NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability

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

Edited By Ramraj Saini | Updated on Sep 14, 2023 07:52 PM IST | #CBSE Class 12th

## NCERT Continuity And Differentiability Class 12 Questions And Answers

NCERT Solutions for Class 12 Maths Chapter 5 are provided here. These NCERT solutions are created bu expert team at careers360 considering the latest syllabus of CBSE 2023-24. Questions based on the topics like continuity, differentiability, and relations between them are covered in the NCERT solutions for class 12 maths chapter 5. In NCERT Class 12 maths book, there are 48 solved examples to understand the concepts of continuity and differentiability class 12. If you are finding difficulties in solving them, you can take help from NCERT maths chapter 5 class 12 solutions. In NCERT class 11 Maths solutions, you have already learned the differentiation of certain functions like polynomial functions and trigonometric functions. In this chapter, you will get NCERT solutions for class 12 maths chapter 5 continuity and differentiability. If you are interested in the chapter 5 class 12 maths NCERT solutions then you can check NCERT solutions for class 12 other subjects.

## Continuity And Differentiability Class 12 NCERT Solutions - Important Formulae

>> Continuity: A function f(x) is continuous at a point x = a if:

• f(a) exists (finite, definite, and real).

• lim(x → a) f(x) exists.

• lim(x → a) f(x) = f(a).

>> Discontinuity: f(x) is discontinuous in an interval if it is discontinuous at any point in that interval.

Algebra of Continuous Functions:

Sum, difference, product, and quotient of continuous functions are continuous.

Differentiation:

The derivative of f(x) at x = a, denoted as f'(a), represents the slope of the tangent line to the graph.

Chain Rule:

If f = v o u, where t = u(x), and if both dt/dx and dv/dx exist, then: df/dx = dv/dt * dt/dx.

Derivatives of Some Standard Functions:

• d/dx(xn) = nxn-1

• d/dx(sin x) = cos x

• d/dx(cos x) = -sin x

• d/dx(tan x) = sec2 x

• d/dx(cot x) = -csc2 x

• d/dx(sec x) = sec x * tan x

• d/dx(csc x) = -csc x * cot x

• d/dx(ax) = ax * ln(a)

• d/dx(ex) = ex

• d/dx(ln x) = 1/x

Mean Value Theorem:

Mean Value Theorem states that if f(x) is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that: f'(c) = (f(b) - f(a)) / (b - a).

Rolle's Theorem:

Rolle's Theorem states that if f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists some c in (a, b) such that f'(c) = 0.

Lagrange's Mean Value Theorem:

Lagrange's Mean Value Theorem states that if f(x) is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a).

Free download Continuity And Differentiability Class 12 NCERT Solutions for CBSE Exam.

## NCERT Continuity And Differentiability Class 12 Questions And Answers (Intext Questions and Exercise)

NCERT Continuity And Differentiability Class 12 Solutions : Excercise: 5.1

Given function is

Hence, function is continous at x = 0

Hence, function is continous at x = -3

Hence, function is continuous at x = 5

Question:2 . Examine the continuity of the function

Given function is

at x = 3

Hence, function is continous at x = 3

Given function is

Our function is defined for every real number say k
and value at x = k ,
and also,

Hence, the function is continuous at every real number

Question:3 b) Examine the following functions for continuity.

Given function is

For every real number k ,
We get,

Hence, function continuous for every real value of x,

Question:3 c) Examine the following functions for continuity.

Given function is

For every real number k ,
We gwt,

Hence, function continuous for every real value of x ,

Question:3 d) Examine the following functions for continuity.

Given function is

for x > 5 , f(x) = x - 5
for x < 5 , f(x) = 5 - x
SO, different cases are their
case(i) x > 5
for every real number k > 5 , f(x) = x - 5 is defined

Hence, function f(x) = x - 5 is continous for x > 5

case (ii) x < 5
for every real number k < 5 , f(x) = 5 - x is defined

Hence, function f(x) = 5 - x is continous for x < 5

case(iii) x = 5
for x = 5 , f(x) = x - 5 is defined

Hence, function f(x) = x - 5 is continous for x = 5

Hence, the function is continuous for each and every real number

GIven function is

the function is defined for all positive integer, n

Hence, the function is continuous at x = n, where n is a positive integer

Given function is

function is defined at x = 0 and its value is 0

Hence , given function is continous at x = 0

given function is defined for x = 1
Now, for x = 1 Right-hand limit and left-hand limit are not equal

R.H.L L.H.L.
Therefore, given function is not continous at x =1
Given function is defined for x = 2 and its value at x = 2 is 5

Hence, given function is continous at x = 2

Given function is

given function is defined for every real number k
There are different cases for the given function
case(i) k > 2

Hence, given function is continuous for each value of k > 2

case(ii) k < 2

Hence, given function is continuous for each value of k < 2

case(iii) x = 2

Right hand limit at x= 2 Left hand limit at x = 2
Therefore, x = 2 is the point of discontinuity

Given function is

GIven function is defined for every real number k
Different cases are their
case (i) k < -3

