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NCERT Solutions For Class 12 Maths Chapter 5 – Continuity and Differentiability Class 12 NCERT Solutions

NCERT Solutions For Class 12 Maths Chapter 5 gives detailed solutions for every question. It was prepared by the best teachers in India. This is the advantage of getting a better score on the exam. Continuity and Differentiability class 12 NCERT pdf will help students get a high score in the exam. This article gives step-wise solutions for better understanding. NCERT Solutions For Class 12 Maths Chapter 5 pdf will help the students as well as students who are preparing for JEE Mains, JEE, and other competitive exams. Class 12 Maths solutions were given in detailed explanation for easy understanding. NCERT Solutions For Class 12 Maths gives you the best results. By preparing for chapter 5 you can easily attempt the questions in the exam.

NCERT Solutions For Class 12 Maths Chapter 5 – Continuity and Differentiability

Section Name Topic Name
5.1 Introduction
5.2 Continuity
5.3 Algebra of Continuous Functions
5.4 Exponential and Logarithmic Functions
5.5 Logarithmic Differentiation
5.6 Derivatives of Implicit Functions in Parametric Forms
5.7 Derivatives of Inverse Trigonometric Functions
5.8 Mean Value Theorem

 

Class 12 Maths NCERT Solutions Chapter 5 Continuity and Differentiability Ex 5.1 Introduction

Question 1:
Prove that the function ncert solutions class 12 maths chapter 5 ex 5.1 q 1is continuous at x=0, at x=-3, and at x=5
Answer:
ncert solutions class 12 maths chapter 5 ex 5.1 q 1(a)
Therefore, f is continuous at x = 0
ncert solutions class 12 maths chapter 5 ex 5.1 q 1(b)

Therefore, is continuous at x = −3
ncert solutions class 12 maths chapter 5 ex 5.1 q 1(c)

Therefore, f is continuous at x = 5


Question 2:
Examine the continuity of the function ncert solutions class 12 maths chapter 5 ex 5.1 q 2
Answer:
ncert solutions class 12 maths chapter 5 ex 5.1 q 2(a)

Thus, f is continuous at x = 3


Question 3:

Examine the following functions for continuity.
(a) ncert solutions class 12 maths chapter 5 ex 5.1 q 3(b) ncert solutions class 12 maths chapter 5 ex 5.1 q 3(a)(c) ncert solutions class 12 maths chapter 5 ex 5.1 q 3(b)(d) ncert solutions class 12 maths chapter 5 ex 5.1 q 3(c)
Answer:
(a) The given function is ncert solutions class 12 maths chapter 5 ex 5.1 q 3
It is evident that f is defined at every real number k and its value at k is k − 5.
It is also observed that, ncert solutions class 12 maths chapter 5 ex 5.1 q 3(d)
ncert solutions class 12 maths chapter 5 ex 5.1 q 3(e)
Hence, f is continuous at every real number and therefore, it is a continuous function
(b) The given function is ncert solutions class 12 maths chapter 5 ex 5.1 q 3(a)
For any real number k ≠ 5, we obtain
ncert solutions class 12 maths chapter 5 ex 5.1 q 3(f)

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.
(c) The given function is ncert solutions class 12 maths chapter 5 ex 5.1 q 3(b)

For any real number c ≠ −5, we obtain
ncert solutions class 12 maths chapter 5 ex 5.1 q 3(g)

Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.
(d) The given function is ncert solutions class 12 maths chapter 5 ex 5.1 q 3(h)

This function f is defined at all points of the real line.

Let c be a point on a real line. Then, c < 5 or c = 5 or c > 5

Case I: c < 5

Then, (c) = 5 − c
ncert solutions class 12 maths chapter 5 ex 5.1 q 3(i)

Therefore, f is continuous at all real numbers less than 5.

Case II : c = 5
Then f(c)= f(5)=(5-5)=0
ncert solutions class 12 maths chapter 5 ex 5.1 q 3(j)

Therefore, is continuous at x = 5

Case III: c > 5
ncert solutions class 12 maths chapter 5 ex 5.1 q 3(k)

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.


