NCERT Solutions Continuity & Differentiability

NCERT Solutions Class 12 Maths Chapter 5 - Continuity & Differentiability

 Table of contents Exercise 5.1 Exercise 5.2 Exercise 5.3 Exercise 5.4 Exercise 5.5 Exercise 5.6 Exercise 5.7

Exercise 5.1

Q1: Prove that the function
Ans:

Therefore, f is continuous at x = 5

Q2: Examine the continuity of the function .
Ans:

Thus, f is continuous at x = 3

Q3: Examine the following functions for continuity.
(a)
(b)
(c)
(d)
Ans:

(c) The given function is
For any real number c ≠ −5, we obtain

Therefore, f is continuous at all real numbers greater than 5.
Hence, f is continuous at every real number and therefore, it is a continuous function.

Q4: Prove that the function   is continuous at x = n, where n is a positive integer.
Ans: The given function is f (x) = xn
It is evident that f is defined at all positive integers, n, and its value at n is nn.

Therefore, f is continuous at n, where n is a positive integer.

Q5: Is the function f defined by continuous at x = 0? At x = 1? At x = 2?
Ans: The given function f is
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.

The right hand limit of f at x = 1 is,

Therefore, f is continuous at x = 2

Q6: Find all points of discontinuity of f, where f is defined by
Ans: The given function f is
It is evident that the given function f is defined at all the points of the real line.
Let c be a point on the real line. Then, three cases arise.
(i) c < 2
(ii) c > 2
(iii) c = 2
Case (i) c < 2

Therefore, f is continuous at all points x, such that x < 2
Case (ii) c > 2

It is observed that the left and right hand limit of f at x = 2 do not coincide.
Therefore, f is not continuous at x = 2
Hence, x = 2 is the only point of discontinuity of f.

Q7: Find all points of discontinuity of f, where f is defined by
Ans: The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x < −3
Case II:

It is observed that the left and right hand limit of f at x = 3 do not coincide.
Therefore, f is not continuous at x = 3
Case V:

Therefore, f is continuous at all points x, such that x > 3
Hence, x = 3 is the only point of discontinuity of f.

Q8: Find all points of discontinuity of f, where f is defined by
Ans: The given function f is

It is observed that the left and right hand limit of f at x = 0 do not coincide.
Therefore, f is not continuous at x = 0
Case III:

Therefore, f is continuous at all points x, such that x > 0
Hence, x = 0 is the only point of discontinuity of f.

Q9: Find all points of discontinuity of f, where f is defined by
Ans: The given function f is

Therefore, the given function is a continuous function.
Hence, the given function has no point of discontinuity.

Q10: Find all points of discontinuity of f, where f is defined by
Ans:
The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x > 1
Hence, the given function f has no point of discontinuity.

Q11: Find all points of discontinuity of f, where f is defined by
Ans: The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x > 2
Thus, the given function f is continuous at every point on the real line.
Hence, f has no point of discontinuity.

Question 12: Find all points of discontinuity of f, where f is defined by
Ans: The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.

Q13: Is the function defined bya continuous function?
Ans: The given function is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observation, it can be concluded that x = 1 is the only point of
discontinuity of f.

Q14: Discuss the continuity of the function f, where f is defined by
Ans: The given function is
The given function is defined at all points of the interval [0, 10].
Let c be a point in the interval [0, 10].
Case I:

Therefore, f is continuous at all points of the interval (1, 3).
Case IV:

Therefore, f is continuous at all points of the interval (3, 10].
Hence, f is not continuous at x = 1 and x = 3

Q15: Discuss the continuity of the function f, where f is defined by
Ans: The given function is
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at all points of the interval (0, 1).
Case IV:

Therefore, f is continuous at all points x, such that x > 1
Hence, f is not continuous only at x = 1

Q16: Discuss the continuity of the function f, where f is defined by
Ans: The given function f is
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:

Therefore, f is continuous at x = −1
Case III:

Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.

Q17: Find the relationship between a and b so that the function f defined by is continuous at x = 3.

Ans: The given function f is
If f is continuous at x = 3, then

Q18: For what value of is the function defined by continuous at x = 0? What about continuity at x = 1?
Ans: The given function f is
If f is continuous at x = 0, then

Therefore, there is no value of λ for which f is continuous at x = 0 At x = 1,
f (1) = 4x + 1 = 4 × 1 + 1 = 5

Therefore, for any values of λ, f is continuous at x = 1

Q19: Show that the function defined by g(x)= x-[x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to x.
Ans: The given function is g(x)= x-[x]
It is evident that g is defined at all integral points.
Let n be an integer.
Then,

It is observed that the left and right hand limits of f at x = n do not coincide.
Therefore, f is not continuous at x = n
Hence, g is discontinuous at all integral points.