Hence, given function is continuous for every value of k < -3

case(ii) k = -3

Hence, given function is continous for x = -3

case(iii) -3 < k < 3

Hence, for every value of k in -3 < k < 3 given function is continous

case(iv) k = 3

Hence . x = 3 is the point of discontinuity

case(v) k > 3

Hence, given function is continuous for each and every value of k > 3

Given function is

if x > 0 ,
if x < 0 ,
given function is defined for every real number k
Now,
case(i) k < 0

Hence, given function is continuous for every value of k < 0
case(ii) k > 0

Hence, given function is continuous for every value of k > 0
case(iii) x = 0

Hence, 0 is the only point of discontinuity

Given function is

if x < 0 ,
Now, for any value of x, the value of our function is -1
Therefore, the given function is continuous for each and every value of x
Hence, no point of discontinuity

Given function is

given function is defined for every real number k
There are different cases for the given function
case(i) k > 1

Hence, given function is continuous for each value of k > 1

case(ii) k < 1

Hence, given function is continuous for each value of k < 1

case(iii) x = 1

Hence, at x = 2 given function is continuous
Therefore, no point of discontinuity

Given function is

given function is defined for every real number k
There are different cases for the given function
case(i) k > 2

Hence, given function is continuous for each value of k > 2

case(ii) k < 2

Hence, given function is continuous for each value of k < 2

case(iii) x = 2

Hence, given function is continuous at x = 2
There, no point of discontinuity

Given function is

given function is defined for every real number k
There are different cases for the given function
case(i) k > 1

Hence, given function is continuous for each value of k > 1

case(ii) k < 1

Hence, given function is continuous for each value of k < 1

case(iii) x = 1

Hence, x = 1 is the point of discontinuity

Question:13. Is the function defined by

Given function is

given function is defined for every real number k
There are different cases for the given function
case(i) k > 1

Hence, given function is continuous for each value of k > 1

case(ii) k < 1

Hence, given function is continuous for each value of k < 1

case(iii) x = 1

Hence, x = 1 is the point of discontinuity

Given function is

GIven function is defined for every real number k
Different cases are their
case (i) k < 1

Hence, given function is continous for every value of k < 1

case(ii) k = 1

Hence, given function is discontinous at x = 1
Therefore, x = 1 is he point od discontinuity

case(iii) 1 < k < 3

Hence, for every value of k in 1 < k < 3 given function is continous

case(iv) k = 3

Hence. x = 3 is the point of discontinuity

case(v) k > 3

Hence, given function is continous for each and every value of k > 3
case(vi) when k < 3

Hence, for every value of k in k < 3 given function is continous

Given function is

Given function is satisfies for the all real values of x
case (i) k < 0

Hence, function is continuous for all values of x < 0

case (ii) x = 0

L.H.L at x= 0

R.H.L. at x = 0

L.H.L. = R.H.L. = f(0)
Hence, function is continuous at x = 0

case (iii) k > 0

Hence , function is continuous for all values of x > 0

case (iv) k < 1

Hence , function is continuous for all values of x < 1

case (v) k > 1

Hence , function is continuous for all values of x > 1

case (vi) x = 1

Hence, function is not continuous at x = 1

Given function is

GIven function is defined for every real number k
Different cases are their
case (i) k < -1

Hence, given function is continuous for every value of k < -1

case(ii) k = -1

Hence, given function is continous at x = -1

case(iii) k > -1

Hence, given function is continous for all values of x > -1

case(vi) -1 < k < 1

Hence, for every value of k in -1 < k < 1 given function is continous

case(v) k = 1

Hence.at x =1 function is continous

case(vi) k > 1

Hence, given function is continous for each and every value of k > 1
case(vii) when k < 1

Hence, for every value of k in k < 1 given function is continuous

Therefore, continuous at all points

Given function is

For the function to be continuous at x = 3 , R.H.L. must be equal to L.H.L.

For the function to be continuous

Given function is

For the function to be continuous at x = 0 , R.H.L. must be equal to L.H.L.

For the function to be continuous

Hence, for no value of function is continuous at x = 0

For x = 1

Hence, given function is continuous at x =1

Given function is

Given is defined for all real numbers k

Hence, by this, we can say that the function defined by is discontinuous at all integral points

Question:20. Is the function defined by continuous at x = ?

Given function is

Clearly, Given function is defined at x =

Hence, the function defined by continuous at x =

Given function is

Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x) + h(x) = sin x + cos x is also a continuous function

Question:21. b) Discuss the continuity of the following functions:

Given function is

Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x) - h(x) = sin x - cos x is also a continuous function

Question:21 c) Discuss the continuity of the following functions:

Given function is

Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x).h(x) = sin x .cos x is also a continuous function

We, know that if two function g(x) and h(x) are continuous then

Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if

Hence, the function is a continuous function
We proved independently that sin x and cos x is a continous function
So, we can say that
cosec x = is also continuous except at
sec x = is also continuous except at
cot x = is also continuous except at

Question:23. Find all points of discontinuity of f, where

Given function is

Hence, the function is continuous
Therefore, no point of discontinuity

Given function is

Given function is defined for all real numbers k
when x = 0

Hence, function is continuous at x = 0
when

Hence, the given function is continuous for all points

Given function is

Given function is defined for all real number
We, know that if two function g(x) and h(x) are continuous then g(x)+h(x) , g(x)-h(x) , g(x).h(x) allare continuous
Lets take g(x) = sin x and h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
Now,
h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
We proved independently that sin x and cos x is continous function
So, we can say that
f(x) = g(x) - h(x) = sin x - cos x is also a continuous function

When x = 0

Hence, function is also continuous at x = 0

Given function is

When

For the function to be continuous

Therefore, the values of k so that the function f is continuous is 6

Given function is

When x = 2
For the function to be continuous
f(2) = R.H.L. = LH.L.