Question 4:
Prove that the function ncert solutions class 12 maths chapter 5 ex 5.1 q 4is continuous at x = n, where n is a positive integer
Answer:

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.
ncert solutions class 12 maths chapter 5 ex 5.1 q 4(a)

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


Question 5:

Is the function f defined by ncert solutions class 12 maths chapter 5 ex 5.1 q 5

continuous at x = 0? At x = 1? At x = 2?
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 5

At x = 0,

It is evident that f is defined at 0 and its value at 0 is 0.
ncert solutions class 12 maths chapter 5 ex 5.1 q 5(a)

Therefore, f is continuous at x = 0

At x = 1,

is defined at 1 and its value at 1 is 1.

The left hand limit of f at x = 1 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 5(b)

The right hand limit of at x = 1 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 5(c)

Therefore, f is not continuous at x = 1

At = 2,

is defined at 2 and its value at 2 is 5.
ncert solutions class 12 maths chapter 5 ex 5.1 q 5(d)

Therefore, f is continuous at = 2


Question 6:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 6
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 6

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
ncert solutions class 12 maths chapter 5 ex 5.1 q 6(a)

Therefore, f is continuous at all points x, such that x < 2

Case (ii) c > 2
ncert solutions class 12 maths chapter 5 ex 5.1 q 6(b)

Therefore, f is continuous at all points x, such that x > 2

Case (iii) c = 2

Then, the left hand limit of at x = 2 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 6(c)

The right hand limit of f at x = 2 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 6(d)

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.


Question 7:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 7
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 7

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 7(a)

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

Case II:
ncert solutions class 12 maths chapter 5 ex 5.1 q 7(b)

Therefore, f is continuous at x = −3

Case III:
ncert solutions class 12 maths chapter 5 ex 5.1 q 7(c)

Therefore, f is continuous in (−3, 3).

Case IV:

If c = 3, then the left hand limit of at x = 3 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 7(d)

The right hand limit of at x = 3 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 7(e)

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 7(f)

Therefore, f is continuous at all points x, such that x > 3

Hence, x = 3 is the only point of discontinuity of f.


Question 8:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 8
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 8
It is known that, ncert solutions class 12 maths chapter 5 ex 5.1 q 8(a)

Therefore, the given function can be rewritten as
ncert solutions class 12 maths chapter 5 ex 5.1 q 8(b)

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 8(c)

Therefore, f is continuous at all points x < 0

Case II:

If c = 0, then the left hand limit of at x = 0 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 8(d)
The right hand limit of at x = 0 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 8(e)

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 8(f)

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

Hence, x = 0 is the only point of discontinuity of f.


Question 9:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 9
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 9
It is known that ncert solutions class 12 maths chapter 5 ex 5.1 q 9(a)

Therefore, the given function can be rewritten as
ncert solutions class 12 maths chapter 5 ex 5.1 q 9(b)
Let c be any real number. Then, ncert solutions class 12 maths chapter 5 ex 5.1 q 9(c)
Also, ncert solutions class 12 maths chapter 5 ex 5.1 q 9(d)

Therefore, the given function is a continuous function.

Hence, the given function has no point of discontinuity.


Question 10:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 10
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 10

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 10(a)

Therefore, f is continuous at all points x, such that x < 1

Case II:
ncert solutions class 12 maths chapter 5 ex 5.1 q 10(b)

The left hand limit of at x = 1 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 10(c)

The right hand limit of at x = 1 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 10(d)

Therefore, f is continuous at x = 1

Case III:
ncert solutions class 12 maths chapter 5 ex 5.1 q 10(e)

Therefore, f is continuous at all points x, such that x > 1

Hence, the given function has no point of discontinuity.


Question 11:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 11
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 11

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 11(a)

Therefore, f is continuous at all points x, such that x < 2

Case II:
ncert solutions class 12 maths chapter 5 ex 5.1 q 11(b)

Therefore, f is continuous at x = 2

Case III:
ncert solutions class 12 maths chapter 5 ex 5.1 q 11(c)

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, has no point of discontinuity.


