NCERT Exemplar Class 12 Maths Solutions Chapter 5 Continuity and Differentiability

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# NCERT Exemplar Class 12 Maths Solutions Chapter 5 Continuity and Differentiability

Edited By Ravindra Pindel | Updated on Sep 15, 2022 04:55 PM IST | #CBSE Class 12th

NCERT solutions for Class 12 Maths chapter 5 continuity and differentiability-Solving the book of NCERT for Class 12 students is a must. NCERT Class 12 Mathematics book covers everything that will come in the board exams and helps in creating a base of the topics. Therefore, if one wants to earn the topic for exams and also for higher education, solving the NCERT questions is a must. We are here to provide you with all the NCERT Exemplar Class 12 Math solutions chapter 5. This is one of the most crucial chapters that need to be covered by the student to score well. That is why we are trying to make sure that the students have the answers with them for practice and reference.

Also, read - NCERT Class 12 Maths Solutions.

Question:1

Examine the continuity of the function

A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit (RHL at x = c) = f(c).
Mathematically we can present it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
is continuous if -

But we can see that,

Similarly, we proceed for RHL-

Question:2

Find which of the functions is continuous or discontinuous at the indicated points:

Given,

We need to check its continuity at x = 2
A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit (RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 2 if -

Similarly, we proceed for RHL-

Now from equation 2, 3 and 4 we can conclude that

Question:3

Find which of the functions is continuous or discontinuous at the indicated points:

Given,

We need to check its continuity at x = 0
A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit (RHL at x = c) = f(c).
Mathematically we can present it as

As this limit can be evaluated directly by putting value of h because it is taking indeterminate form (0/0)
As we know,

Again, using sandwich theorem, we get -

And,
f (0) = 5 …(4)
From equation 2, 3 and 4 we can conclude that

∴ f(x) is discontinuous at x = 0

Question:4

Find which of the functions is continuous or discontinuous at the indicated points:

at x = 2

Given,

We need to check its continuity at x = 2
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 2 if -

Then,

Similarly, we proceed for RHL-

From the above equation 2 , 3 and 4 we can say that

∴ f(x) is continuous at x = 2

Question:5

Find which of the functions is continuous or discontinuous at the indicated points:

at x = 4

Given,
...(1)
We need to check its continuity at x = 4
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 4 if -

Clearly,

Similarly, we proceed for RHL-

∴ f(x) is discontinuous at x = 4

Question:6

at x = 0

Given,

We need to check its continuity at x = 0
A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit (RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 4 if -

As cos (1/h) is going to be some finite value from -1 to 1 as h→0
∴ LHL = 0 × (finite value) = 0 …(2)
Similarly, we proceed for RHL-

And,
f(0) = 0 {using eqn 1} …(4)
From the above equation 2 , 3 and 4 we can conclude that

∴ f(x) is continuous at x = 0

Question:7

Find which of the functions is continuous or discontinuous at the indicated points:
Check continuity at x =a

Given,

We need to check its continuity at x = a
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = a if -

Clearly,

As sin (-1/h) is going to be some finite value from -1 to 1 as h→0
∴ LHL = 0 × (finite value) = 0 …(2)
Similarly we proceed for RHL-
.
h > 0 as defined above.
∴ |h| = h

As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ RHL = 0 × (finite value) = 0 …(3)
And,
f(a) = 0 {using eqn 1} …(4)
From equation 2, 3 and 4 we can conclude that

Question:8

Given,

We need to check its continuity at x = 0
A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 4 if -

Similarly, we proceed for RHL-

And,
f(0) = 0 {using eqn 1} …(4)
from equation 2 , 3 and 4 we can conclude that

∴ f (x) is discontinuous at x = 0

Question:9

Find which of the functions is continuous or discontinuous at the indicated points:

Given,

We need to check its continuity at x = 1
A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 1 if -

Similarly, we proceed for RHL-

And,

Question:10

Find which of the functions is continuous or discontinuous at the indicated points:
at x = 1

Given,

We need to check its continuity at x = 1
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now according to above theory-
f(x) is continuous at x = 1 if -

Then,

Similarly, we proceed for RHL-

From equation 2,3 and 4 we can conclude that

F(x) is continuous at x=1

Question:11

Find the value of k so that the function f is continuous at the indicated point:

Given,

We need to find the value of k such that f(x) is continuous at x = 5.
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 5.

As we have to find k so pick out a combination so that we get k in our equation.
In this question we take LHL = f(5)

Question:12

Find the value of k so that the function f is continuous at the indicated point:

Given,

We need to find the value of k such that f(x) is continuous at x = 2.
A function f(x) is said to be continuous at x = c if,
Left hand limit (LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 2.

Now to find k we have to pick out a combination so that we get k in our equation.
In this question we take LHL = f(5)

As the limit can't be evaluated directly as it is taking 0 / 0 form.
So, use the formula:

Divide the numerator and denominator by -h to match with the form in formula-

Question:13

Find the value of k so that the function f is continuous at the indicated point:
at x= 0

Given,

We need to find the value of k such that f(x) is continuous at x = 0.
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 0.

Now to find k pick out a combination using which we get k in our equation.
In this question we take LHL = f(0)

We can’t find the limit directly, because it is taking 0/0 form.
thus, we will rationalize it.

Question:14

Find the value of k so that the function f is continuous at the indicated point:
at x= 0

Given,

We need to find the value of k such that f(x) is continuous at x = 0.
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 0.

to find k we have to pick out a combination so that we get k in our equation.
In this question we take LHL = f(0)

As this limit can be evaluated directly by putting value of h because it is taking indeterminate form (0/0)
Thus, we use sandwich or squeeze theorem according to which -

Dividing and multiplying by to match the form in formula we have-

Question:15

Prove that the function f defined by

remains discontinuous at x=0, regardless the choice of k.

Given,

We need to prove that f(x) is discontinuous at x = 0 irrespective of the value of k.
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically, we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now, we need to prove that f(x) is discontinuous at x = 0 irrespective of the value of k
If we show that,

Then there will not be involvement of k in the equation & we can easily prove it.
So let’s take LHL first -

Now Let's find RHL,

From the equation 2 and 3, conclude that
LHL ≠ RHL
Hence,
f(x) is discontinuous at x = 0 irrespective of the value of k.

Question:16

Find the values of a and b such that the function f defined by

is a continuous function at x = 4.