Q20: Is the function defined by   continuous at x = p?
Ans: The given function is
It is evident that f is defined at x = p

Therefore, the given function f is continuous at x = π

Q21: Discuss the continuity of the following functions.
(a) f (x) = sin x + cos x
(b) f (x) = sin x − cos x
(c) f (x) = sin x × cos x

Ans: It is known that if g and h are two continuous functions, then
g+h, g-h, and g.h are also continuous.
It has to proved first that g (x) = sin x and h (x) = cos x are continuous functions.
Let g (x) = sin x
It is evident that g (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0

Therefore, g is a continuous function.
Let h (x) = cos x
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0
h (c) = cos c

Therefore, h is a continuous function.
Therefore, it can be concluded that
(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function
(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function
(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function.

Q22: Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
Ans: It is known that if g and h are two continuous functions, then

It has to be proved first that g (x) = sin x and h (x) = cos x are continuous functions. Let g (x) = sin x
It is evident that g (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + h
Therefore, g is a continuous function.
Let h (x) = cos x
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number.
Put x = c + h
If x → c, then h → 0
Therefore, h (x) = cos x is continuous function.

Therefore, cotangent is continuous except at x = np, n Î Z

Q23: Find the points of discontinuity of f, where
Ans:

Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at all points of
the real line.
Thus, f has no point of discontinuity.

Q24:Determine if f defined byis a continuous function?
Ans: The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:

Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at every point of the real line.
Thus, f is a continuous function.

Q25: Examine the continuity of f, where f is defined by
Ans: The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:

Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at every point of
the real line.
Thus, f is a continuous function.

Q26: Find the values of k so that the function f is continuous at the indicated point.

Ans:

Therefore, the required value of k is 6.
Question 27: Find the values of k so that the function f is continuous at the indicated point.

Ans: The given function is
The given function f is continuous at x = 2, if f is defined at x = 2 and if the value of f at x = 2 equals the limit of f at x = 2
It is evident that f is defined at x = 2 and f(2) = k(2)2 = 4k

Q28: Find the values of k so that the function f is continuous at the indicated point.

Ans: The given function is
The given function f is continuous at x = p, if f is defined at x = p and if the value of f at
x = p equals the limit of f at x = p

Q29: Find the values of k so that the function f is continuous at the indicated point.

Ans: The given function f is
The given function f is continuous at x = 5, if f is defined at x = 5 and if the value of f at
x = 5 equals the limit of f at x = 5

Q30: Find the values of a and b such that the function defined by, is a continuous function.
Ans: The given function f is
It is evident that the given function f is defined at all points of the real line.
If f is a continuous function, then f is continuous at all real numbers.
In particular, f is continuous at x = 2 and x = 10
Since f is continuous at x = 2, we obtain

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.

Q31: Show that the function defined by f(x) = cos (x2) is a continuous function.
Ans: The given function is f (x) = cos (x2)
This function f is defined for every real number and f can be written as the composition
of two functions as,
f = g o h, where g (x) = cos x and h (x) = x2

Therefore, g (x) = cos x is continuous function.
h (x) = x2
Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k2

Therefore, h is a continuous function.
It is known that for real-valued functions g and h, such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.
Therefore,   is a continuous function.

Q32: Show that the function defined by  is a continuous function.
Ans: The given function is
This function f is defined for every real number and f can be written as the composition of two functions as,

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

Therefore, g is continuous at all points x, such that x > 0.

Q33: Examine that sin|x| is a continuous function.
Ans:
This function f is defined for every real number and f can be written as the composition
of two functions as,

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

Therefore, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all
points. h (x) = sin x
It is evident that h (x) = sin x is defined for every real number.
Let c be a real number. Put x = c +k
If x → c, then k → 0 h (c) = sin c

Therefore, h is a continuous function.
It is known that for real-valued functions g and h, such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.
is a continuous function.

Q34: Find all the points of discontinuity of f defined by .
Ans: The given function is
The two functions, g and h, are defined as

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

From the above three observations, it can be concluded that g is continuous at all

Case II:

Clearly, h is defined for every real number.
Let c be a real number.
Case I:

Therefore, h is continuous at all points x, such that x < −1
Case III:

Therefore, h is continuous at x = −1
From the above three observations, it can be concluded that h is continuous at all points of the real line. g and h are continuous functions. Therefore, f = g − h is also continuous function.
Therefore, f has no point of discontinuity.

Exercise 5.2

Continuity & Differentiability

Question 1: Differentiate the functions with respect to x.

Question 2: Differentiate the functions with respect to x. cos(sinx)

Thus, f is a composite function of two functions.

