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RD Sharma Solution CLass 12 Mathematics Chapter 2 Functions

Chapter 2 – Functions Exercise Ex. 2.1

Question 1

Give an example of function which is one-one but not onto.

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2

Give an example of a function which is not one-one but onto.

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 3

Given an example of a function which is neither one-one nor onto.

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 15
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Classify the following functions as injection, surjection or bijection:

f : R → R, defined by f(x) = sin2x + cos2x

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 21
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 22
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 22
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 23
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 23
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 24
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 24
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 25
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 25
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 26
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 26
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 27
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 27
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 28
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 28
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 29

If A = { 1, 2, 3}, show that a ono-one function f : A → A must be onto.

Solution 29

Since f is one-one, three elements of {1, 2, 3} must be taken to 3 different elements of the co-domain {1, 2, 3} under f.

Hence, f has to be onto.

Question 30

If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.

Solution 30

Suppose f is not one-one.

Then, there exists two elements, say 1 and 2 in the domain whose image in the co-domain is same.

Also, the image of 3 under f can be only one element.

Therefore, the range set can have at most two elements of the co-domain {1, 2, 3}

i.e f is not an onto function,  a contradiction.

Hence, f must be one-one.

Question 31
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 31
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 32
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 32
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 33
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 33
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 34
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 34
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 35
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 35
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 37

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 37

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Chapter 2 – Functions Exercise Ex. 2.2

Question 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 1
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 4
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 5
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 6
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 8
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 13
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 14
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 15

Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and

h(z)  = sin z for all x, y, z ϵ N.

show tht ho (gof) = (hog) of.

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 16

Given Examples of two functions f : N → N and g ; N → N such that gof is onto but f is not onto.

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 17
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 18
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 19
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 19
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Chapter 2 – Functions Exercise Ex. 2.3

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 10
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 19

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

fof

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

fofof

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

(f0f0f) (38)

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 22

Let f be a real function given by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Find each of the following

f2Also, show that fof ≠ f2 .

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Chapter 2 – Functions Exercise Ex. 2.5

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Thus, h is a bijection and is invertible.

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 4

Consider Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions and Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions defined as Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions.

Show that Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions are invertible. Find Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions and show that Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 4

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

f-1 = {(3, 1), (5, 2), (7, 3), (9, 4)}

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

f-1 0g-1 = {(7, 1), (23, 2), (47, 3), (79, 4)} ……. (A)

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Now (A) & (B) we have (g0f)-1 = f-10g-1

Question 6

Show that the function Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions is invertible. Also find Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions.

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 7
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 9
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 10

Consider f : R+ → [-5,] given by f(x) = 9×2 + 6x – 5. Show that f is invertible with

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 11
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Solution 12
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions
Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Question 14

Let A = R – {3} and B = R – {1}. Consider the function Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions defined by Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions. Show that f is one-one and onto and hence find Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions.

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15

Consider the function f ; R+
[-9, ∞]
given by

f(x) = 5x2
+ 6x – 9.

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions 

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Chapter 2 – Functions Exercise Ex. 2VSAQ

Question 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Figure (a)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Figure (b)

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Figure (a)
Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Figure (b)

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 3

If A = {1, 2, 3} B = {a, b}, write total number of functions from A to B.

Solution 3

A = {1, 2, 3} B = {a, b}

The total number of functions is 8

Question 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 36

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1,4), (2,5), (3,6)} be a function from A to B.

State the whether f is one – one or onto

Solution 36

It is given that

A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1,4), (2,5), (3,6)}

The function f is one-one from A to B

Chapter 2 – Functions Exercise MCQ

Question 1

Let A = {x ε R : – 1 ≤ x ≤ 1} = B and C = {x ε R : x ≥ 0} and let S = {(x, y) ε A × B : x2 + y2 = 1} and s0 = {(x, y) ε A × C : x2 + y2 = 1}. Then

 