Given,
…(1)
We need to find the value of a & b such that f(x) is continuous at x = 4.
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
Now, let’s assume that f(x) is continuous at x = 4.

to find a & b, we have to pick out a combination so that we get a or b in our equation.
In this question first we take LHL = f(4)

{using equation 1}

h > 0 as defined in theory above.

a - 1 = a + b
b = -1
Now, taking other combination,
RHL = f(4)

{using equation 1}

h > 0 as defined in theory above.
|h| = h

⇒ b + 1 = a + b
∴ a = 1
Hence,
a = 1 and b = -1

Question:17

Given,

we have to find: Points discontinuity of composite function f(f(x))

As f(x) is not defined at x = -2 as denominator becomes 0, at x = -2.

x = -2 is a point of discontinuity

And f(f(x)) is not defined at x = -5/2 as denominator becomes 0, at x = -5/2.
∴ x = -5/2 is another point of discontinuity
Thus f (f(x)) has 2 points of discontinuity at x = -2 and x = -5/2

Question:18

Find all points of discontinuity of the function .

We have to find: Points discontinuity of function f(t) where
As t is not defined at x = 1 as denominator becomes 0, at x = 1.
x = 1 is a point of discontinuity
f(t) =

The f(t) is not going to be defined whenever denominator is 0 and thus will give a point of discontinuity.

∴ Solution of the following equation gives other points of discontinuities.

Hence,
f(t) is discontinuous at x = 1, x = 2 and x = 1/2

Question:19

Show that the function f(x) = |sin x + cos x| is continuous at x = .

Given,

We need to prove that f(x) is continuous at x = π

A function f(x) is said to be continuous at x = c if,

Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).

Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)

Now according to above theory-

f(x) is continuous at x = π if -

Now,

LHL =

⇒ LHL = {using eqn 1}

Similarly, we proceed for RHL-

{using eqn 1}

Now from equation 2, 3 and 4 we can conclude that

∴ f(x) is continuous at x = π is proved

Question:20

Examine the differentiability of f, where f is defined by

.

Given,
…(1)
We need to check whether f(x) is continuous and differentiable at x = 2
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
Left hand derivative (LHD at x = c) = Right hand derivative (RHD at x = c) = f(c).
Mathematically we can represent it as-

Finally, we can state that for a function to be differentiable at x = c

Checking for the continuity:
Now according to above theory-
f(x) is continuous at x = 2 if -

Note: As [.] represents greatest integer function which gives greatest integer less than the number inside [.].
E.g. [1.29] = 1; [-4.65] = -5 ;  = 9
[2-h] is just less than 2 say 1.9999 so [1.999] = 1
⇒ LHL = (2-0) ×1
∴ LHL = 2 …(2)
Similarly,
RHL =
⇒ RHL =

And, f(2) = (2-1)(2) = 2 …(4) {using equation 1}
From equation 2,3 and 4 we observe that:

∴ f(x) is continuous at x = 2. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to above theory-
f(x) is differentiable at x = 2 if -

LHD =
⇒ LHD =
Note: As [.] represents greatest integer function which gives greatest integer less than the number inside [.].
E.g. [1.29] = 1 ; [-4.65] = -5 ;  = 9
[2-h] is just less than 2 say 1.9999 so [1.999] = 1

⇒ LHD =
⇒ LHD =

Now,
RHD =
⇒ RHD =
⇒ RHD =
∴ RHD =

Clearly from equation 5 and 6, we can conclude that-
(LHD at x=2) ≠ (RHD at x = 2)
∴ f(x) is not differentiable at x = 2

Question:21

Examine the differentiability of f, where f is defined by

Given,

We need to check whether f(x) is continuous and differentiable at x = 0
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
Left hand derivative(LHD at x = c) = Right hand derivative(RHD at x = c) = f(c).
Mathematically we can represent it as-

Finally, we can state that for a function to be differentiable at x = c

Checking for the continuity:
Now according to above theory-
f(x) is continuous at x = 0 if -

As sin (-1/h) is going to be some finite value from -1 to 1 as h→0
∴ LHL = × (finite value) = 0
∴ LHL = 0 …(2)
Similarly,

As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ RHL = (finite value) = 0 …(3)
And, f(0) = 0 {using equation 1} …(4)
From equation 2,3 and 4 we observe that:

∴ f(x) is continuous at x = 0. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to above theory-
f(x) is differentiable at x = 0 if -

As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ LHD = 0×(some finite value) = 0
∴ LHD = 0 …(5)
Now,

As sin (1/h) is going to be some finite value from -1 to 1 as h→0
∴ RHD = 0×(some finite value) = 0
∴ RHD = 0 …(6)
Clearly from equation 5 and 6, we can conclude that-
(LHD at x=0) = (RHD at x = 0)
∴ f(x) is differentiable at x = 0

Question:22

Examine the differentiability of f, where f is defined by

Given,

We need to check whether f(x) is continuous and differentiable at x = 2
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-
.
Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
Left hand derivative(LHD at x = c) = Right hand derivative(RHD at x = c) = f(c).
Mathematically we can represent it as-

Finally, we can state that for a function to be differentiable at x = c

Checking for the continuity:
Now according to above theory-
f(x) is continuous at x = 2 if -

And, f(2) = 1 + 2 = 3 {using equation 1} …(4)
From equation 2,3 and 4 we observe that:

∴ f(x) is continuous at x = 2. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to above theory-
f(x) is differentiable at x = 2 if -

Clearly from equation 5 and 6,we can conclude that-
(LHD at x=2) ≠ (RHD at x = 2)
∴ f(x) is not differentiable at x = 2

Question:23

Show that f(x) = |x - 5| is continuous but not differentiable at x = 5.

Given,
f(x) = |x - 5| …(1)
We need to prove that f(x) is continuous but not differentiable at x = 5
A function f(x) is said to be continuous at x = c if,
Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).
Mathematically we can represent it as-

Where h is a very small number very close to 0 (h→0)
And a function is said to be differentiable at x = c if it is continuous there and
Left hand derivative(LHD at x = c) = Right hand derivative(RHD at x = c) = f(c).
Mathematically we can represent it as-

Finally, we can state that for a function to be differentiable at x = c

Checking for the continuity:
Now according to above theory-
f(x) is continuous at x = 5 if -

And, f(5) = |5-5| = 0 {using equation 1} …(4)
From equation 2,3 and 4 we observe that:

∴ f(x) is continuous at x = 5. So we will proceed now to check the differentiability.
Checking for the differentiability:
Now according to above theory
f(x) is differentiable at x = 2 if -

As h > 0 as defined in theory above.
∴ |-h| = h

Now,

As h>0 as defined in theory above.

Clearly from equation 5 and 6,we can conclude that-
(LHD at x=5) ≠ (RHD at x = 5)
∴ f(x) is not differentiable at x = 5 but continuous at x = 5.
Hence proved.

