In NCERT Solutions for Class 11 maths chapter 2 relations and functions, we learn about  ordered pair, cartesian product of sets, relations, representation of a relation, function as a special kind of relation, function as a correspondence, equal functions, real functions, domain of real functions, some standard real functions and their graphs, operations on real functions.

Class 11 Maths NCERT Solutions Chapter 2 Ex 2.1

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NCERT Solutions for Class 11 Maths Chapter 2 Ex 2.2

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Class 11 Maths NCERT Solutions Chapter 2 Ex 2.3

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Class 11 Maths NCERT Miscellaneous Solutions Chapter 2

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Exercise 2.1

Q.1: If (,) = (,), what is the value of a and b?

Q:2. If the set X has 4 elements and the set Y = {2, 3, 4, 5}, then find the number of elements in X × Y

Q.3: If A = {8, 9} and B = {4, 5, 2}, what is the value of A × B and B × A?

Q.4: State whether the given statements are True or False. If the statement is false, write that statement correctly.

(i). If X = {a, b} and Y = {b, a}, then X × Y = {(a, b), (b, a)}

(ii). If P and Q are non – empty sets, then P × Q is a non – empty set of ordered pairs (a, b) such that x ∈ P and b ∈ Q.

(iii). If M = {2, 3}, N = {4, 5}, then M × (N ∩Ø ) = Ø.

Q.5: If M = {-2, 2}, then find M × M × M.

Q.6: If X × Y = {(a, m), (a, n), (b, m), (b, n)}. Find X and Y.

Q.7: Let P = {2, 3}, Q = {2, 3, 4, 5}, R = {6, 7} and S = {6, 7, 8, 9}. Verify the following:

(i). P×(Q∩R) = (P×Q)∩(P×R)

(ii). P × R is a subset of Q × S

Q.8: Let P = {2, 3} and Q = {4, 5}. Find P × Q and then find how many subsets will P × Q have? List them.

Q.9: Let M and N be two sets where n (M) = 3 and n (N) = 2. If (a, 1), (b, 2), (c, 1) are in M × N, find M and N, where a, b and c are different elements.

Q.10: The Cartesian product Z × Z has 9 elements among which are found (-2, 0) and (0, 2). Find the set Z and also the remaining elements of Z × Z.

Exercise 2.2 

Q.1: Let X = {1, 2, 3, 4, . . . . . 14}. Define a relation Z from X to X by Z= {(a, b): 3a – b = 0, where a, b ∈ X}. Find its co – domain, domain and range.

Q.2: Define a relation Z on the set N of natural no. by Z = {(a, b): b = a + 5, a is a natural no less than 4; a, b ∈ N}. Give this relationship in the roaster form. Find the domain and the range.

Q.3: M = {1, 2, 3, 5} and N = {4, 6, 9}. Define a relation Z from M to N by Z = {(a, b): the difference between a and b is odd; a ∈ M, b ∈ N}. Find Z in roster form.

The figure given below shows a relationship between the sets A and B. Find the following relation:

(i) In set-builder form

(ii) In roster form.

What is its range and domain?

Q.5: Let X = {1, 2, 3, 4, 6}. Let Z be the relation on X defined by {(p, q): p, q ∈ X, q is divisible by p}.

(i) Write Z in the roster form

(ii) Find domain of Z

(iii) Find range of Z

Q.6: Find the range and domain of the relation Z defined by Z = {(a, a + 5): a ∈ {0, 1, 2, 3, 4, 5}}.

Q.7: Find the relation Z = {(a, a³): a is a prime number less than 10} in the roster form.

Q.8: Let X = {a, b, c} and Y = {11, 12}. Find the no. of relations from X to Y.

Q.9: Let Z be the relation on P defined by Z = {(x, y): x, y ∈ P, x – y is an integer}. Find the range and domain of Z.

Exercise 2.3 

Q.1: Which of the given relations are functions? Answer with reason. If it is a function, find its range and domain.

(i) {(12, 11), (15, 11), (18, 11), (1, 11), (4, 11), (7, 11)}

(ii) {(12, 11), (14, 12), (16, 13), (18, 14), (0, 15), (2, 16), (4, 17)}

(iii) {(11, 13), (11, 15), (12, 15)}

Q.2: Find the range and domain of the given real function:

(i) f(y) = -|y|

(ii) f(y) = √9–y²

Q.3: A function f is f(y) = 3y – 6. Find the values of the following:

(i) f(1)

(ii) f(8)

(iii) f(-2)

Q.4: The function ‘f’ which shows temperature in degree Celsius into temperature in degree Fahrenheit is expressed as: .

Find for the following values:

(i) f(0)

(ii) f(28)

(iii) f(-10)

(iv) The value of C, when f(C) = 212

Q.5: Calculate range of the given functions:

(i) f(y) = 2 – 3y, y ∈ R, y > 0.

(ii) f(y) = y2+2, is a real no.

(iii) f(y)  = y, y is a real no.

Miscellaneous Exercise

Q-1:  The relation ‘m’ is defined by:

m (y) = y²,  0≤y≤5

          = 5y,  5≤y≤30

The relation ‘n’ is defined by

n (y) = y²,  0≤y≤4

         = 5y,  4≤y≤30

Now, prove that ‘m’ is a function and ‘n’ is not a function.

Q-2: If g(y) = y² then, Find

 Find the domain for the function given below:

Q-4: Find the range and domain of the function given below:

g(y)=√(y–5)

Q-5: Find the range and domain of the function:   g(y) = |y – 4|

Q-6: A function from ‘R into R’ is given below. Find the range of ‘g’.

Q-7: Assume that function ‘m’ and ‘n’ is defined from: R→R.

m (y) = y + 2, n(y) = 3y – 2

Find m + n, m – n and

Q-8: Let g = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from ‘Z to Z’ defined by g(y) = uy + v, for some integers u,v. Find u,v.

Q-9: Let ‘f’ be a relation from ‘N to N’ defined by f = {(x,y): x,yϵN and x = y²}. Find out which of the following is true and which one is false.

1.(x,y)ϵf,(y,z)ϵf⇒(x,z)ϵf.

2.(x,x)ϵf,forallxϵN

3.(x,y)ϵf⇒(y,x)ϵf

Also justify your answer.

Q-10: Assume U = {1, 2, 3, 4}, V = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2,9), (3,1), (4, 5), (2, 11)}. Find out which of the following is true and which one is false.

(1). ‘f’ is a function from U to V.

(2). ‘f’ is a relation from U to V.

Justify your answer.

Q-11: Assume ‘g’ be the subset of ‘Z to Z’ defined by f = {(xy, x + y): x,yϵZ}. Is ‘g’ a function from ‘Z to Z’, also justify your answer.

Q-12: Assume ‘X’ = {5,7,9, 10, 11, 12, 13} and let ‘g’: X→N be defined by g(n) = The highest prime factor of ‘n’. Find the range of ‘g’.