(a) S defines a function from A to B

(b) S0 defines a function from A To C

(c) S0 defines a function from A to B

(d) S defines a function from A to C

Solution 1

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a)  injective

(b) surjective

(c) bijective

(d) none of these

Solution 2

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 3

If f : A → B given by 3f(x) + 2-x = 4 is a bijection, then

 

(a) A = {x ε R : – 1 < x < ∞], B = {x ε R ; 2 < x < 4]

(b) A = {x ε R : – 3 < x < ∞}, B = {x ε R ; 0 < x < 4}

(c) A = {x ε R : – 2 < x ε R : 0 < x < 4}

(d) none of these

 

Solution 3

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 4

 The function f : R → R defined by f(x) = 2x + 2|x| is

 

(a) one-one and onto

(b) many-one and onto

(c) one-one and into

(d) many-one and into

Solution 4

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) f is one-one but not onto

(b) f is onto but one-one

(c) f is both one-one and into

(d) none of these

Solution 5

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 6

The function f : A → B defined by f(x) = -x2 + 6x – 8 is a bijection, if

 

(a) A = ( -∞, 3] and B = ( -∞, 1]

(b) A = [-3, ∞) and B = (-∞, 1]

(c) A = ( -∞, 3] and B = [1, ∞)

(d) A = [3, ∞) and B = [1, ∞)

Solution 6

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 7

Let A = {x ε R : – 1 ≤ x ≤ 1} = B. then, the mapping f : A → B given by f(x) = x|x| is 

 

(a) injective but not surjective

(b) surjective but not injective

(c) bijective

(d) none of these

Solution 7

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 8

Let f : R  → R be given by f(x) = [x]2 + [x + 1] – 3, where [x] denotes the greatest integer less than or equal to x. Then, f(x) is

 

(a) many-one and onto

(b) Many-one and into

(c) one-one and into

(d) one-one and onto

Solution 8

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 9

let m be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then the function f : M → R defined by f(A) = |A| for every A ε M, is

 

(a)  one-one and onto

(b) neither one-one nor onto

(c) one-one but-not onto

(d) onto but not one-one

Solution 9

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) one-one and onto

(b) one-one but not onto

(c) onto but not one-one

(d) neither one-one nor onto

Solution 10

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 11

The range of the function f (x) = 7-xPx-3 is

 

(a) {1, 2, 3, 4, 5}

(b) {1, 2, 3, 4, 5, 6}

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

(d) {1, 2, 3}

Solution 11

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 12

A function f from the set of natural numbers to integers defined by

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

is

 

(a) neither one-one nor onto

(b) one-one but not onto

(c) onto but not one-one

(d) one-one and onto both

Solution 12

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 13

Let f be an injective map with domain {x, y, z} and range {1, 2, 3} such that exactly one of the following statements is correct and the remaining are false.

f(x) = 1, f(y) ≠ 1, f(z) ≠ 2.

The value of f-1 (1) is

 

(a) x

(b) y

(c) z

(d) none of these

Solution 13

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 14

Which of the following functions from Z to itself are bijections?

 

(a) f(x) = x3

(b)  f(x) = x + 2

(c) f(x) = 2x + 1

(d) f(x) = x2 + x

Solution 14

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 15

Which of the following from A = {x : -1 ≤ x ≤ 1} to itself are bijections?

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 15

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 16

Let A = {x : -1 ≤ x ≤ 1} and f; A → A such that f(x) = x|x|, then f is

 

(a) a bijection

(b) injective but not surjective

(c) surjective but not injective

(d) neither injective nor surjective

Solution 16

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) R

(b) [0, 1]

(c) (0, 1]

(d) [0, 1)

Solution 17

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 18

if a function f : [2, ∞) → B defined by f(x) = x2 – 4x + 5 is a bijection, then B =

 

(a) R

(b) [1, ∞)

(c) [4, ∞)

(d) [5, ∞)

Solution 18

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 19

The function f : R → R defined by f(x) = (x – 1)(x – 2)(x – 3) is

 