Question:24

A function satisfies the equation f(x + y) = f(x) f(y) for all x, y. Suppose that the function is differentiable at x = 0 and f’(0) = 2. Prove that f’(x) = 2f(x).

Given f(x) is differentiable at x = 0 and f(x) ≠ 0
And f(x + y) = f(x)f(y) also f’(0) = 2
To prove: f’(x) = 2f(x)
As we know that,

As
f(x + y) = f(x)f(y)
put x = y = 0

Question:25

Differentiate each of the following w.r.t. x

Given:
Let Assume
Now, Taking Log on both sides we get,

Now, substitute the value of y

Question:26

Differentiate each of the following w.r.t. x

We have been given
Let us Assume
Now, taking log on both sides, we get

since we know,

since we know,

Now, Differentiate w.r.t x

Hence,

Question:27

Differentiate each of the following w.r.t. x

Given:

Now, Differentiate w.r.t t

And,
Now, differentiate w.r.t x

Now, using chain rule, we get

Substitute the value of t

Question:28

Differentiate each of the following w.r.t. x

Differentiate the given function w.r.t x
Let Assume

Let
Let Assume
Let Assume
Differentiate both side w.r.t x

Now, Bu using chain rule we get, Differentiation of

Hence, This the differentiation of given function.

Question:29

Differentiate each of the following w.r.t. x

Given:

We have

Differentiate w.r.t x

Question:31

Differentiate each of the following w.r.t. x

We have given
Let us Assume
And

Now, differentiate w.r.t v

And,
Now, again differentiate w.r.t. w

And, we know,
So, differentiate w w.r.t. x we get

Now, using chain rule we get,

Substitute the value of v and w

Hence, dy/dy is the differentiation of function.

Question:32

Differentiate each of the following w.r.t. x

Let us Assume
Now, differentiate y w.r.t x

Question:33

Differentiate each of the following w.r.t. x

Given
Let us assume

Substitute the value of t

Hence, this is the differentiation of .

Question:34

Differentiate each of the following w.r.t. x

Given:
To Find: Differentiate w.r.t x
We have
Let
Now, Taking Log on both sides, we get
Log y = cos x.log(sin x)
Now, Differentiate both side w.r.t. x

Substitute the value of y, we get

Question:35

Differentiate each of the following w.r.t. x

It is given

Taking log both side, we get

Differentiate w.r.t x

Question:36

Differentiate each of the following w.r.t. x

We have given,
Let us Assume,
Taking log both side

Differentiate w.r.t x

Question:38

Differentiate each of the following w.r.t. x

We have

Differentiate w.r.t. x

Question:39

Differentiate each of the following w.r.t. x

We have given
Let us Assume (sec x +tan x) =t
So,
Now, differentiate w.r.t t

Question:40

Differentiate each of the following w.r.t. x

We have given

Now, divide by cos x in both numerator and denominator

Question:44

Find of each of the functions expressed in parametric form in

We have given, two parametric equation,

Now, differentiate both equation w.r.t x
We know,
So,

and,

Now,

Question:45

Find of each of the functions expressed in parametric form

We have two equations
Now, differentiate w.r.t θ
So,

Also,

Question:46

We have given,
Now, differentiate both the equation w.r.t. x then we get

Question:47

Find dy/dx of each of the functions expressed in parametric form in

We have given two parametric equation:
Let us Assume t= tan θ

Question:48

Find dy/dx of each of the functions expressed in parametric form

We have given,
Now, differentiate w.r.t t

Also,

Question:49

If and , prove that

Taking log on both sides to get,

Taking log on both sides we get,
log y = sin 2t

Differentiating w.r.t x,

Question:50

If , show that

Differentiate w.r.t t

Also,
y = bcos2t

Now, for dy/dx

Hence Proved.

Question:51

If , find

Differentiate w.r.t t in both equation
x = 3sint - sin3t

Question:52

Differentiate w.r.t. sinx.

Let us Assume,

Question:53

Differentiate w.r.t. when

We have
Let us Assume,

Hence Differentiation of w.r.t. is .

Question:54

Find when x and y are connected by the relation given

We have,

Use chain rule and quotient rule to get:
Chain Rule
f(x) = g(h(x))
f’(x) = g’(h(x))h’(x)
By Quotient Rule

On differentiating both the sides with respect to x, we get

Question:55

Find when x and y are connected by the relation given

We have,
sec(x + y) = xy
By the rules given below:
Chain Rule
f(x) = g(h(x))
f’(x) = g’(h(x))h’(x)
Product rule:

Question:58

If , then show that

Given:
Differentiating the above with respect to x, we get

Now, we again differentiate eq (i) with respect to y, we get,

Now, multiplying Eq. (ii) and (iii), we get

Hence Proved

Question:59

If prove that

Given:
Taking log on both the sides, we get

Hence Proved.

Question:60

If , prove that

Given:.

Taking log on both the sides, we get

Hence Proved

Question:61

If Show that

Given:
⇒ y = (cos x)y
Taking log both the sides, we get
log y = y log(cos x)
On differentiating both the sides, we get

Hence Proved

Question:62

If , prove that .

Given:

Hence Proved.

Question:63

If prove that

Given:

Now, we know that

Now, we use some trigonometry formulas,

Hence Proved

Question:64

Given:

Question:65

Verify the Rolle’s theorem for each of the functions
in [0, 1].

Given:

Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:

On expanding
Since, f(x) is a polynomial and we know that, every polynomial function is continuous for all

Hence, condition 1 is satisfied.
Condition 2:

Since, f(x) is a polynomial and every polynomial function is differentiable for all

Hence, condition 2 is satisfied.
Condition 3:

Hence, f(0) = f(1)
Hence, condition 3 is also satisfied.
Now, let us show that such that f’(c) = 0

On differentiating above with respect to x, we get

Put x = c in above equation, we get

, all the three conditions of Rolle’s theorem are satisfied
f’(c) = 0

On factorising, we get

Thus, Rolle’s theorem is verified.

Question:66

Verify the Rolle’s theorem for each of the functions

Given:
Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:

Since, f(x) is a trigonometric function and trigonometric function is continuous everywhere

Hence, condition 1 is satisfied.
Condition 2:

On differentiating above with respect to x, we get

f(x) is differentiable at
Hence, condition 2 is satisfied.
Condition 3:

Hence, condition 3 is also satisfied.
Now, let us show that c ∈ such that f’(c) = 0

Put x = c in above equation, we get

all the three conditions of Rolle’s theorem are satisfied
f’(c) = 0

or Now, cos 2c = 0

Thus, Rolle’s theorem is verified.

Question:67

Verify the Rolle’s theorem for each of the functions
in [- 1, 1].

