Important Questions for CBSE Class 11 Maths Chapter 2 - Relations and Functions
CBSE Class 11 Maths Chapter-2 Important Questions - Free PDF Download
1 Marks Questions
1. Find a and b if (a – 1, b + 5) = (2, 3)If A = {1,3,5}, B = {2,3} find : (Question-2, 3)
Ans. a = 3, b = –2
2. A × B
Ans. A × B = {(1,2), (1,3), (3,2), (3,3), (5,2), (5,3)}
3. B × A Let A = {1,2}, B = {2,3,4}, C = {4,5}, find (Question- 4,5)
Ans. B × A = { (2,1), (2,3), (2,5), (3,1), (3,3), (3,5)}
4. A × (B ∩ C)
Ans. {(1,4), (2,4)}
5. A × (B ∪ C)
Ans. {(1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5)}
6. If P = {1,3}, Q = {2,3,5}, find the number of relations from A to B
Ans. 
= 64
7. If A = {1,2,3,5} and B = {4,6,9}, R = {(x, y) : |x – y| is odd, x ∈ A, y ∈ B} Write R in roster form
Which of the following relations are functions. Give reason.
Ans. R = { (1,4), (1,6), (2,9), (3,4), (3,6), (5,4), (5,6)}
8. R = { (1,1), (2,2), (3,3), (4,4), (4,5)}
Ans. Not a function because 4 has two images.
9. R = { (2,1), (2,2), (2,3), (2,4)}
Ans. Not a function because 2 does not have a unique image.
10. R = { (1,2), (2,5), (3,8), (4,10), (5,12), (6,12)} Which of the following arrow diagrams represent a function? Why?
Ans. Function
11.
Ans. Function
12.
Let f and g be two real valued functions, defined by, f(x) = x2, g(x) = 3x +2.
Ans. Not a function
13. (f + g)(–2)
Ans. 0
14. (f – g)(1)
Ans. -4
15. (fg)(–1)
Ans. -1
16. 
Ans. 0
17. If f(x) = x3, find the value of,
Ans. 31
18. Find the domain of the real function, f(×) = 
Ans. (–∞, –2] ∪ [2, ∞)
19. Find the domain of the function, f (×) = 
Find the range of the following functions, (Question- 20,21)
Ans. R – {2,3}
20. f (x) = 
Ans. (–∞, –0] ∪ [1, ∞)
21. f(x) = 
+ 2
Ans. [2,∞)
22. Find the domain of the relation, R = { (x, y) : x, y ϵ Z, xy = 4} Find the range of the following relations : (Question-23, 24)
Ans. {–4, –2, –1,1,2,4}
23. R = {(a,b) : a, b ϵ N and 2a + b = 10}
Ans. {2,4,6,8}
24.R = 
Ans. 
25.If the ordered Pairs
and 
are equal, find
and 
Ans. 
26.
And
are two sets Then no. of relations of
have.
Ans. 64
27.Let 
then Range of function
Ans. 
28.A real function 
is defined by 
Then the Value of 
Ans. -11
29.If 
and 
form the sets
and
are these two Cartesian products equal?
Ans. Given and by definition of cartesion product, we set
and 
By definition of equality of ordered pains the pair 
is not equal to the pair 
therefore 
30..If
and are finite sets such that 
and 
find the number of relations fromto
Ans. Linen 
the number of subsets of 
then the number of subsets of 
Since every subset of is a relation from A to B therefore the number of relations from A to B = 2mk
31.Let 
be a function from z to z defined by 
for same integers a and b determine a and b.
Ans. Given 
Since 
Subtracting (i) from(ii) we set a=2
Substituting a=2 is (ii) we get 2+b=1
b = -1
Hence a = 2, b = -1
32.Express 
as the set of ordered pairs
Ans. Since 
and 
Put 
For anther values of 
we do not get 
Hence the required set of ordered peutes is 
33.If 
find 
Ans. 
34.Functionis defined by
find 
Ans. 
35.Let
be a linear function from 
intofind
Ans. 
36.If the ordered pairs
and 
are equal, find &
Ans. 
37.Let 
and
be the relation, is one less than from tothen find domain and Range of
Ans. Given 
and is the relation ‘is one less than’ from to therefore 
Domain of 
and range of 
38.Letbe a relation from 
todefine by
.
Is the following true 
implies 
Ans. No; let 
As 
so 
but 
so 
39.Letbe the set of natural numbers and the relationbe define inby =
what is the domain, co domain and range of? Is this relation a function?
Ans. Given 
Domain of 
co domain of and Range of is the set of even natural numbers.
Since every natural number has 
unique image 
therefore, the relation is a function.
40.Let 
and 
list the element of
Ans. 
