of has a unique value of

Therefore, the relation is a function.

Domain of function = {2, 4, 6, 8, 10, 12, 14}

Range of function = {1, 2, 3, 4, 5, 6, 7}

(iii) Given: Relation is {(1, 3), (1, 5), (2, 5)}

This relation is not a function because there is an element 1 which is associated to two elements 3 and 5.

2. Find the domain and range of the following real functions:

(i)

(ii)

Ans. (i) Given: . The function is defined for all real values of

Domain of the function = R

Now, when then

When

Therefore, for all real values of

Range of function =

(ii) Given: .

The function is not defined when

Domain of function =

= =

Range of function =

3. A function is defined by Write down the values of:

(i)

(ii)

(iii)

Ans. Given:

(i) Putting

(ii) Putting

(iii) Putting

4. The function which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by Find:

(i)

(ii)

(iii)

(iv) The value of C when

Ans. Given:

(i) Putting C = 0,

(ii) Putting C = 28,

(iii) Putting C = –10,

(iv) Putting = 212,

5. Find the range of each of the following functions:

(i)

(ii) is a real number.

(iii) is a real number.

Ans. (i) Given:, R and

Range of function

= =

(ii) Given: R

for R

Range of function

= =

(iii) Given:, R

Range of function = R

Miscellaneous Exercise

1. The relation is defined by . The relation is defined by. Show that is a function and is not a function.

Ans. Given: and

At and

It is observed that takes unique value at each point in its domain [0, 10]. Therefore, is a function.

Now, and

At and

Therefore, does not have unique value at

Hence, is not a function.

2. If find

Ans. Given:

and

3. Find the domain of the function

Ans. Given:

is a rational function of

assumes real values of all except for those values of for which

Domain of function = R – {2, 6}

4. Find the domain and range of the real function defined by

Ans. Given: assumes real values if

Domain of

For

Range of = all real numbers 0 =

5. Find the domain and range of the real function defined by

Ans. Given:

The function is defined for all values of

Domain of = R

When ,

When,

Range of = All real numbers 0 =

6. Let be a function from R into R. Determine the range of

Ans. Here

Putting

Now, will be real if

Range of

7. Let be defined respectively

by Find and

Ans. Given: and

Now,

And

8. Let be a function from Z to Z defined by for some integers Determine

Ans. Given: and

Now

……….(i)

And

……….(ii)

Solving eq. (i) and (ii), we get and

9. Let R be a relation from N to N defined by R = Are the following true:

(i) R for all N

(ii) R implies R

(iii) R, R implies R

Ans. Given: R =

(i) No, (3, 3) R because

(ii) No, (9, 3) R but (3, 9) R

(iii) No, (81, 9) R but (81, 3) R

10. 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) is a relation from A to B.

(ii) is a function from A to B.

Justify your answer in each case.

Ans. (i) Here A = {1, 2, 3, 4} and B = {1, 5, 9, 11, 15, 16}

= {(1, 1), (1, 5), (1, 9), (1, 11), (1, 15), (1, 16), (2, 1), (2, 5), (2, 9), (2, 11),

(2, 15), (2, 16), (3, 1), (3, 5), (3, 9), (3, 11), (3, 15), (3, 16), (4, 1), (4, 5),

(4, 9), (4, 11), (4, 15), (4, 16)}

= {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}

Now, (1, 5), (2, 9), (3, 1), (4, 5), (2, 11)

is a relation from A to B.

11. Let be a subset of defined by Is a function from Z to Z? Justify your answer.

Ans. We observed that = 4 and = 4

and

(4, 5) and (4, 4)

It shows that is not a function from Z to Z.

12. Let A = {9, 10, 11, 12 13} and let be defined by the highest prime factor of Find the range of

Ans. Here A = {9, 10, 11, 12, 13}

For

[and 3 is highest prime factor of 9]

For

[]

For

[]

For

[]

For

[]

Range of = {5, 11, 3, 13}

= {3, 5, 11, 13}

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Class 11 Maths Chapter 2 Relations And Functions

NCERT Solutions For Class 11 Maths Chapter 2 Relations And Functions

Class 11 Maths Chapter 2 Relations And Functions

NCERT Solutions For Class 11 Maths Chapter 2 Relations And Functions

Class 11 Maths Chapter 2 Relations And Functions

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