Given:
Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:

Since, f(x) is a logarithmic function and logarithmic function is continuous for all values of x.

Hence, condition 1 is satisfied.
Condition 2:

Hence, condition 2 is satisfied.
Condition 3:

Hence, condition 3 is also satisfied.
Now, let us show that such that f(c)=0

Put x=c in above equation, we get

Question:68

Verify the Rolle’s theorem for each of the functions
in [-3, 0].

Given:

Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:

Since, f(x) is multiplication of algebra and exponential function and is defined everywhere in its domain.
is continuous at x ∈ [-3,0]
Hence, condition 1 is satisfied.
Condition 2:
On differentiating f(x) with respect to x, we get,

⇒ f(x) is differentiable at [-3,0]
Hence, condition 2 is satisfied.
Condition 3:

all the three conditions of Rolle’s theorem are satisfied
f’(c) = 0

Thus, Rolle’s theorem is verified.

Question:69

Verify the Rolle’s theorem for each of the functions

Given:
Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:
Firstly, we have to show that f(x) is continuous.
Here, f(x) is continuous because f(x) has a unique value for each x ∈ [-2,2]
Condition 2:
Now, we have to show that f(x) is differentiable

So, f(x) is differentiable on (-2,2)
Hence, Condition 2 is satisfied.
Condition 3:

Now, we have to show that f(a) = f(b)
so, f(a) = f(-2)

and

Hence, condition 3 is satisfied
Now, let us show that

On differentiating above with respect to x, we get

Put x=c in above equation, we get,

Thus, all the three conditions of Rolle's theorem is satisfied. Now we have to see that there exist such that

Hence, Rolle’s theorem is verified.

Question:70

Discuss the applicability of Rolle’s theorem on the function given by

Given:
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:
At x = 1

and

Hence, condition 1 is satisfied.
Condition 2:
Now, we have to check f(x) is differentiable

On differentiating with respect to x, we get
or

Now, let us consider the differentiability of f(x) at x = 1
LHD ⇒ f(x) = 2x = 2(1) = 2
RHD ⇒ f(x) = -1 = -1
LHD ≠ RHD
∴ f(x) is not differentiable at x = 1
Thus, Rolle’s theorem is not applicable to the given function.

Question:71

Find the points on the curve y = (cosx - 1) in [0, ], where the tangent is parallel to x-axis.

Given: Equation of curve, y = cos x - 1
Firstly, we differentiate the above equation with respect to x, we get

Given tangent to the curve is parallel to the x - axis
This means, Slope of tangent = Slope of x - axis

Hence, the tangent to the curve is parallel to the x -axis at
(π, -2)

Question:72

Using Rolle’s theorem, find the point on the curve y = x(x - 4), where the tangent is parallel to x-axis.

Given: y = x(x - 4)

Now, we have to show that f(x) verify the Rolle’s Theorem
First of all, Conditions of Rolle’s theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
c) f(a) = f(b)
If all three conditions are satisfied then there exist some ‘c’ in (a,b) such that f’(c) = 0
Condition 1:
On expanding we get
since, is a polynomial and we know that, every polynomial function is continuous for all

Hence, condition 1 is satisfied.
Condition 2

is differentiable at [0,4]
Hence, condition 2 is satisfied.
Condition 3:

Hence, condition 3 is also satisfied.
Now, there is atleast one value of c ∈ (0,4)
Given tangent to the curve is parallel to the x - axis
This means, Slope of tangent = Slope of x - axis

Hence, the tangent to the curve is parallel to the x -axis at (2, -4).

Question:73

Verify mean value theorem for each of the functions given

Given:
Now, we have to show that f(x) verify the Mean Value Theorem
First of all, Conditions of Mean Value theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that

Here,

On differentiating above with respect to x, we get

Hence, f(x) is differentiable in (1,4)
We know that,
Differentiability Continuity
Hence, f(x) is continuous in (1,4)
Thus, Mean Value Theorem is applicable to the given function
Now,

Now, let us show that there exist c ∈ (0,1) such that

but
So, value of
Thus, Mean Value Theorem is verified.

Question:74

Verify mean value theorem for each of the functions given

Given:
Now, we have to show that f(x) verify the Mean Value Theorem
First of all, Conditions of Mean Value theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that

Condition 1:

Since, f(x) is a polynomial and we know that, every polynomial function is continuous for all x ∈ R
is continuous at x ∈ [0,1]
Hence, condition 1 is satisfied.
Condition 2:

Since, f(x) is a polynomial and every polynomial function is differentiable for all x ∈ R

⇒ f(x) is differentiable at [0,1]
Hence, condition 2 is satisfied.
Thus, Mean Value Theorem is applicable to the given function
Now,

On differentiating above with respect to x, we get

Put x=c in above equation, we get

By Mean Value Theorem,

Thus, Mean Value Theorem is verified.

Question:75

Verify mean value theorem for each of the functions given
in

Given:
f(x) = sinx - sin2x in [0,π]
Now, we have to show that f(x) verify the Mean Value Theorem
First of all, Conditions of Mean Value theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that

Condition 1:
f(x) = sinx - sin 2x
Since, f(x) is a trigonometric function and we know that, sine function are defined for all real values and are continuous for all x ∈ R
⇒ f(x) = sinx - sin 2x is continuous at x ∈ [0,π]
Hence, condition 1 is satisfied.
Condition 2:
f(x) = sinx - sin 2x
f’(x) = cosx - 2 cos2x

⇒ f(x) is differentiable at [0,π]
Hence, condition 2 is satisfied.
Thus, Mean Value Theorem is applicable to the given function
Now,

Now, let us show that there exist c ∈ (0,1) such that

On differentiating above with respect to x, we get

Put x=c in above equation, we get
f’(c) = cos(c) - 2cos2c …(i)
By Mean Value Theorem,

Now, to find the factors of the above equation, we use

Thus, Mean Value Theorem is verified.

Question:76

Verify mean value theorem for each of the functions given
in [1,5]

Given:
Now, we have to show that f(x) verify the Mean Value Theorem
First of all, Conditions of Mean Value theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that

Condition 1:
Firstly, we have to show that f(x) is continuous.
Here, f(x) is continuous because f(x) has a unique value for each x ∈ [1,5]
Condition 2:
Now, we have to show that f(x) is differentiable

∴ f’(x) exists for all x ∈ (1,5)
So, f(x) is differentiable on (1,5)
Hence, Condition 2 is satisfied.
Thus, mean value theorem is applicable to given function.
Now,

Now, we will find f(a) and f(b)

and

Now, let us show that such that

On differentiating above with respect to x, we get

Put x=c in above equation, we get

By Mean Value theorem,

Hence, Mean Value Theorem is verified.

