# 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.

## Class 11 Maths Chapter 2 Relations And Functions

Exercise 2.1

1. If  find the values of  and

Ans. Here

and

and

and

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

A and  B

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

A and  A

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

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:

At

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, (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.

## Class 11 Maths Chapter 2 Relations And Functions

Exercise 2.1

1. If  find the values of  and

Ans. Here

and

and

and

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

A and  B

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

A and  A

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

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:

At

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,

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

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.

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}

Benefits of NCERT Solution for Class 11

NCERT Solution for Class 11 contains extremely important points, and for each chapter, each concept has been simplified to make it easier to remember and increase your chances of achieving excellent exam results. Exam Preparation References Here are some tips on how these solutions can help you prepare for the exam.

1. This helps students solve many of the problems in each chapter and encourages them to make their concepts more meaningful.
2. NCERT Solution for Class 11 encourage you to update your knowledge and refine your concepts so that you can get good results in the exam.
3. These NCERT Solution For Class 11 are the best exam materials, allowing you to learn more about your week and your strengths. To get good results in the exam, it is important to overcome your weaknesses.
4. Most of the questions in the exam are formulated in a similar way to NCERT textbooks. Therefore, students should review the solutions in each chapter in order to better understand the topic.
5. It is free of cost.

Tips & Strategies for Class 11 Exam Preparation

1. Plan your course and syllabus and make time for revision
2. Please refer to the NCERT solution available on the cbsestudyguru website to clarify your concepts every time you prepare for the exam.
3. Use the cbsestudyguru learning app to start learning to successfully pass the exam. Provide complete teaching materials, including resolved and unresolved tasks.
4. It is important to clear all your doubts before the exam with your teachers or Alex (an Al study Bot).
5. When you read or study a chapter, write down algorithm formulas, theorems, etc., and review them quickly before the exam.
6. Practice an ample number of question papers to make your concepts stronger.
7. Take rest and a proper meal.  Don’t stress too much.

Why opt for cbsestudyguru NCERT Solution for Class 11 ?

• cbsestudyguru provide NCERT Solutions for all subjects at your fingertips.
• These solutions are designed by subject matter experts and provide solutions to every NCERT textbook questions.
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