Hence, the values of k so that the function f is continuous at x= 2 is

Given function is

When x =
For the function to be continuous
f( ) = R.H.L. = LH.L.

Hence, the values of k so that the function f is continuous at x= is

Given function is

When x = 5
For the function to be continuous
f(5) = R.H.L. = LH.L.

Hence, the values of k so that the function f is continuous at x= 5 is

Given continuous function is

The function is continuous so

By solving equation (i) and (ii)
a = 2 and b = 1
Hence, values of a and b such that the function defined by is a continuous function is 2 and 1 respectively

Given function is

given function is defined for all real values of x
Let x = k + h
if

Hence, the function is a continuous function

Given function is

given function is defined for all values of x
f = g o h , g(x) = |x| and h(x) = cos x
Now,

g(x) is defined for all real numbers k
case(i) k < 0

Hence, g(x) is continuous when k < 0

case (ii) k > 0

Hence, g(x) is continuous when k > 0

case (iii) k = 0

Hence, g(x) is continuous when k = 0
Therefore, g(x) = |x| is continuous for all real values of x
Now,
h(x) = cos x
Let suppose x = c + h
if

Hence, function is a continuous function
g(x) is continuous , h(x) is continuous
Therefore, f(x) = g o h is also continuous

Question:33 . Examine that sin | x| is a continuous function.

Given function is
f(x) = sin |x|
f(x) = h o g , h(x) = sin x and g(x) = |x|
Now,

g(x) is defined for all real numbers k
case(i) k < 0

Hence, g(x) is continuous when k < 0

case (ii) k > 0

Hence, g(x) is continuous when k > 0

case (iii) k = 0

Hence, g(x) is continuous when k = 0
Therefore, g(x) = |x| is continuous for all real values of x
Now,
h(x) = sin x
Let suppose x = c + h
if

Hence, function is a continuous function
g(x) is continuous , h(x) is continuous
Therefore, f(x) = h o g is also continuous

Given function is

Let g(x) = |x| and h(x) = |x+1|
Now,

g(x) is defined for all real numbers k
case(i) k < 0

Hence, g(x) is continuous when k < 0

case (ii) k > 0

Hence, g(x) is continuous when k > 0

case (iii) k = 0

Hence, g(x) is continuous when k = 0
Therefore, g(x) = |x| is continuous for all real values of x

Now,

g(x) is defined for all real numbers k
case(i) k < -1

Hence, h(x) is continuous when k < -1

case (ii) k > -1

Hence, h(x) is continuous when k > -1

case (iii) k = -1

Hence, h(x) is continuous when k = -1
Therefore, h(x) = |x+1| is continuous for all real values of x
g(x) is continuous and h(x) is continuous
Therefore, f(x) = g(x) - h(x) = |x| - |x+1| is also continuous

NCERT class 12 maths chapter 5 question answer: Excercise: 5.2

Given function is

when we differentiate it w.r.t. x.
Lets take . then,

(By chain rule)

Now,

Given function is

Lets take then,

( By chain rule)

Now,

Given function is

when we differentiate it w.r.t. x.
Lets take . then,

(By chain rule)

Now,

Given function is

when we differentiate it w.r.t. x.
Lets take . then,

take . then,

(By chain rule)

Now,

Given function is

We know that,

and
Lets take
Then,

(By chain rule)

-(i)
Similarly,

-(ii)
Now, put (i) and (ii) in

Given function is

Differentitation w.r.t. x is

Lets take
Our functions become,
and
Now,
( By chain rule)

-(i)
Similarly,

-(ii)
Put (i) and (ii) in

Give function is

Let's take
Now, take

Differentiation w.r.t. x
-(By chain rule)

So,
( Multiply and divide by and multiply and divide by )

Let us assume :

Differentiating y with respect to x, we get :

or

or

Given function is

We know that any function is differentiable when both
and are finite and equal
Required condition for function to be differential at x = 1 is

Now, Left-hand limit of a function at x = 1 is

Right-hand limit of a function at x = 1 is

Now, it is clear that
R.H.L. at x= 1 L.H.L. at x= 1
Therefore, function is not differentiable at x = 1

Given function is

We know that any function is differentiable when both
and are finite and equal
Required condition for function to be differential at x = 1 is

Now, Left-hand limit of the function at x = 1 is

Right-hand limit of the function at x = 1 is

Now, it is clear that
R.H.L. at x= 1 L.H.L. at x= 1 and L.H.L. is not finite as well
Therefore, function is not differentiable at x = 1
Similary, for x = 2
Required condition for function to be differential at x = 2 is