Question:77

Find a point on the curve ,where the tangent is parallel to the chord joining the points (3, 0) and (4, 1).

Given: Equation of curve,
Firstly, we differentiate the above equation with respect to x, we get

Given tangent to the curve is parallel to the chord joining the points (3, Q) and (4,1)
i.e

Put in , we have

Hence, the tangent to the curve is parallel to chord joining the points (3,0) and (4,1) at

Question:78

Using mean value theorem, prove that there is point on the curve between the points A(1, 0) and B(2, 1), where tangent is parallel to the chord AB. Also, find that point.

Given: in [1,2]
Now, we have to show that f(x) verify the Mean Value Theorem
First of all, Conditions of Mean Value theorem are:
a) f(x) is continuous at (a,b)
b) f(x) is derivable at (a,b)
If both the conditions are satisfied, then there exist some ‘c’ in (a,b) such that

Condition 1:

since, f(x) is a polynomial and we know that, every polynomial function is continuous for all
is continuous at
Hence, condition 1 is satisfied.
Condition 2

since, f(x) is a polynomial and every polynomial function is differentiable for all

is differentiable at [1, 2]
Hence, condition 2 is satisfied.
Thus, Mean Value Theorem is applicable to the given function.
Now,

Then, there exist such that

Put x=c in equation, we get

By Mean Value Theorem,

So, value of

Thus, Mean Value Theorem is verified.
Put in given equation we have

Question:79

Find the values of p and q so that

Is differentiable at x = 1.

Given that,

Is differentiable at x = 1.
We know that, f(x) is differentiable at x =1 ⇔ Lf’(1) = Rf’(1).

Since, Lf’(1) = Rf’(1)
∴ 5 = q (i)
Now, we know that if a function is differentiable at a point,it is necessarily continuous at that point.
⇒ f(x) is continuous at x = 1.
⇒ f(1-) = f(1+) = f(1)
⇒ 1+3+p = q+2 = 1+3+p
⇒ p-q = 2-4 = -2
⇒ q-p = 2
Now substituting the value of ‘q’ from (i), we get
⇒ 5-p = 2
⇒ p = 3
∴ p = 3 and q = 5

Question:80

A.If prove that
B. If prove that

A.
We have,

Taking log on both sides, we get
Taking log on both sides, we get

Differentiating both sides w.r.t x, we get

Hence proved.
B.
We have,

Hence Proved

Question:81

If x = sint and y = sin pt, prove that

We have,

Hence proved.

Question:82

Find

We have,

Putting

Taking log on both sides, we get

Differentiating w.r.t x, we get

Now,

Differentiating w.r.t x, we get

Now, y=u+v

Question:83

If f(x) = 2x and g(x) = then which of the following can be a discontinuous function.
A. f(x) + g(x)
B. f(x) - g(x)
C. f(x) . g(x)
D.

We know that if two functions f(x) and g(x) are continuous then {f(x) +g(x)},{f(x)-g(x)}, {f(x).g(x)} and are continuous.
Since, f(x) = 2x and are polynomial functions, they are continuous everywhere.
⇒ {f(x) +g(x)},{f(x)-g(x)}, {f(x).g(x)} are continuous functions.
for,
now, f(x) = 0
⇒ 4x = 0
⇒ x = 0
is discontinuous at x=0.

Question:90

Let . Then
A. f is everywhere differentiable
B. f is everywhere continuous but not differentiable
C. f is everywhere continuous but not differentiable at
D. None of these

B)
Given that,
Let g(x) = sinx and h(x) = |x|
Then, f(x) = hog(x)
We know that, modulus function and sine function are continuous everywhere.
Since, composition of two continuous functions is a continuous function.
Hence, f(x) = hog(x) is continuous everywhere.
Now, v(x)=|x| is not differentiable at x=0.

So, f(x) is not differentiable where
We know that at
Hence, f(x) is everywhere continuous but not differentiable

Question:91

Question:92

If then is equal to

A)

Question:93

The derivative of w.r.t is

A)

Question:94

If , then is
A.
B.
C.
D.

B)

Question:95

A)

Question:96

B.
C. 2
D. None of these

B)
Mean Value Theorem states that, Let f : [a, b] → R be a continuous function on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that

Since, f(x) is a polynomial function it is continuous on [1,3] and differentiable on (1,3).
Now, as per Mean value Theorem, there exists at least one c ∈ (1,3), such that

Question:97

Fill in the blanks in each of the
An example of a function which is continuous everywhere but fails to be differentiable exactly at two points is _______.

Consider, f(x) = |x-1| + |x-2|
Let’s discuss the continuity of f(x).
We have, f(x) = |x-1| + |x-2|

When x<1, we have f(x) = -2x+3, which is a polynomial function and polynomial function is continuous everywhere.
When 1≤x<2, we have f(x) = 1, which is a constant function and constant function is continuous everywhere.
When x≥2, we have f(x) = 2x-3, which is a polynomial function and polynomial function is continuous everywhere.
Hence, f(x) = |x-1| + |x-2| is continuous everywhere.
Let’s discuss the differentiability of f(x) at x=1 and x=2.
We have

⇒ f(x) is not differentiable at x=2.
Thus, f(x) = |x-1| + |X-2| is continuous everywhere but fails to be differentiable exactly at two points x=1 and x=2.

Question:100

Question:102

State True or False for the statements
Rolle’s theorem is applicable for the function f(x) = |x - 1| in [0, 2].

False
As per Rolle’s Theorem, Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b), such that f(a) = f(b), where a and b are some real numbers. Then there exists some c in (a, b) such that f’(c) = 0.
We have, f(x) = |x - 1| in [0, 2].
Since, polynomial and modulus functions are continuous everywhere f(x) is continuous
Now, x-1=0
⇒ x=1
We need to check if f(x) is differentiable at x=1 or not.
We have,

⇒ Lf’(1) ≠ Rf’(1)
⇒ f(x) is not differentiable at x=1.
Hence, Rolle’s theorem is not applicable on f(x) since it is not differentiable at x=1 ∈ (0,2).

Question:103

State True or False for the statements
If f is continuous on its domain D, then |f| is also continuous on D.

True.
Given that, f is continuous on its domain D.
Let a be an arbitrary real number in D. Then f is continuous at a.

Now,

If |f| is continuous at x=a.
since a is an arbitrary point in D. Therefore |f| is continuous in D.

Question:104

State True or False for the statements
The composition of two continuous functions is a continuous function.