41.Let be the subset of 
defined by
. Is a function from 
Justify your answer
Ans. Is not a function from Q to Z
One element 
have two images
is not function
42.The function 
which maps temperature in Celsius into temperature in Fahrenheit is defined by 
Ans. 
43.If 
Prove that 
Ans. 
44.If and are two sets containing
and 
elements respectively how many different relations can be defined fromto?
Ans. 
4 Marks Questions
1. Let A = {1,2,3,4}, B = {1,4,9,16,25} and R be a relation defined from A to B as, R = {(x, y) : x ϵ A, y ϵ B and y = x2}
(a) Depict this relation using arrow diagram.
(b) Find domain of R.
(c) Find range of R.
(d) Write co-domain of R.
Ans.
(b) {1,2,3,4}
(c) {1,4,9,16}
(d) {1,4,9,16,25}
2. Let R = { (x, y) : x, y ϵ N and y = 2x} be a relation on N. Find :
(i) Domain
(ii) Codomain
(iii) Range
Is this relation a function from N to N
Ans. (i) N
(ii) N
(iii) Set of even natural numbers
yes, R is a function from N to N.
3. Find the domain and range of, f(x) = |2x – 3| – 3
Ans. Domain is R
Range is [–3, ∞)
4. Draw the graph of the Constant function, f : R ϵ R; f(x) = 2 x ϵ R. Also find its domain and range.
Ans. Domain = R
Range = {2}
5.Let 
then
(i) Find the domain and the range of R (ii) Write R as a set of ordered pairs.
Ans. (i)Given 
and 
Put
for all other values of 
we do not get 
Domain of 
and range of 
(ii) 
as a set of ordered pairs can be written as
6.Let R be a relation from Q to Q defined by 
show that 
Ans. 
(i)
(ii)
(iii)
7.
Ans. 
8.Find the domain and the range of the function
Also find 
and the numbers which are associated with the number 43 m its range.
Ans. 
For 
must be real number
must be a real number
Which is a real number for every 
let 
We know that for all 
which are associated with the number 
in 
9.If 
Ans. 
10.Find the domain and the range of the function 
Ans. 
11.Let a relation 
then
(i) write domain of R
(ii) write range of R
(iii) write R the set builder form
(iv) represent R by an arrow diagram
Ans. Given 
(i) Domain of 
(ii)Rang of 
(iii)R in the builder from can be written as
(iv) The reaction R can be represented by the arrow diagram are shown.
12.Let 
and 
(i) find 
(ii) write R in roster form
(iii) write domain & range of R
(iv) represent R by an arrow diagram
Ans. (i)
(ii)
(iii)Domain of 
and range of 
(iv)The relation R can be represented by the are arrow diagram are shown.
13.The cartesian product 
has a elements among which are found 
and 
find the set and the remaining elements of
Ans. Let 
Given 
Given 
and 
Also 
and 
This 
but 
Therefore 
The remaining elements of are 
14.Find the domain and the range of the following functions 
Ans. Given
For 
must be a real number
Must be a real number
For 
let 
As 
15.Let 
and 
be two real functions. Find the following functions 
Ans. Given and we note that 
and 
so there functions have the same Domain
(i)
for
(ii)
for all
(iii)
for all
(iv)
(v)
for all
16.Find the domain and the range of the following functions
Ans. (i)Given 
For must be a real number
must be a real number
set of all real number except
For 
Must be a real number 
Set of all real number except 
(ii)Given 
For must be a real number 
Must be a real number
For let 
But 
for all 
Multiply both sides by 
a positive real number
(iii)Given 
For 
must be a real number
Must be a real number
Set of all real number except 
For let 
But 
for all 
But 
Multicity bath sides by 
a positive real number
Either 
or 
but 
17.If 
and 
find 
Ans. 
18.For non empty sets 
and 
prove that 
Ans. First we assume that 
Then 
and 
This, when 
then 
Conversely, Let 
and let be 
Then,
for same 
19.Let
be
given fixed positive integer. let 
show that is an equivalence relation on Z.
Ans. 
(i)
(ii)Let 
Then
is divisible by
is divisible by
is divisible
Then 
So is symmetric.
(iii) Let and 
is divisible by and 
is divisible by
is divisible by
is divisible by
and 
So, is transitive this is reflexive symmetric and transitive Hence, is an equivalence relation and
.
20.Let 
and
let be the relation, is greater than from to 
Write as a set of ordered pairs. find domain 
and range
Ans. 
Domain of R 
Range of R 
21.Define modulus function Draw graph.
Ans. let
for each
then
we know that
for all 
dom
and range
set of non negative real number
Drawing the graph of modulus function defined by
We have
| 3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| 3 | 2 | 1 | 0 | 1 | 2 | 3 | 4 |
Scale: 5 small divisions = 1 unit
On a graph paper, we plot the points 
and
Join them successively to obtain the graph lines AO and OG, as show in the figure above.