Question 12:

Find all points of discontinuity of f, where f is defined by
ncert solutions class 12 maths chapter 5 ex 5.1 q 12
Answer:
The given function f is ncert solutions class 12 maths chapter 5 ex 5.1 q 12

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:
ncert solutions class 12 maths chapter 5 ex 5.1 q 12(a)

Therefore, f is continuous at all points x, such that x < 1

Case II:

If c = 1, then the left hand limit of f at x = 1 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 12(b)

The right hand limit of f at = 1 is,
ncert solutions class 12 maths chapter 5 ex 5.1 q 12(c)

It is observed that the left and right hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:
ncert solutions class 12 maths chapter 5 ex 5.1 q 12(d)

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.


Question 13:
Is the function defined by
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13
a continuous function?
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13

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:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13(a)

Therefore, f is continuous at all points x, such that x < 1

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13(c)

The left hand limit of at x = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13(d)

The right hand limit of f at = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13(e)

It is observed that the left and right-hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 13(f)

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



Question 14:

Discuss the continuity of the function f, where f is defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14
Answer:
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:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(a)

Therefore, f is continuous in the interval [0, 1).

Case II:
If c=1 then f(3)=3

The left hand limit of at x = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(b)

The right hand limit of f at = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(c)

It is observed that the left and right hand limits of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(d)

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

Case IV:
If c=3, then f(c)=5

The left hand limit of at x = 3 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(e)

The right hand limit of f at = 3 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(f)

It is observed that the left and right hand limits of f at x = 3 do not coincide.

Therefore, f is not continuous at x = 3

Case V:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 14(g)

Therefore, f is continuous at all points of the interval (3, 10].

Hence, is not continuous at = 1 and = 3


Question 15:

Discuss the continuity of the function f, where f is defined by

NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(a)

Therefore, f is continuous at all points x, such that x < 0

Case II:
If c=0, then f(c)=f(0)=0

The left hand limit of at x = 0 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(b)

The right hand limit of f at = 0 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(c)

Therefore, f is continuous at x = 0

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(d)

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

Case IV:
If c=1, then f(c)=f(1)=0

The left hand limit of at x = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(e)

The right hand limit of f at = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(f)

It is observed that the left and right hand limits of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1

Case V:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(g)

Therefore, f is continuous at all points x, such that x > 1

Hence, is not continuous only at = 1


Question 16:

Discuss the continuity of the function f, where f is defined by
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(h)
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 15(h)

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16

Therefore, f is continuous at all points x, such that x < −1

Case II:
If c=-1, then f(c)=f(-1)=-2

The left hand limit of at x = −1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16(a)

The right hand limit of f at = −1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16(b)

Therefore, f is continuous at x = −1

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16(c)

Therefore, f is continuous at all points of the interval (−1, 1).

Case IV:
If c=1, then f(c)=f(1)=2×1=2

The left hand limit of at x = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16(d)

The right hand limit of f at = 1 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16(e)

Therefore, f is continuous at x = 2

Case V:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 16(f)

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.


Question 17:
Find the relationship between a and b so that the function f defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 17
is continuous at = 3.
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 17
If f is continuous at x = 3, then
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 17(a)
Therefore, from (1), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 17(b)
Therefore, the required relationship is given by, NCERT Solutions class 12 maths chapter 5 ex 5.1 q 17(c)


Question 18:
For what value of is the function defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 18
continuous at x = 0? What about continuity at x = 1?
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 18
If f is continuous at x = 0, then
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 18(a)

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
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 18(b)
Therefore, for any values of λ, f is continuous at x = 1


Question 19:
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.
Answer:
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,
g(n)=n-[n]=n-n=0
The left hand limit of at x = n is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 19

The right hand limit of f at n is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 19(a)

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.


Question 20:
Is the function defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 20

continuous at =

π?
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 20

It is evident that f is defined at =

π.
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 20(a)
Therefore, the given function f is continuous at = π


Question 21:

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
Answer:
It is known that if and are two continuous functions, then g+ h, g- h, and g.h

are also continuous.

It has to proved first that g (x) = sin and h (x) = cos x are continuous functions.