True.
Let f be a function defined by f(x) = |1-x + |x||.
Let g(x) = 1-x + |x| and h(x) = |x| be two functions defined on R.
Then,
hog(x) = h(g(x))
⇒ hog(x) = h(1-x + |x|)
⇒ hog(x) = |(1-x + |x|)|
⇒ hog(x) = f(x) ∀ x ∈ R.
Since (1-x) is a polynomial function and |X| is modulus function are continuous in R.
⇒ g(x) = 1-x + |x| is everywhere continuous.
⇒ h(x) = |x| is everywhere continuous.
Hence, f = hog is everywhere continuous.

Question:106

State True or False for the statements
If f .g is continuous at x = a, then f and g are separately continuous at x = a.

False
Let and
which is a constant function and continuous everywhere.
But, is discontinuous at

Students can use NCERT Exemplar Class 12 Math solutions chapter 5 PDF download, prepared by experts, for better understanding of concepts and topics of probability. The topics and subtopics are mentioned below.

## Subtopics of NCERT Exemplar Class 12 Math Solutions Chapter 5 Continuity and Differentiability

The sub-topics covered in this chapter are:

• Introduction
• Continuity
• Algebra of continuous functions
• Differentiability
• Derivatives of composite functions
• Derivatives of implicit functions
• Derivatives of inverse trigonometric functions
• Logarithmic and exponential functions
• Logarithmic differentiation
• Derivatives of functions in parametric forms
• Second-order derivative
• Mean value theorem

## What will you learn from NCERT Exemplar Class 12 Math Solutions Chapter 5?

In Class 12 Math NCERT exemplar solutions chapter 5, students will learn about continuity, differential, limits and their properties in detail. Other than learning about continuity and discontinuity in detail, one will also cover Heine's definition, Cauchy's theorem, and algebra of functions. In a separate subtopic of differentiability, one will learn about differentiability concept, fundamentals, derivatives and its types, relations between differentiability and continuity. NCERT exemplar solutions for Class 12 Math chapter 5 will also cover implicit functions, composite functions, inverse trigonometric functions and its derivatives.

NCERT exemplar class 12 Math solutions chapter 5 also includes logarithmic and exponential functions, their concept and their differentiability. The topics also cover logarithmic differentiation in detail and all its fundamentals. One will also get a closer look at parametric forms of functions and their derivatives and how to solve the questions. Other than all this, one will learn two of the most major advanced calculus theorems, Mean Value Theorem and Rolle's Theorem.

We have a team of highly experienced teachers, who have solved the NCERT exemplar Class 12 Math solutions chapter 5 continuity and differentiability. Our teachers have solved the questions in the simplest way; so that every student can understand the answer in a single get-go. The answers are exhaustive, meaning every step is mentioned as per the exam requirement. No shortcuts and no tricks are used in solving the questions, as per the NCERT syllabus. One will learn some of the most crucial theorems of calculus mathematics. Better understanding will help in higher education and also in getting a clear idea about advanced calculus.

## NCERT Exemplar Class 12 Maths Solutions

 Chapter 1 Relations and Functions Chapter 2 Inverse Trigonometric Functions Chapter 3 Matrices Chapter 4 Determinants Chapter 6 Applications of Derivatives Chapter 7 Integrals Chapter 8 Applications of Integrals Chapter 9 Differential Equations Chapter 10 Vector Algebra Chapter 11 Three dimensional Geometry Chapter 12 Linear Programming Chapter 13 Probability

## Important Topics To Cover for NCERT Exemplar Class 12 Math Solutions Chapter 5 Continuity and Differentiability

Continuity and differentiability is a chapter, which every student should be well prepared for. NCERT exemplar Class 12 Math chapter 5 solutions not only helps in understanding calculus better but also helps in other subjects like physics. This chapter explains the base of the continuity function and differentiability function in detail.

In NCERT exemplar Class 12 Math solutions chapter 5, students will learn that Continuity is characteristic of the function. This characteristic shows that the function can have an unbroken and continuous wave. Differentiability, on the other hand, is a function, which shows that the derivative of the domain exists at every single point. Differentiability shows that there can be a slope to the graph at each point.

### NCERT Exemplar Class 12 Solutions

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### Also, check NCERT Solutions for questions given in the book:

 Chapter 1 Relations and Functions Chapter 2 Inverse Trigonometric Functions Chapter 3 Matrices Chapter 4 Determinants Chapter 5 Continuity and Differentiability Chapter 6 Application of Derivatives Chapter 7 Integrals Chapter 8 Application of Integrals Chapter 9 Differential Equations Chapter 10 Vector Algebra Chapter 11 Three Dimensional Geometry Chapter 12 Linear Programming Chapter 13 Probability

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Yes, you can download the NCERT exemplar Class 12 Maths chapter 5 solutions from the link given in the page.

2. Are these solutions helpful for board examinations?

Yes, the NCERT exemplar solutions for Class 12 Maths chapter 5 are prepared to help students prepare well for board exams.

3. Do I have to make notes of this chapter?

You can prepare notes for this chapter by highlighting or underlining the important points which will make it easier for you to read for quick revision.

4. How many questions are there in this chapter?

The chapter has only 1 exercise with 107 questions in total that are all problem based with variable weightage. ## Upcoming School Exams

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### Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

hello,

Yes you can appear for the compartment paper again since CBSE gives three chances to a candidate to clear his/her exams so you still have two more attempts. However, you can appear for your improvement paper for all subjects but you cannot appear for the ones in which you have failed.

Good Luck

Hello dear,

If you was not able to clear 1st compartment and now you giving second compartment so YES, you can go for your improvement exam next year but if a student receives an improvement, they are given the opportunity to retake the boards as a private candidate the following year, but there are some requirements. First, the student must pass all of their subjects; if they received a compartment in any subject, they must then pass the compartment exam before being eligible for the improvement.

As you can registered yourself as private candidate for giving your improvement exam of 12 standard CBSE(Central Board of Secondary Education).For that you have to wait for a whole year which is bit difficult for you.

Positive side of waiting for whole year is you have a whole year to preparing yourself for your examination. You have no distraction or something which may causes your failure in the exams. In whole year you have to stay focused on your 12 standard examination for doing well in it. By this you get a highest marks as a comparison of others.

Believe in Yourself! You can make anything happen

All the very best.

Hello Student,

I appreciate your Interest in education. See the improvement is not restricted to one subject or multiple subjects  and  we cannot say if improvement in one subject in one year leads to improvement in more subjects in coming year.

You just need to have a revision of all subjects what you have completed in the school. have a revision and practice of subjects and concepts helps you better.

All the best.

If you'll do hard work then by hard work of 6 months you can achieve your goal but you have to start studying for it dont waste your time its a very important year so please dont waste it otherwise you'll regret.

Yes, you can take admission in class 12th privately there are many colleges in which you can give 12th privately.