Now, Left-hand limit of the function at x = 2 is

Right-hand limit of the function at x = 1 is

Now, it is clear that
R.H.L. at x= 2 L.H.L. at x= 2 and L.H.L. is not finite as well
Therefore, function is not differentiable at x = 2

NCERT class 12 maths chapter 5 question answer: Exercise: 5.3

Question:1. Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:2. Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:3. Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:4. Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:5. Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:6 Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:7 . Find dy/dx in the following:

Given function is

Now, differentiation w.r.t. x is

Question:8. Find dy/dx in the following:

Given function is

We can rewrite it as

Now, differentiation w.r.t. x is

Question:9 Find dy/dx in the following:

Given function is

Lets consider
Then,

Now,

Our equation reduces to

Now, differentiation w.r.t. x is

Question:10. Find dy/dx in the following:

Given function is

Lets consider
Then,

Now,

Our equation reduces to

Now, differentiation w.r.t. x is

Question:11. Find dy/dx in the following:

Given function is

Let's consider
Then,

Now,

Our equation reduces to

Now, differentiation w.r.t. x is

Question:12 . Find dy/dx in the following:

Given function is

We can rewrite it as

Let's consider
Then,

Now,

Our equation reduces to

Now, differentiation w.r.t. x is

Question:13. Find dy/dx in the following:

Given function is

We can rewrite it as

Let's consider
Then,

Now,

Our equation reduces to

Now, differentiation w.r.t. x is

Question:14 . Find dy/dx in the following:

Given function is

Lets take
Then,

And

Now, our equation reduces to

Now, differentiation w.r.t. x

Question:15 . Find dy/dx in the following:

Given function is

Let's take
Then,

And

Now, our equation reduces to

Now, differentiation w.r.t. x

NCERT class 12 maths chapter 5 question answer: Exercise 5.4

Question:1. Differentiate the following w.r.t. x:

Given function is

We differentiate with the help of Quotient rule

Question:2 . Differentiate the following w.r.t. x:

Given function is

Let
Then,

Now, differentiation w.r.t. x
-(i)

Put this value in our equation (i)

Question:3 . Differentiate the following w.r.t. x:

Given function is

Let
Then,

Now, differentiation w.r.t. x
-(i)

Put this value in our equation (i)

Question:4. Differentiate the following w.r.t. x:

Given function is

Let's take
Now, our function reduces to

Now,
-(i)
And

Put this value in our equation (i)

Question:5 . Differentiate the following w.r.t. x:

Given function is

Let's take
Now, our function reduces to

Now,
-(i)
And

Put this value in our equation (i)

Question:6 . Differentiate the following w.r.t. x:

Given function is

Now, differentiation w.r.t. x is

Question:7 . Differentiate the following w.r.t. x:

Given function is

Lets take
Now, our function reduces to

Now,
-(i)
And

Put this value in our equation (i)

Question:8 Differentiate the following w.r.t. x:

Given function is

Lets take
Now, our function reduces to

Now,
-(i)
And

Put this value in our equation (i)

Question:9. Differentiate the following w.r.t. x:

Given function is

We differentiate with the help of Quotient rule

Question:10. Differentiate the following w.r.t. x:

Given function is

Lets take
Then , our function reduces to

Now, differentiation w.r.t. x is
-(i)
And

Put this value in our equation (i)

Class 12 Maths Chapter 5 NCERT solutions: Exercise: 5.5

Question:1 Differentiate the functions w.r.t. x.

Given function is

Now, take log on both sides

Now, differentiation w.r.t. x

Question:2. Differentiate the functions w.r.t. x.

Given function is

Take log on both the sides

Now, differentiation w.r.t. x is

Question:3 Differentiate the functions w.r.t. x.

Given function is

take log on both the sides

Now, differentiation w.r.t x is

Question:4 Differentiate the functions w.r.t. x.

Given function is

Let's take
take log on both the sides

Now, differentiation w.r.t x is

Similarly, take
Now, take log on both sides and differentiate w.r.t. x

Now,

Question:5 Differentiate the functions w.r.t. x.

Given function is

Take log on both sides

Now, differentiate w.r.t. x we get,

Question:6 Differentiate the functions w.r.t. x.

Given function is

Let's take
Now, take log on both sides

Now, differentiate w.r.t. x
we get,

Similarly, take
Now, take log on both sides

Now, differentiate w.r.t. x
We get,

Now,

Question:7 Differentiate the functions w.r.t. x.

Given function is

Let's take
Now, take log on both the sides

Now, differentiate w.r.t. x
we get,

Similarly, take
Now, take log on both sides

Now, differentiate w.r.t. x
We get,

Now,

Question:8 Differentiate the functions w.r.t. x.

Given function is

Lets take
Now, take log on both the sides

Now, differentiate w.r.t. x
we get,

Similarly, take
Now, differentiate w.r.t. x
We get,

Now,

Given function is Now, take
Now, take log on both sides

Now, differentiate it w.r.t. x
we get,

Similarly, take
Now, take log on both the sides

Now, differentiate it w.r.t. x
we get,

Now,

Question:10 Differentiate the functions w.r.t. x.