Let (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 21

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 be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 21(a)

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


Question 22:
Discuss the continuity of the cosine, cosecant, secant and cotangent functions:
Answer:
It is known that if and are two continuous functions, then
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 22

It has to be proved first that g (x) = sin and h (x) = cos x are continuous functions.

Let (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let be a real number. Put x = c + h

If x
c, then h
→ 0
NCRET Solutions class 12 maths chapter 5 ex 5.1 q 22

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 be a real number. Put x = c + h

If x ® c, then h ® 0

(c) = cos c
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 22(a)

Therefore, h (x) = cos x is continuous function.

It can be concluded that,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 22(b)

Therefore, cosecant is continuous except at np, Î Z
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 22(c)
Therefore, secant is continuous except at NCERT Solutions class 12 maths chapter 5 ex 5.1 q 22(d)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 22(e)
Therefore, cotangent is continuous except at np, Î Z



Question 23:

Find the points of discontinuity of f, where
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 23
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 23

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 23(a)

Therefore, f is continuous at all points x, such that x < 0

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 23(b)

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

Case III:
If c=0, then f(c)=f(0)=0+1=1
The left hand limit of f at x = 0 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 23(c)

The right hand limit of f at x = 0 is,
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 23(d)

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.


Question 24:
Determine if f defined by
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 24
is a continuous function?
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 24

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 24(a)

Therefore, f is continuous at all points ≠ 0

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 24(b)
⇒-x2≤x2sin1x≤x2
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 24(c)

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.


Question 25:
Examine the continuity of f, where f is defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 25
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 25

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 25(a)

Therefore, f is continuous at all points x, such that x ≠ 0

Case II:
If c=0, then f(0)=-1
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 25(b)

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.


Question 26:
Find the values of so that the function f is continuous at the indicated point. NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(a)
The given function f is continuous at NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(b)if f is defined at NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(b)and if the value of the f at NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(b)equals the limit of f at NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(b)
It is evident that is defined at NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(b)and NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(c)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(d)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 26(e)

Therefore, the required value of k is 6.


Question 27:
Find the values of so that the function f is continuous at the indicated point. NCERT Solutions class 12 maths chapter 5 ex 5.1 q 27at x=2
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 27
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 is defined at x = 2 and NCERT Solutions class 12 maths chapter 5 ex 5.1 q 27(a)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 27(b)
Therefore, the required value of NCERT Solutions class 12 maths chapter 5 ex 5.1 q 27(c)


Question 28:
Find the values of so that the function f is continuous at the indicated point. NCERT Solutions class 12 maths chapter 5 ex 5.1 q 28at x=n
Answer:
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
It is evident that is defined at x = p and NCERT Solutions class 12 maths chapter 5 ex 5.1 q 28(a)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 28(b)
Therefore, the required value of NCERT Solutions class 12 maths chapter 5 ex 5.1 q 28(c)


Question 29:
Find the values of so that the function f is continuous at the indicated point.
NCERT Solutins class 12 maths chapter 5 ex 5.1 q 29at x=5
Answer:
The given function is NCERT Solutins class 12 maths chapter 5 ex 5.1 q 29
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
It is evident that is defined at x = 5 and NCERT Solutions class 12 maths chapter 5 ex 5.1 q 29(a)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 29(b)
Therefore, the required value of NCERT Solutins class 12 maths chapter 5 ex 5.1 q 29(c)


Question 30:
Find the values of a and b such that the function defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 30
is a continuous function.
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 30

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 = 2 and = 10

Since f is continuous at = 2, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 30(a)
Since f is continuous at = 10, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 30(b)

On subtracting equation (1) from equation (2), we obtain

8a = 16

⇒ a = 2

By putting a = 2 in equation (1), we obtain

2 × 2 + b = 5

⇒ 4 + b = 5

⇒ b = 1

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


Question 31:
Show that the function defined by f (x) = cos (x2) is a continuous function.
Answer:

The given function is (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
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 31

It has to be first proved that (x) = cos x and h (x) = x2 are continuous functions.

It is evident that g is defined for every real number.