Put t = u (x) = sin x

By chain rule,
Alternate method

Question 3: Differentiate the functions with respect to x.

sin(ax + b)

Alternate method

Question 4: Differentiate the functions with respect to x.

Hence, by chain rule, we obtain

Question 5: Differentiate the functions with respect to x.

The given function is

Put y = p (x) = cx + d

Question 6: Differentiate the functions with respect to x.

Question 7:  Differentiate the functions with respect to x.

Question 8: Differentiate the functions with respect to x.

Clearly, f is a composite function of two functions, u and v, such that

Alternate method

Question 9:
Prove that the function f given by  is not differentiable at x = 1.

The given function is

It is known that a function f is differentiable at a point x = c in its domain if both

are finite and equal.

To check the differentiability of the given function at x = 1,

consider the left hand limit of f at x = 1

Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1

Question 10:
Prove that the greatest integer function   defined by is not differentiable at x = 1 and x = 2.

The given function f is
It is known that a function f is differentiable at a point x = c in its domain if both

are finite and equal.

To check the differentiability of the given function at x = 1, consider the left hand limit  of f at x = 1

Since the left and right hand limits of f at x = 1 are not equal, f is not differentiable at x = 1

To check the differentiability of the given function at x = 2, consider the left hand limit of f at x = 2

Since the left and right hand limits of f at x = 2 are not equal, f is not differentiable at x = 2

Exercise 5.3

Continuity & Differentiability

Question 1:

Question 2:

Question 3:

Question 4:

The given relationship is
Differentiating this relationship with respect to x, we obtain

Question 5:

Find

The given relationship is x2 +xy +y2 = 100

Differentiating this relationship with respect to x, we obtain

[Derivative of constant function is 0]

Question 6:

Question 7:

Using chain rule, we obtain

Question 8:

Find

The given relationship is

Differentiating this relationship with respect to x, we obtain

Question 9:

Find

Therefore, by quotient rule, we obtain

Question 10:

Question 11:

The given relationship is,

On comparing L.H.S. and R.H.S. of the above relationship, we obtain  tany/2 = x

Differentiating this relationship with respect to x, we obtain

Question 12:

The given relationship is

From (1), (2), and (3), we obtain

Differentiating this relationship with respect to x, we obtain

Question 13:

Question 14:

Find

Differentiating this relationship with respect to x, we obtain

Question 15:

Find dy/dx

Differentiating this relationship with respect to x, we obtain

Exercise 5.4

Q1: Differentiate the following w.r.t. x:
Ans: Let y =
By using the quotient rule, we obtain

Q2: Differentiate the following w.r.t. x: esin-1x
Ans: Let y = esin-1x
By using the chain rule, we obtain

Q3: Differentiate the following w.r.t. x:
Ans: Let y =
By using the chain rule, we obtain

Q4: Differentiate the following w.r.t. x: sin(tan-1e-x)
Ans: Let y = sin(tan-1e-x)
By using the chain rule, we obtain

Q5: Differentiate the following w.r.t. x: log(cosex)
Ans: Let y = log(cosex)
By using the chain rule, we obtain

Q6: Differentiate the following w.r.t. x:

Ans:

Q7: Differentiate the following w.r.t. x:
Ans: Let

Q8: Differentiate the following w.r.t. x:

Ans: Let y =
By using the chain rule, we obtain

Q9: Differentiate the following w.r.t. x:

Ans: Let y =
By using the quotient rule, we obtain

Q10: Differentiate the following w.r.t. x: , x>0
Ans: Let y =
By using the chain rule, we obtain

Exercise 5.5

Continuity & Differentiability

Question 1: Differentiate the function with respect to x.

Question 2: Differentiate the function with respect to x.

Let y =
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 3: Differentiate the function with respect to x.

Let y =
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 4: Differentiate the function with respect to x. xx - 2sinx

Let y =

u = xx
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 5: Differentiate the function with respect to x.

Let y =
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 6: Differentiate the function with respect to x.

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Therefore, from (1), (2), and (3), we obtain

Question 7: Differentiate the function with respect to x.

Let y =

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 8: Differentiate the function with respect to x.

Let y =

Differentiating both sides with respect to x, we obtain

Therefore, from (1), (2), and (3), we obtain

Question 9: Differentiate the function with respect to x.

Let y =

Differentiating both sides with respect to x, we obtain

Question 10: Differentiate the function with respect to x.

Let y =

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 11: Differentiate the function with respect to x.

Differentiating both sides with respect to x, we obtain

Differentiating both sides with respect to x, we obtain

Question 12:
Find
of function .

Differentiating both sides with respect to x, we obtain

Question 13:

Find

Differentiating both sides with respect to x, we obtain

Question 14:
Find  of function  .

Differentiating both sides, we obtain

Question 15:
Find
of function .