Given function is

Take
Take log on both the sides

Now, differentiate w.r.t. x
we get,

Similarly,
take
Now. differentiate it w.r.t. x
we get,

Now,

Question:11 Differentiate the functions w.r.t. x.

Given function is

Let's take
Now, take log on both sides

Now, differentiate w.r.t. x
we get,

Similarly, take
Now, take log on both the sides

Now, differentiate w.r.t. x
we get,

Now,

Given function is

Now, take
take log on both sides

Now, differentiate w.r.t x
we get,

Similarly, take
Now, take log on both sides

Now, differentiate w.r.t. x
we get,

Now,

Given function is

Now, take
take log on both sides

Now, differentiate w.r.t x
we get,

Similarly, take
Now, take log on both sides

Now, differentiate w.r.t. x
we get,

Now,

Given function is

Now, take log on both the sides

Now, differentiate w.r.t x

By taking similar terms on the same side
We get,

Given function is

Now, take take log on both the sides

Now, differentiate w.r.t x

By taking similar terms on same side
We get,

Given function is

Take log on both sides

NOW, differentiate w.r.t. x

Therefore,
Now, the vale of is

Given function is

Now, we need to differentiate using the product rule

Given function is

Multiply both to obtain a single higher degree polynomial

Now, differentiate w.r.t. x
we get,

Given function is

Now, take log on both the sides

Now, differentiate w.r.t. x
we get,

And yes they all give the same answer

It is given that u, v and w are the functions of x
Let
Now, we differentiate using product rule w.r.t x
First, take
Now,
-(i)
Now, again by the product rule

Put this in equation (i)
we get,

Hence, by product rule we proved it

Now, by taking the log
Again take
Now, take log on both sides

Now, differentiate w.r.t. x
we get,

Hence, we proved it by taking the log

Class 12 Maths Chapter 5 NCERT solutions: Exercise:5.6

Given equations are

Now, differentiate both w.r.t t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t
We get,

Similarly,

Now,

Given equations are

Now, differentiate both w.r.t
We get,

Similarly,

Now,

Question:11 If , show that

Given equations are

differentiating with respect to x

Class 12 Maths Chapter 5 NCERT solutions: Exercise: 5.7

Given function is

Now, differentiation w.r.t. x

Now, second order derivative

Therefore, the second order derivative is

Given function is

Now, differentiation w.r.t. x

Now, the second-order derivative is

Therefore, second-order derivative is

Given function is

Now, differentiation w.r.t. x

Now, the second-order derivative is

Therefore, the second-order derivative is

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is

Therefore, second order derivative is

Given function is

Now, differentiation w.r.t. x

Now, the second-order derivative is

Therefore, the second-order derivative is

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is

Therefore, second order derivative is

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is

Therefore, second order derivative is

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is

Therefore, second order derivative is

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is

Therefore, second order derivative is

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is
Using Quotient rule

Therefore, second order derivative is

Question:11 If prove that

Given function is

Now, differentiation w.r.t. x

Now, the second-order derivative is

Now,

Hence proved

Question:12 If Find in terms of y alone.

Given function is

Now, differentiation w.r.t. x

Now, second order derivative is
-(i)
Now, we want in terms of y

Now, put the value of x in (i)

Question:13 If , show that

Given function is

Now, differentiation w.r.t. x

-(i)
Now, second order derivative is
By using the Quotient rule

-(ii)
Now, from equation (i) and (ii) we will get
Now, we need to show

Put the value of from equation (i) and (ii)

Hence proved

Question:14 If , show that

Given function is

Now, differentiation w.r.t. x
-(i)
Now, second order derivative is
-(ii)
Now, we need to show

Put the value of from equation (i) and (ii)

Hence proved

Question:15 If , show that

Given function is

Now, differentiation w.r.t. x
-(i)
Now, second order derivative is

-(ii)
Now, we need to show

Put the value of from equation (ii)

Hence, L.H.S. = R.H.S.
Hence proved

Question:16 If show that

Given function is

We can rewrite it as

Now, differentiation w.r.t. x
-(i)
Now, second order derivative is
-(ii)
Now, we need to show

Put value of from equation (i) and (ii)

Hence, L.H.S. = R.H.S.
Hence proved

Question:17 If show that

Given function is

Now, differentiation w.r.t. x
-(i)
Now, the second-order derivative is
By using the quotient rule
-(ii)
Now, we need to show

Put the value from equation (i) and (ii)

Hence, L.H.S. = R.H.S.
Hence proved

Class 12 Maths Chapter 5 NCERT solutions: Excercise: 5.8

Question:1 Verify Rolle’s theorem for the function

According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then their exist a such that
If all these conditions are satisfies then we can verify Rolle's theorem
Given function is

Now, being a polynomial function, is both continuous in [-4,2] and differentiable in (-4,2)
Now,

Similalrly,

Therefore, value of and value of f(x) at -4 and 2 are equal
Now,
According to roll's theorem their is point c , such that
Now,

And
Hence, Rolle's theorem is verified for the given function

According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then their exist a such that
If all these conditions are satisfied then we can verify Rolle's theorem
Given function is