Let c be a real number.

Then, g (c) = cos c
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 31(a)

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
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 31(b)

Therefore, h is a continuous function.

It is known that for real valued functions and h, such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.
Therefore, NCERT Solutions class 12 maths chapter 5 ex 5.1 q 31(c) is a continuous function.


Question 32:
Show that the function defined by NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32is a continuous function.
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32
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 NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(a)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(b)
It has to be first proved that NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(a) are continuous functions.
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(c)

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(d)

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

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(e)

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

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(f)

Therefore, g is continuous at x = 0

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

(x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(g)

Therefore, h (x) = cos x is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.
Therefore, NCERT Solutions class 12 maths chapter 5 ex 5.1 q 32(h)
continuous function.


Question 33:
Examine that NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33is a continuous function.
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(a)
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 NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(b)
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(c)
It has to be proved first that NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(b) are continuous functions.
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(d)

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(e)

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

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(f)

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

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(g)

Therefore, g is continuous at x = 0

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

(x) = sin x

It is evident that h (x) = sin x is defined for every real number.

Let be a real number. Put x = c + k

If x → c, then k → 0

(c) = sin c
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(h)

Therefore, h is a continuous function.

It is known that for real valued functions and h,such that (h) is defined at c, if is continuous at and if is continuous at (c), then (g) is continuous at c.
Therefore, NCERT Solutions class 12 maths chapter 5 ex 5.1 q 33(i)
continuous function.


Question 34:
Find all the points of discontinuity of defined by
NCERT Solutins class 12 maths chapter 5 ex 5.1 q 34
Answer:
The given function is NCERT Solutins class 12 maths chapter 5 ex 5.1 q 34

The two functions, g and h, are defined as
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 34(a)

Then, f = − h

The continuity of g and is examined first.
NCERT Solutions class 12 maths cahpter 5 ex 5.1 q 34(b)

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 34(c)

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

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 34(d)

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

Case III:
NCERT Solutions class 12 maths cahpter 5 ex 5.1 q 34(e)

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points
NCERT Solutins class 12 maths cahpter 5 ex 5.1 q 34(f)

Clearly, h is defined for every real number.

Let be a real number.

Case I:
NCERT Solutins class 12 maths cahpter 5 ex 5.1 q 34(g)

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

Case II:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 34(h)

Therefore, h is continuous at all points x, such that x > −1

Case III:
NCERT Solutions class 12 maths chapter 5 ex 5.1 q 34(i)

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, g − is also a continuous function.

Therefore, has no point of discontinuity.


NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.2 Continuity

Question 1:

Differentiate the functions with respect to x.
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 1
Answer:

Let f(x)=sinx2+5, ux=x2+5, and v(t)=sint

 

Then, vou=vux=vx2+5=tanx2+5=f(x)

Thus, f is a composite of two functions.
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 1(a)

Alternate method
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 1(b)


Question 2:
Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 2
Answer:
NCERT Solutions class 12 maths cahpter 5 ex 5.2 q 2(a)

Thus, is a composite function of two functions.

Put t = u (x) = sin x
NCERT Solutions class 12 maths cahpter 5 ex 5.2 q 2(b)
By chain rule, NCERT Solutions class 12 maths cahpter 5 ex 5.2 q 2(c)
Alternate method
NCERT Solutions class 12 maths cahpter 5 ex 5.2 q 2(d)


Question 3:
Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 3
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 3(a)

Thus, is a composite function of two functions, u and v.

Put t = u (x) = ax + b
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 3(b)

Hence, by chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 3(c)

Alternate method
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 3(d)


Question 4:

Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 4
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 4(a)

Thus, is a composite function of three functions, u, v, and w.
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 4(b)

Hence, by chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 4(c)

Alternate method
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 4(d)


Question 5:

Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5(a)where g (x) =
sin (ax + b) and

h (x) = cos (cx d)
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5(b)

∴ is a composite function of two functions, u and v.
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5(c)

Therefore, by chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5(d)

h is a composite function of two functions, p and q.