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

 Option 1) Option 2) Option 3) Option 4) A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

 Option 1) 2.45×10−3 kg Option 2)  6.45×10−3 kg Option 3)  9.89×10−3 kg Option 4) 12.89×10−3 kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

 Option 1) Option 2) Option 3) Option 4) A particle is projected at 600   to the horizontal with a kinetic energy . The kinetic energy at the highest point

 Option 1) Option 2) Option 3) Option 4) In the reaction, Option 1) at STP  is produced for every mole consumed Option 2) is consumed for ever produced Option 3) is produced regardless of temperature and pressure for every mole Al that reacts Option 4) at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, will contain 0.25 mole of oxygen atoms?

 Option 1) 0.02 Option 2) 3.125 × 10-2 Option 3) 1.25 × 10-2 Option 4) 2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

 Option 1) decrease twice Option 2) increase two fold Option 3) remain unchanged Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

 Option 1) Molality Option 2) Weight fraction of solute Option 3) Fraction of solute present in water Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

 Option 1) twice that in 60 g carbon Option 2) 6.023 × 1022 Option 3) half that in 8 g He Option 4) 558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

 Option 1) less than 3 Option 2) more than 3 but less than 6 Option 3) more than 6 but less than 9 Option 4) more than 9
##### Bio Medical Engineer

The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.

4 Jobs Available

Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.

4 Jobs Available
##### GIS Expert

GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.

3 Jobs Available
##### Database Architect

If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi

3 Jobs Available
##### Ethical Hacker

A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.

3 Jobs Available
##### Data Analyst

The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.

Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.

3 Jobs Available
##### Geothermal Engineer

Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.

3 Jobs Available
##### Geotechnical engineer

The role of geotechnical engineer starts with reviewing the projects needed to define the required material properties. The work responsibilities are followed by a site investigation of rock, soil, fault distribution and bedrock properties on and below an area of interest. The investigation is aimed to improve the ground engineering design and determine their engineering properties that include how they will interact with, on or in a proposed construction.

The role of geotechnical engineer in mining includes designing and determining the type of foundations, earthworks, and or pavement subgrades required for the intended man-made structures to be made. Geotechnical engineering jobs are involved in earthen and concrete dam construction projects, working under a range of normal and extreme loading conditions.

3 Jobs Available
##### Finance Executive

A career as a Finance Executive requires one to be responsible for monitoring an organisation's income, investments and expenses to create and evaluate financial reports. His or her role involves performing audits, invoices, and budget preparations. He or she manages accounting activities, bank reconciliations, and payable and receivable accounts.

3 Jobs Available
##### Investment Banker

An Investment Banking career involves the invention and generation of capital for other organizations, governments, and other entities. Individuals who opt for a career as Investment Bankers are the head of a team dedicated to raising capital by issuing bonds. Investment bankers are termed as the experts who have their fingers on the pulse of the current financial and investing climate. Students can pursue various Investment Banker courses, such as Banking and Insurance, and Economics to opt for an Investment Banking career path.

3 Jobs Available
##### Bank Branch Manager

Bank Branch Managers work in a specific section of banking related to the invention and generation of capital for other organisations, governments, and other entities. Bank Branch Managers work for the organisations and underwrite new debts and equity securities for all type of companies, aid in the sale of securities, as well as help to facilitate mergers and acquisitions, reorganisations, and broker trades for both institutions and private investors.

3 Jobs Available
##### Treasurer

Treasury analyst career path is often regarded as certified treasury specialist in some business situations, is a finance expert who specifically manages a company or organisation's long-term and short-term financial targets. Treasurer synonym could be a financial officer, which is one of the reputed positions in the corporate world. In a large company, the corporate treasury jobs hold power over the financial decision-making of the total investment and development strategy of the organisation.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Underwriter

An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.

3 Jobs Available
##### Bank Probationary Officer (PO)

A career as Bank Probationary Officer (PO) is seen as a promising career opportunity and a white-collar career. Each year aspirants take the Bank PO exam. This career provides plenty of career development and opportunities for a successful banking future. If you have more questions about a career as  Bank Probationary Officer (PO), what is probationary officer or how to become a Bank Probationary Officer (PO) then you can read the article and clear all your doubts.

3 Jobs Available
##### Operations Manager

Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.

3 Jobs Available
##### Transportation Planner

A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.

3 Jobs Available
##### Conservation Architect

A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.

2 Jobs Available
##### Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Structural Engineer

A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.

2 Jobs Available
##### Architect

Individuals in the architecture career are the building designers who plan the whole construction keeping the safety and requirements of the people. Individuals in architect career in India provides professional services for new constructions, alterations, renovations and several other activities. Individuals in architectural careers in India visit site locations to visualize their projects and prepare scaled drawings to submit to a client or employer as a design. Individuals in architecture careers also estimate build costs, materials needed, and the projected time frame to complete a build.

2 Jobs Available
##### Landscape Architect

Having a landscape architecture career, you are involved in site analysis, site inventory, land planning, planting design, grading, stormwater management, suitable design, and construction specification. Frederick Law Olmsted, the designer of Central Park in New York introduced the title “landscape architect”. The Australian Institute of Landscape Architects (AILA) proclaims that "Landscape Architects research, plan, design and advise on the stewardship, conservation and sustainability of development of the environment and spaces, both within and beyond the built environment". Therefore, individuals who opt for a career as a landscape architect are those who are educated and experienced in landscape architecture. Students need to pursue various landscape architecture degrees, such as M.Des, M.Plan to become landscape architects. If you have more questions regarding a career as a landscape architect or how to become a landscape architect then you can read the article to get your doubts cleared.

2 Jobs Available
##### Plumber

An expert in plumbing is aware of building regulations and safety standards and works to make sure these standards are upheld. Testing pipes for leakage using air pressure and other gauges, and also the ability to construct new pipe systems by cutting, fitting, measuring and threading pipes are some of the other more involved aspects of plumbing. Individuals in the plumber career path are self-employed or work for a small business employing less than ten people, though some might find working for larger entities or the government more desirable.

2 Jobs Available
##### Orthotist and Prosthetist

Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.

6 Jobs Available
##### Pathologist

A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.

5 Jobs Available
##### Veterinary Doctor

A veterinary doctor is a medical professional with a degree in veterinary science. The veterinary science qualification is the minimum requirement to become a veterinary doctor. There are numerous veterinary science courses offered by various institutes. He or she is employed at zoos to ensure they are provided with good health facilities and medical care to improve their life expectancy.

5 Jobs Available
##### Gynaecologist

Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.

4 Jobs Available
##### Oncologist

An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.