It is clear that Given function is not continuous for each and every point in [5,9]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,

Now,
R.H.L. at x = n ,

We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, the function is not differential in (5,9)
Hence, Rolle's theorem is not applicable for given function ,

According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then their exist a such that
If all these conditions are satisfies then we can verify Rolle's theorem
Given function is

It is clear that Given function is not continuous for each and every point in [-2,2]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,

Now,
R.H.L. at x = n ,

We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (-2,2)
Hence, Rolle's theorem is not applicable for given function ,

According to Rolle's theorem function must be
a ) continuous in given closed interval say [x,y]
b ) differentiable in given open interval say (x,y)
c ) f(x) = f(y)
Then there exist a such that
If all these conditions are satisfied then we can verify Rolle's theorem
Given function is

Now, being a polynomial , function is continuous in [1,2] and differentiable in(1,2)
Now,

And

Therefore,
Therefore, All conditions are not satisfied
Hence, Rolle's theorem is not applicable for given function ,

It is given that
is a differentiable function
Now, f is a differential function. So, f is also a continuous function
We obtain the following results
a ) f is continuous in [-5,5]
b ) f is differentiable in (-5,5)
Then, by Mean value theorem we can say that there exist a c in (-5,5) such that

Now, it is given that does not vanish anywhere
Therefore,

Hence proved

Condition for M.V.T.
If
a ) f is continuous in [a,b]
b ) f is differentiable in (a,b)
Then, there exist a c in (a,b) such that

It is given that
and interval is [1,4]
Now, f is a polynomial function , is continuous in[1,4] and differentiable in (1,4)
And

and

Then, by Mean value theorem we can say that their exist a c in (1,4) such that

Now,

And
Hence, mean value theorem is verified for the function

Condition for M.V.T.
If
a ) f is continuous in [a,b]
b ) f is differentiable in (a,b)
Then, their exist a c in (a,b) such that

It is given that
and interval is [1,3]
Now, f being a polynomial function , is continuous in[1,3] and differentiable in (1,3)
And

and

Then, by Mean value theorem we can say that their exist a c in (1,4) such that

Now,

And
Hence, mean value theorem is varified for following function and is the only point where f '(c) = 0

According to Mean value theorem function
must be
a ) continuous in given closed interval say [a,b]
b ) differentiable in given open interval say (a,b)
Then their exist a such that

If all these conditions are satisfies then we can verify mean value theorem
Given function is

It is clear that Given function is not continuous for each and every point in [5,9]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,

Now,
R.H.L. at x = n ,

We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (5,9)
Hence, Mean value theorem is not applicable for given function ,

Similaly,
Given function is

It is clear that Given function is not continuous for each and every point in [-2,2]
Now, lets check differentiability of f(x)
L.H.L. at x = n ,

Now,
R.H.L. at x = n ,

We can clearly see that R.H.L. is not equal to L.H.L.
Therefore, function is not differential in (-2,2)
Hence, Mean value theorem is not applicable for given function ,

Similarly,
Given function is

Now, being a polynomial , function is continuous in [1,2] and differentiable in(1,2)
Now,

And

Now,

Now,

And
Therefore, mean value theorem is applicable for the function

NCERT class 12 continuity and differentiability ncert solutions Miscellaneous Excercise

Given function is

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

Given function is

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

Given function is

Take, log on both the sides

Now, differentiation w.r.t. x is
By using product rule

Therefore, differentiation w.r.t. x is

Given function is

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

Given function is

Now, differentiation w.r.t. x is
By using the Quotient rule

Therefore, differentiation w.r.t. x is

Given function is

Now, rationalize the [] part

Given function reduces to

Now, differentiation w.r.t. x is

Therefore, differentiation w.r.t. x is

Given function is

Take log on both sides

Now, differentiate w.r.t.

Therefore, differentiation w.r.t x is

Question:8 , for some constant a and b.

Given function is

Now, differentiation w.r.t x

Therefore, differentiation w.r.t x

Question: 9

Given function is

Take log on both the sides

Now, differentiate w.r.t. x

Therefore, differentiation w.r.t x is

Question:10 , for some fixed a > 0 and x > 0

Given function is

Lets take

Now, take log on both sides

Now, differentiate w.r.t x
-(i)
Similarly, take
take log on both the sides

Now, differentiate w.r.t x
-(ii)

Similarly, take
take log on both the sides

Now, differentiate w.r.t x
-(iii)

Similarly, take
take log on both the sides

Now, differentiate w.r.t x
-(iv)
Now,

Put values from equation (i) , (ii) ,(iii) and (iv)

Therefore, differentiation w.r.t. x is

Question: 11

Given function is

take
Now, take log on both the sides

Now, differentiate w.r.t x
-(i)
Similarly,
take
Now, take log on both the sides

Now, differentiate w.r.t x
-(ii)
Now

Put the value from equation (i) and (ii)