Put y = p (x) = cx d
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5(e)
Therefore, by chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 5(f)


Question 6:

Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 6
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.2 q 6
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 6(a)


Question 7:

Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 7
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 7(a)



Question 8:

Differentiate the functions with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.2 q 8
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 8(a)

Clearly, is a composite function of two functions, and v, such that
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 8(b)

By using chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 8(c)

Alternate method
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 8(d)


Question 9:

Prove that the function given by NCERT Solutions class 12 maths chapter 5 ex 5.2 q 9is not differentiable at x = 1.
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.2 q 9
It is known that a function f is differentiable at a point x = c in its domain if both
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 9(b)

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
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 9(c)
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 9(d)
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 NCERT Solutions class 12 maths chapter 5 ex 5.2 q 10

is not

differentiable at x = 1 and x = 2.
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.2 q 10
It is known that a function f is differentiable at a point x = c in its domain if both
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 10(a)

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
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 10(b)

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
NCERT Solutions class 12 maths chapter 5 ex 5.2 q 10(c)
Since the left and right hand limits of f at x = 2 are not equal, f is not differentiable at x = 2

NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.3 Algebra of Continuous Functions

Question 1:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 1
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 1(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 1(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 1(b)


Question 2:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 2
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 2(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 2(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 2(b)


Question 3:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 3
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 3(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 3(b)
Using chain rule, we obtain NCERT Solutions class 12 maths chapter 5 ex 5.3 q 3(c)
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 3(d)

From (1) and (2), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 3(e)


Question 4:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 4
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 4(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 4(a)
Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 4(b)


Question 5:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 5
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 5(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 5(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 5(b)[Derivative of constant function is 0] NCERT Solutions class 12 maths chapter 5 ex 5.3 q 5(c)


Question 6:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 6
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 6(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 6(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 6(b)


Question 7:
Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 7
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 7(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 7(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 7(b)

Using chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 7(c)

From (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 7(d)


Question 8:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 8
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 8(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 8(a)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 8(b)


Question 9:
Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 9
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 9(a)
Answer:
We have,y = sin-12×1 + x2put x = tan θ ⇒ θ = tan-1xNow,    y = sin-12 tan θ1 + tan2θ⇒y = sin-1sin 2θ, as sin 2θ=2 tan θ1 + tan2θ⇒y = 2θ,  as sin-1sin x=x⇒y = 2 tan-1x⇒dydx = 2 × 11 + x2, because dtan-1xdx=11 + x2⇒dydx = 21 + x2


Question 10

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10(b)
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10(d)…..(1)

It is known that, NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10(e)

Comparing equations (1) and (2), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10(f)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 10(g)


Question 11:
Find NCERT Solutions class 12 maths chapter 5 ex 5.3 q 11
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 11(a)
Answer:

The given relationship is,
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 11(b)
On comparing L.H.S. and R.H.S. of the above relationship, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 11(c)

Differentiating this relationship with respect to x, we obtain

sec2y2.ddxy2=ddxx

 

⇒sec2y2×12dydx=1

 

⇒dydx=2sec2y2

 

⇒dydx=21+tan2y2

dydx=21+x2


Question 12:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(b)
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(c)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(d)

Using chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(e)

From (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(f)

Alternate method
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(g)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 12(h)


Question 13:

Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 13
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 13(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 13(b)
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 13(c)
Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 13(d)


Question 14:
Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 14
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 14(a)
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 14(b)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 14(c)


Question 15:
Find
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 15
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.3 q 15(a)
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 15(b)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.3 q 15(c)

NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.4 Exponential and Logarithmic Functions

Question 1:

Differentiate the following w.r.t. x:
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 1
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.4 q 1

By using the quotient rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 1(a)


Question 2:

Differentiate the following w.r.t. x: NCERT Solutions class 12 maths chapter 5 ex 5.4 q 2
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.4 q 2

By using the chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 2(a)


Question 3:
Differentiate the following w.r.t. x: NCERT Solutions class 12 maths chapter 5 ex 5.4 q 3
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.4 q 3

By using the chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 3(a)


Question 4:

Differentiate the following w.r.t. x: NCERT Solutions class 12 maths chapter 5 ex 5.4 q 4
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.4 q 4