3 Jobs Available
##### Surgical Technologist

When it comes to an operation theatre, there are several tasks that are to be carried out before as well as after the operation or surgery has taken place. Such tasks are not possible without surgical tech and surgical tech tools. A single surgeon cannot do it all alone. It’s like for a footballer he needs his team’s support to score a goal the same goes for a surgeon. It is here, when a surgical technologist comes into the picture. It is the job of a surgical technologist to prepare the operation theatre with all the required equipment before the surgery. Not only that, once an operation is done it is the job of the surgical technologist to clean all the equipment. One has to fulfil the minimum requirements of surgical tech qualifications.

3 Jobs Available
##### Critical Care Specialist

A career as Critical Care Specialist is responsible for providing the best possible prompt medical care to patients. He or she writes progress notes of patients in records. A Critical Care Specialist also liaises with admitting consultants and proceeds with the follow-up treatments.

2 Jobs Available
##### Ophthalmologist

Individuals in the ophthalmologist career in India are trained medically to care for all eye problems and conditions. Some optometric physicians further specialize in a particular area of the eye and are known as sub-specialists who are responsible for taking care of each and every aspect of a patient's eye problem. An ophthalmologist's job description includes performing a variety of tasks such as diagnosing the problem, prescribing medicines, performing eye surgery, recommending eyeglasses, or looking after post-surgery treatment.

2 Jobs Available
##### Actor

For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.

4 Jobs Available

Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.

A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.

3 Jobs Available
##### Acrobat

Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.

3 Jobs Available
##### Video Game Designer

Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages. Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.

3 Jobs Available
##### Talent Agent

The career as a Talent Agent is filled with responsibilities. A Talent Agent is someone who is involved in the pre-production process of the film. It is a very busy job for a Talent Agent but as and when an individual gains experience and progresses in the career he or she can have people assisting him or her in work. Depending on one’s responsibilities, number of clients and experience he or she may also have to lead a team and work with juniors under him or her in a talent agency. In order to know more about the job of a talent agent continue reading the article.

If you want to know more about talent agent meaning, how to become a Talent Agent, or Talent Agent job description then continue reading this article.

3 Jobs Available
##### Videographer

Careers in videography are art that can be defined as a creative and interpretive process that culminates in the authorship of an original work of art rather than a simple recording of a simple event. It would be wrong to portrait it as a subcategory of photography, rather photography is one of the crafts used in videographer jobs in addition to technical skills like organization, management, interpretation, and image-manipulation techniques. Students pursue Visual Media, Film, Television, Digital Video Production to opt for a videographer career path. The visual impacts of a film are driven by the creative decisions taken in videography jobs. Individuals who opt for a career as a videographer are involved in the entire lifecycle of a film and production.

2 Jobs Available
##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Visual Communication Designer

Individuals who want to opt for a career as a Visual Communication Designer will work in the graphic design and arts industry. Every sector in the modern age is using visuals to connect with people, clients, or customers. This career involves art and technology and candidates who want to pursue their career as visual communication designer has a great scope of career opportunity.

2 Jobs Available
##### Copy Writer

In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.

5 Jobs Available
##### Journalist

Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.

3 Jobs Available
##### Publisher

For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.

3 Jobs Available
##### Vlogger

In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion. Ever since internet cost got reduced the viewership for these types of content has increased on a large scale. Therefore, the career as vlogger has a lot to offer. If you want to know more about the career as vlogger, how to become a vlogger, so on and so forth then continue reading the article. Students can visit Jamia Millia Islamia, Asian College of Journalism, Indian Institute of Mass Communication to pursue journalism degrees.

3 Jobs Available
##### Editor

Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.

3 Jobs Available

Advertising managers consult with the financial department to plan a marketing strategy schedule and cost estimates. We often see advertisements that attract us a lot, not every advertisement is just to promote a business but some of them provide a social message as well. There was an advertisement for a washing machine brand that implies a story that even a man can do household activities. And of course, how could we even forget those jingles which we often sing while working?

2 Jobs Available
##### Photographer

Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.

2 Jobs Available
##### Social Media Manager

A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available

A Team Leader is a professional responsible for guiding, monitoring and leading the entire group. He or she is responsible for motivating team members by providing a pleasant work environment to them and inspiring positive communication. A Team Leader contributes to the achievement of the organisation’s goals. He or she improves the confidence, product knowledge and communication skills of the team members and empowers them.

2 Jobs Available
##### Quality Systems Manager

A Quality Systems Manager is a professional responsible for developing strategies, processes, policies, standards and systems concerning the company as well as operations of its supply chain. It includes auditing to ensure compliance. It could also be carried out by a third party.

2 Jobs Available
##### Merchandiser

A career as a merchandiser requires one to promote specific products and services of one or different brands, to increase the in-house sales of the store. Merchandising job focuses on enticing the customers to enter the store and hence increasing their chances of buying a product. Although the buyer is the one who selects the lines, it all depends on the merchandiser on how much money a buyer will spend, how many lines will be purchased, and what will be the quantity of those lines. In a career as merchandiser, one is required to closely work with the display staff in order to decide in what way a product would be displayed so that sales can be maximised. In small brands or local retail stores, a merchandiser is responsible for both merchandising and buying.

2 Jobs Available
##### Procurement Manager

The procurement Manager is also known as  Purchasing Manager. The role of the Procurement Manager is to source products and services for a company. A Procurement Manager is involved in developing a purchasing strategy, including the company's budget and the supplies as well as the vendors who can provide goods and services to the company. His or her ultimate goal is to bring the right products or services at the right time with cost-effectiveness.

2 Jobs Available
##### Production Planner

Individuals who opt for a career as a production planner are professionals who are responsible for ensuring goods manufactured by the employing company are cost-effective and meets quality specifications including ensuring the availability of ready to distribute stock in a timely fashion manner.

2 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### .NET Developer

.NET Developer Job Description: A .NET Developer is a professional responsible for producing code using .NET languages. He or she is a software developer who uses the .NET technologies platform to create various applications. Dot NET Developer job comes with the responsibility of  creating, designing and developing applications using .NET languages such as VB and C#.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### DevOps Architect

A DevOps Architect is responsible for defining a systematic solution that fits the best across technical, operational and and management standards. He or she generates an organised solution by examining a large system environment and selects appropriate application frameworks in order to deal with the system’s difficulties.

2 Jobs Available
##### Cloud Solution Architect

Individuals who are interested in working as a Cloud Administration should have the necessary technical skills to handle various tasks related to computing. These include the design and implementation of cloud computing services, as well as the maintenance of their own. Aside from being able to program multiple programming languages, such as Ruby, Python, and Java, individuals also need a degree in computer science.

2 Jobs Available