Therefore, differentiation w.r.t x is

Question:12 Find dy/dx if

Given equations are

Now, differentiate both y and x w.r.t t independently

And

Now

Therefore, differentiation w.r.t x is

Question:13 Find dy/dx if

Given function is

Now, differentiatiate w.r.t. x

Therefore, differentiatiate w.r.t. x is 0

Question:14 If

Given function is

Now, squaring both sides

Now, differentiate w.r.t. x is

Hence proved

Given function is

- (i)
Now, differentiate w.r.t. x
-(ii)
Now, the second derivative

Now, put values from equation (i) and (ii)

Now,

Which is independent of a and b
Hence proved

Question:16 If , with , prove that

Given function is

Now, Differentiate w.r.t x

Hence proved

Question:17 If and find

Given functions are
and
Now, differentiate both the functions w.r.t. t independently
We get

Similarly,

Now,

Now, the second derivative

Therefore,

Given function is

Now, differentiate in both the cases

And

In both, the cases f ''(x) exist
Hence, we can say that f ''(x) exists for all real x
and values are

Given equation is

We need to show that for all positive integers n
Now,
For ( n = 1)
Hence, true for n = 1
For (n = k)
Hence, true for n = k
For ( n = k+1)

Hence, (n = k+1) is true whenever (n = k) is true
Therefore, by the principle of mathematical induction we can say that is true for all positive integers n

Given function is

Now, differentiate w.r.t. x

Hence, we get the formula by differentiation of sin(A + B)

Consider f(x) = |x| + |x +1|
We know that modulus functions are continuous everywhere and sum of two continuous function is also a continuous function
Therefore, our function f(x) is continuous
Now,
If Lets differentiability of our function at x = 0 and x= -1
L.H.D. at x = 0

R.H.L. at x = 0

R.H.L. is not equal to L.H.L.
Hence.at x = 0 is the function is not differentiable
Now, Similarly
R.H.L. at x = -1

L.H.L. at x = -1

L.H.L. is not equal to R.H.L, so not differentiable at x=-1

Hence, exactly two points where it is not differentiable

Question:22 If , prove that

Given that

We can rewrite it as

Now, differentiate w.r.t x
we will get

Hence proved

Given function is Now, differentiate w.r.t x we will get
-(i)
Now, again differentiate w.r.t x

-(ii)
Now, we need to show that

Put the values from equation (i) and (ii)

Hence proved

If you are looking for continuity and differentiability class 12 NCERT solutions of exercises then these are listed below.

## Topics of NCERT class 12 maths chapter 5 Continuity and Differentiability

5.1 Introduction

5.2 Continuity

5.2.1 Algebra of continuous functions

5.3. Differentiability

5.3.1 Derivatives of composite functions

5.3.2 Derivatives of implicit functions

5.3.3 Derivatives of inverse trigonometric functions

5.4 Exponential and Logarithmic Functions

5.5. Logarithmic Differentiation

5.6 Derivatives of Functions in Parametric Form

The mathematical definition of Continuity and Differentiability -

Let f be a real function and c be a point in the domain of f. Then f is continuous at c if . A function f is differentiable at point c in its domain if it is continuous at point c. A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b].

'Continuity and differentiability' is one of the very important and time-consuming chapters of the NCERT Class 12 maths syllabus. It contains 8 exercises with 121 questions and also 23 questions in the miscellaneous exercise. In this article, you will find all NCERT solutions for class 12 maths chapter 5 continuity and differentiability including miscellaneous exercises.

NCERT exemplar solutions class 12 maths chapter 5

## NCERT class 12 maths ch 5 question answer - Topics

The main topics covered in chapter 5 maths class 12 are:

• Continuity

A function is continuous at a given point if the left-hand limit, right-hand limit and value of function exist and are equal. In this class 12 NCERT topics elaborate concepts related to continuity, point of discontinuity, algebra of continuous function. Continuity and Differentiability class 12 solutions include a comprehensive module of quality questions.

• Differentiability

This ch 5 maths class 12 discuss differentiability concepts of different functions including derivatives of composite functions, derivatives of implicit functions, derivatives of inverse trigonometric functions. To get command on these concepts you can refer to NCERT solutions for class 12 maths chapter 5.

• Exponential and Logarithmic Functions

This ch 5 maths class 12 also includes concepts of exponential and logarithmic functions including natural log and their graphical representation. maths class 12 chapter 5 also contains fundamental properties of the logarithmic function. You can refer to class 12 NCERT solutions for questions about these concepts.

• Logarithmic Differentiation

this class 12 ncert chapter discusses a special technique of differentiation known as logarithmic differentiation. to get command of these concepts you can go through the NCERT solution for class 12 maths chapter 5.

• Derivatives of Functions in Parametric Forms

concepts to differentiate a function which is not implicit and explicit but given in the parametric form are explained in this chapter. Continuity and Differentiability class 12 solutions include problems to understand the concepts.

ch 5 maths class 12 also discuss in detail the concepts of second-order derivative, mean value theorem, Rolle's theorem. for questions on these concepts, you can browse NCERT solutions for class 12 chapter 5.

Topics enumerated in class 12 NCERT are very important and students are suggested to go through all the concepts discussed in the topics. Questions related to all the above topics are covered in the NCERT solutions for class 12 maths chapter 5

## NCERT solutions for class 12 maths - Chapter wise

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

## Tips to Use NCERT Solutions for Class 12 Maths Chapter 5

NCERT solutions for class 12 maths chapter 5 continuity and differentiability are very helpful in the preparation of this chapter. But here are some tips to get command on this chapter.