(a) one-one but not onto

(b) onto but not one-one

(c) both one and onto

(d) neither one-one nor onto

Solution 19

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) bijection

(b) injection but not a surjection

(c) surjection but not an injection

(d) neither an injection nor a surjection

Solution 20

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) f is a bijection

(b) f is an injection only

(c) f is surjection on only

(d) f is neither an injection nor a surjection

Solution 21

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) f is one-one onto

(b) f is one-one into

(c) f is many one onto

(d) f is many one into

Solution 22

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) one-one but not onto

(b) one-one and onto

(c) onto but not one-one

(d) neither one-one nor onto

Solution 23

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) one-one but not onto

(b) many-one but onto

(c) one-one and onto

(d) neither one-one nor onto

Solution 24

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 25

The function f ; R → R, f(x) = x2 is

 

(a) injective but not surjective

(b) surjective but not injective

(c) injective as well as surjective

(d) neither injective nor surjective

Solution 25

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) neither one-one nor onto

(b) one-one but not onto

(c) onto but not one-one

(d) one-one and onto both

Solution 26

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 27

Which of the following functions from A = {x ε R : – 1 ≤ x  ≤ 1} to itself are bijections?

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 27

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) onto but not one-one

(b) one-one but not onto

(c) one-one and onto

(d) neither one-one nor onto

Solution 28

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 29

The function f ; R → R defined by f (x) = 6x + 6|x| is

 

(a) one-one and onto

(b) many one and onto

(c) one-one and into

(d) many one and into

Solution 29

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 30

Let f(x) = x2 and g(x) = 2x. Then the solution set of the equation fog (x) = gof (x) is

 

(a) R

(b) {0}

(c) {0, 2}

(d) none of these

Solution 30

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 31

if f : R → R is given by f(x) = 3x – 5, then f-1(x)

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 31

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 32

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 33

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 34

Let A = {x ε R : x ≥ 1}. The inverse of the function f ; A → A given by f(x) = 2x(x – 1), is

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 34

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 35

Let A = {x ε R : x ≤ 1} and f : A → 1 A be defined as f (x) = x(2 – x). Then, f-1 (x) is

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 35

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsANSWER PENDING

Question 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 36

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 37

If the function f: R→R be such that f(x) = x – [x], where [x] denotes the greatest integer less than or equal to x, then f-1(x) is

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 37

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsANSWER PENDING

Question 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 38

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

(a) x

(b) 1

(c) f(x)

(d) g(x)

Solution 39

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 40

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 40

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 41

The distinct linear functions which map [-1, 1] onto [0, 2] are

 

(a) f(x) = x + 1, g(x) = -x + 1

(b) f(x) = x – 1, g(x) = x + 1

(c) f(x) = – x – 1, g(x) = x – 1

(d) none of these

Solution 41

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 42

Let f: [2, ∞) → X be defined by f(x) = 4x – x2. Then, f is invertible, if X=

 

(a) [2, ∞)

(b) ( -∞, 2]

(c) (-∞, 4]

(d) [4, ∞)

Solution 42

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsANSWER PENDING

Question 43

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 43

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 44

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 44

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

(a) 2x – 3

(b) 2x + 3

(c) 2x2 + 3x + 1

(d) 2x2 – 3x – 1

Solution 45

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 46

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 46

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 47

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 47

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 48

Let f(x) = x3 be a function with domain {0, 1, 2, 3}. Then domain of f-1 is

 

(a) {3, 2, 1, 0}

(b) {0, -1, -2, -3}

(c) {0, 1, 8, 27}

(d) {0, -1, -8, -27}

Solution 48

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 49

Let f : R → R be given by f(x) = x2 – 3. Then domain of f-1 is

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 49

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 50

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 50

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 51

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 51

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 52

Let A = {1, 2, …, n} and B = {a, b}. Then the number of subjections  from A into B is

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 52

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 53

If the set A contains 5 elements and the set B 6 elements, then the number of one-one and onto mappings from A to B is

(a) 720

(b) 120

(c) 0

(d) none of these

Solution 53

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - FunctionsRd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 54

If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is

 

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 54

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Question 55

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

Solution 55

Rd-sharma Solutions Cbse Class 12-science Mathematics Chapter - Functions

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