Differentiating both sides with respect to x, we obtain

Question 16:
Find the derivative of the function given by
and
hence find
.

The given relationship is
Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

Question 17:
Differentiate
in three ways mentioned below
(i) By using product rule.
(ii) By expanding the product to obtain a single polynomial.
(iii By logarithmic differentiation.
Do they all give the same answer?

Let y =
(i)

(ii)

( iii)

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to x, we obtain

From the above three observations, it can be concluded that all the results of   are same.

Question 18: If u, v and w are functions of x, then show that

in two ways-first by repeated application of product rule, second by logarithmic
differentiation.

Let
By applying product rule, we obtain

Exercise 5.6

Continuity & Differentiability

Question 1:
If x and y are connected parametrically by the equation, without eliminating the

Question 2: If x and y are connected parametrically by the equation, without eliminating the
parameter, find
.
x = a cos θ, y = b cos θ

The given equations are x = a cos θ and y = b cos θ

Question 3: If x and y are connected parametrically by the equation, without eliminating the
parameter, find
.
x = sin t, y = cos 2t

The given equations are x = sin t and y = cos 2t

Question 4:
If x and y are connected parametrically by the equation, without eliminating the

.

Question 5: If x and y are connected parametrically by the equation, without eliminating the

The given equations are

Question 6: If x and y are connected parametrically by the equation, without eliminating the

The given equations are

Question 7: If x and y are connected parametrically by the equation, without eliminating the

parameter,  find  .

The given equations are

Question 8: If x and y are connected parametrically by the equation, without eliminating the

parameter, find

The given equations are

Question 9: If x and y are connected parametrically by the equation, without eliminating the
parameter, find
.

Question 10: If x and y are connected parametrically by the equation, without eliminating the

parameter, find  .

Question 11:
If

Hence, proved.

Exercise 5.7

Continuity & Differentiability

Question 1: Find the second order derivatives of the function.

Let y =
Then,

Question 2: Find the second order derivatives of the function. x20

Let y = x20
Then,

Question 3: Find the second order derivatives of the function. x.cos x

Let y = x.cos x
Then,

Question 4: Find the second order derivatives of the function. log x

Let y = log x
Then,

Question 5: Find the second order derivatives of the function. x3 log x

Let y = x3 log x
Then,

Question 6: Find the second order derivatives of the function.

Let y =

Question 7: Find the second order derivatives of the function.

Let y =
Then,

Question 8: Find the second order derivatives of the function.

Let y =
Then,

Question 9: Find the second order derivatives of the function.

Let y =
Then,

Question 10: Find the second order derivatives of the function.

Let y =
Then,

Question 11: If

It is given that,
Then,

Hence, proved.

Question 12:
If
in terms of y alone.

It is given that,
Then,

Question 13:
If

It is given that,
Then,

Hence, proved.

Question 14:
If
, show that

It is given that,
Then,

Hence, proved.

Question 15:
If
, show that

It is given that,
Then,

Hence, proved.

Question 16:
If
, show that

The given relationship is

Taking logarithm on both the sides, we obtain

Differentiating this relationship with respect to x, we obtain

Hence, proved.

Question 17:
If
, show that

The given relationship is
Then,

Hence, proved.

The document NCERT Solutions Class 12 Maths Chapter 5 - Continuity & Differentiability is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on NCERT Solutions Class 12 Maths Chapter 5 - Continuity & Differentiability

 1. What is the definition of continuity in calculus?
Ans. Continuity in calculus refers to a function that is uninterrupted and connected on its entire domain without any breaks, jumps, or holes. Mathematically, a function f(x) is continuous at a point x = a if the limit of f(x) as x approaches a exists and is equal to f(a).
 2. How can we determine if a function is continuous at a specific point?
Ans. To determine if a function is continuous at a specific point, we need to check three conditions: 1. The function must be defined at that point. 2. The limit of the function as x approaches the point must exist. 3. The value of the function at that point must be equal to the limit.
 3. What is the importance of continuity and differentiability in calculus?
Ans. Continuity and differentiability are essential concepts in calculus as they help us understand the behavior of functions and their derivatives. Continuity ensures that a function is smooth and connected, while differentiability indicates the existence of the derivative at a point, which provides information about the rate of change of the function.
 4. How does the concept of continuity relate to the concept of differentiability?
Ans. Continuity is a necessary condition for differentiability. If a function is differentiable at a point, it must also be continuous at that point. However, continuity does not guarantee differentiability, as a function can be continuous without having a derivative at a specific point (e.g., sharp corners or cusps).
 5. Can a function be differentiable without being continuous?
Ans. No, a function cannot be differentiable at a point if it is not continuous at that point. Differentiability implies continuity, so if a function has a derivative at a specific point, it must also be continuous at that point.

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