• You should make sure that concepts related to 'limit' are clear to you as it forms the base for continuity.
• First, go for the theorem and solved examples of continuity given in the NCERT textbook then try to solve exercise questions. You may find some difficulties in solving them. Go through the NCERT solutions for class 12 maths chapter 5 continuity and differentiability, it will help you to understand the concepts in a much easy way.
• This chapter seems very easy but at the same time, the chances of silly mistakes are also high. So, it is advised to understand the theory and concepts properly before practicing questions of NCERT.
• Once you are good in continuity, then go for the differentiability. Practice more and more questions to get command on it.
• Differentiation is mostly formula-based, so practice NCERT questions, it won't take much effort to remember the formulas.

Also check,

Happy learning!!!

1. What are the important topics in chapter Continuity and Differentiability?

Basic concepts of continuity and differentiability, derivatives of composite functions, derivatives of implicit functions, derivatives of inverse trigonometric functions, exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric form are the important topics in this chapter. Practice these class 12 maths ch 5 question answer to command the concepts.

2. What are the reasons to opt for ncert continuity and differentiability solution?

The maths chapter 5 class 12 NCERT solutions created by the experts at Careers360 offer numerous advantages to students preparing for their board exams. These solutions provide comprehensive explanations of each topic, which help students achieve high scores. Additionally, the solutions are based on the latest CBSE syllabus for the 2022-23 academic year. Furthermore, these solutions also assist students in preparing for other competitive exams such as JEE Main and JEE Advanced. For ease, Students can study continuity and differentiability pdf both online and offline

3. Which is the best book for CBSE class 12 maths ?

NCERT is the best book for CBSE class 12 maths. Most of the questions in CBSE board exam are directly asked from NCERT textbook. All you need to do is rigorous practice of all the problems given in the NCERT textbook.

4. How many exercises are included in ncert solutions class 12 maths chapter 5?

According to the given information, there are 8 exercises in NCERT Solutions for maths chapter 5 class 12 . The following is the number of questions in each exercise:

1. Exercise 5.1: 34 questions

2. Exercise 5.2: 10 questions

3. Exercise 5.3: 15 questions

4. Exercise 5.4: 10 questions

5. Exercise 5.5: 18 questions

6. Exercise 5.6: 11 questions

7. Exercise 5.7: 17 questions

8. Exercise 5.8: 6 questions

Additionally, there is a Miscellaneous Exercise with 23 questions.

5. What is the weightage of the chapter Continuity and Differentiability for CBSE board exam ?

Generally, Continuity and differntiability has 9% weightage in the 12th board final examination. if you want to obtain meritious marks or full marks then you should have good command on concepts that can be developed by practice therefore you should practice NCERT solutions and NCERT exercise solutions. ## 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
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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
##### 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
##### 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
##### 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
##### 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
##### 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
##### 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
##### 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
##### 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
##### 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
##### 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
##### 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
##### Chemical Pathologist

Are you searching for a chemical pathologist job description? A chemical pathologist is a skilled professional in healthcare who utilises biochemical laboratory tests to diagnose disease by analysing the levels of various components or constituents in the patient’s body fluid.

2 Jobs Available
##### Biochemical Engineer

A Biochemical Engineer is a professional involved in the study of proteins, viruses, cells and other biological substances. He or she utilises his or her scientific knowledge to develop products, medicines or ways to improve quality and refine processes. A Biochemical Engineer studies chemical functions occurring in a living organism’s body. He or she utilises the observed knowledge to alter the composition of products and develop new processes. A Biochemical Engineer may develop biofuels or environmentally friendly methods to dispose of waste generated by industries.

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
##### 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

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
##### Producer

An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.

They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.

2 Jobs Available
##### Fashion Blogger

Fashion bloggers use multiple social media platforms to recommend or share ideas related to fashion. A fashion blogger is a person who writes about fashion, publishes pictures of outfits, jewellery, accessories. Fashion blogger works as a model, journalist, and a stylist in the fashion industry. In current fashion times, these bloggers have crossed into becoming a star in fashion magazines, commercials, or campaigns.

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
##### 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
##### 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
##### 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
##### Travel Journalist

The career of a travel journalist is full of passion, excitement and responsibility. Journalism as a career could be challenging at times, but if you're someone who has been genuinely enthusiastic about all this, then it is the best decision for you. Travel journalism jobs are all about insightful, artfully written, informative narratives designed to cover the travel industry. Travel Journalist is someone who explores, gathers and presents information as a news article.

2 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
##### SEO Analyst

An SEO Analyst is a web professional who is proficient in the implementation of SEO strategies to target more keywords to improve the reach of the content on search engines. He or she provides support to acquire the goals and success of the client’s campaigns.

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
##### QA Manager

Quality Assurance Manager Job Description: A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes.

2 Jobs Available

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.

2 Jobs Available
##### Reliability Engineer

Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment.

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
##### 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
##### 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