22.Let 
Show that
is function, while g is notfunction.
Ans. Each element in 
has a unique image under 
But, 
and
So
is not a function
23.Let 
and 
write how many subsets will have? List them.
Ans. 
16 Subsets of have
24.Let 
and
verify that 
Ans. 
Part-I
Part-II
25.Find the domain and the range of the relation defined by 
Ans. 
26.Find the linear relation between the components of the ordered pairs of the relation where 
Ans. Given 
Let 
be the linear relation between the components of
Since 
Also 
Subtracting 
from 
, we get 
Subtracting 
is , we get 
Subtracting there values of a and b in 
we get
which is the required linear relation between the components of the given relation.
27.Let
define a relation from toby
(i) write in the roaster form
(ii) write down the domain, co-domain and range of
(iii) Represent by an arrow diagram
Ans. (i)
(ii) Domain 
co domain 
range 
(iii)
28.A relation
is defined by 
where 
(i) list the elements of
(ii) is a function?
Ans. 
(i)
(ii)We note that each element of the domain of has a unique image; therefore, the relationis a function.
29.If
Prove that 
Ans. 
30.Let
be defined by
for all
where
and 
write the relation in the roster farm. It a function?
Ans. 
is a function because different elements of 
have different imager in y
31.Determine a quadratic function 
defined by 
Ans. 
Multiplying eq. (i) by 3 and eq. (ii) by 2
32.Find the domain and the range of the function defied by 
Ans.
For Df , 
must be a real no.
Domain of = set of all real numbers
33.Find the domain and the range of 
Ans.
34. If
Ans.
(i) 
(ii)
6 Marks Questions
1.Draw the graphs of the following real functions and hence find their range
Ans. Given
Let 
(Fig for Answer 11)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| 2 | 1 | 0.5 | 0.25 |
Plot the points shown is the above table and join there points by a free hand drawing.
Portion of the graph are shown the right margin
From the graph, it is clear that 
This function is called reciprocal function.
2.If 
Prove that 
Ans. Ifprove that 
3.Draw the graphs of the following real functions and hence find their range
Ans. (i)Given
, which is first degree equation in 
and hence it represents a straight line. Two points are sufficient to determine straight lint uniquely
Table of values
| 0 | 1 |
y | -1 | 1 |
A portion of the graph is shown in the figure from the graph, it is clear that y takes all real values. It therefore that 
(ii)Given 
Let 
i.e 
which is a first degree equation is and hence it represents a straight line. Two points are sufficient to determine a straight line uniquely
Table of values
| -1 | 0 |
y | 0 | 1 |
A portion of the graph is shown is the figure from the graph it is clear that y takes all real values except 2. It fallows that 
4.Let f be a function defined by 
(i) find the image of 3 under 
(ii) find 
(iii) find 
such that 
Ans. Given 
(i)
(ii)
(iii)
5.The function
is the formula to connect 
to Fahrenheit
units find 
interpret the result is each case
Ans.
6.Draw the graph of the greatest integer function,
Ans. Clearly, we have
| …… |
|
|
|
|
| …... |
| …… |
|
|
|
|
| …... |
7.Find the domain and the range of the following functions:
Ans. (i)Given 
For 
must be a real number
Must be a real number
either 
For 
let 
As square root of a real number is always non-negative, 
On squaring (i), we get 
but 
for all 
which is true for all 
also
(ii)Given 
For 
must be a real number
must be a real number
For let 
As square root of real number is always non-negative,
Squaring 
we get
but for all 
but
(iii)Given 
For must be a real number
must be a real number
For 
let 
Also as the square root of a real number is always non-negative, 
on squaring we get
But for all 
(Multiply bath sides by 
a positive real number)
either 
or 
8.Draw the graphs of the following real functions and hence find range: 
Ans.
Given 
Let 
| -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| 16 | 9 | 4 | 1 | 0 | 1 | 4 | 9 | 16 |
Plot the points
And join these points by a free hand drawing. A portion of the graph is shown in sigma (next)
From the graph, it is clear that 
takes all non-negative real values, if follows that 
9.Define polynomial function. Draw the graph of 
find domain and range
Ans. A function 
define by
And 
is non negative integer is called polynomial function
Graph of
| 0 | 1 | 2 | -1 | -2 |
| 0 | 1 | 8 | -1 | -8 |
Domain of 
Range of
10.(a) If 
are two sets such that 
and some elements of 
are 
than find
(b) Find domain of the function 
Ans. (a)Given A and B are two sets such that
Some elements of are
(b)
0 Comments