By using the chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 4(a)


Question 5:

Differentiate the following w.r.t. x: NCERT Solutions class 12 maths chapter 5 ex 5.4 q 5
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.4 q 5

By using the chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 5(a)


Question 6:

Differentiate the following w.r.t. x: NCERT Solutions class 12 maths chapter 5 ex 5.4 q 6
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 6(a)



Question 7: 

Differentiate the following w.r.t. x: NCERT Solutions class 12 chapter 5 ex 5.4 q 7
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.4 q 7(a)
then, NCERT Solutions class 12 maths chapter 5 ex 5.4 q 7(b)

By differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 7(c)


Question 8:

Differentiate the following w.r.t. x:
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 8
Answer:
Let y= log (log x)

By using the chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 8(a)
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 8(b)x>1


Question 9:

Differentiate the following w.r.t. x:
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 9
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.4 q 9(a)

By using the quotient rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 9(b)


Question 10:

Differentiate the following w.r.t. x:
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 10
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.4 q 10(a)

By using the chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.4 q 10(b)


NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.5 Logarithmic Differentiation

Question 1:

Differentiate the function with respect to x.
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 1
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.5 q 1

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 1(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 1(b)


Question 2:

Differentiate the function with respect to x.
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 2
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.5 q 2

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 2(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 2(b)


Question 3:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 3
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.5 q 3

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 3(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 3(b)


Question 4:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 4
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 4(a)

xx

Taking logarithm on both the sides, we obtain
log u= x log x

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 4(b)

v = 2sin x

Taking logarithm on both the sides with respect to x, we obtain
log v=sin x. log 2

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 4(c)


Question 5:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 5
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.5 q 5

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 5(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 5(b)


Question 6:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 6
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 6(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 6(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 6(c)

Therefore, from (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 6(d)


Question 7:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 7
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 7(a)

= (log x)x
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 7(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 7(c)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 7(d)

Therefore, from (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 7(e)


Question 8:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 8
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 8(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 8(b)

Therefore, from (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 8(c)


Question 9:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 9
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 9(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 9(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 9(c)

From (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 9(d)


Question 10:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 10
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.5 q 10
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 10(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 10(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 10(c)

From (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 10(d)


Question 11:

Differentiate the function with respect to x. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 11
Answer:
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 11(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 11(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 11(c)

From (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 11(d)


Question 12:
Find NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12of function. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12(a)
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12(a)
Let xy = u and yx = v

Then, the function becomes u v = 1
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12(c)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12(d)

From (1), (2), and (3), we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12(e)


Question 13:
Find NCERT Solutions class 12 maths chapter 5 ex 5.5 q 13of function NCERT Solutions class 12 maths chapter 5 ex 5.5 q 13
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.5 q 13

Taking logarithm on both the sides, we obtain
x log y= y log x

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 13(a)


Question 14:
Find NCERT Solutions class 12 maths chapter 5 ex 5.5 q 12of function. NCERT Solutions class 12 maths chapter 5 ex 5.5 q 14
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.5 q 14

Taking logarithm on both the sides, we obtain
y log cos x= x log cos y

Differentiating both sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 14(a)


Question 15:
Find NCERT Solutions class 12 maths chapter 5 ex 5.5 q 15of function NCERT Solutions class 12 maths chapter 5 ex 5.5 q 15(a)
Answer:
The given function is NCERT Solutions class 12 maths chapter 5 ex 5.5 q 15(a)

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 15(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 15(c)


Question 16:
Find the derivative of the function given by NCERT Solutions class 12 maths chapter 5 ex 5.5 q 16and hence find NCERT Solutions class 12 maths chapter 5 ex 5.5 q 16(a)
Answer:
The given relationship is NCERT Solutions class 12 maths chapter 5 ex 5.5 q 16

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 16(b)
Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 16(c)


Question 18:
If uv and w are functions of x, then show that NCERT Solutions class 12 maths chapter 5 ex 5.5 q 18

in two ways-first by repeated application of product rule, second by logarithmic differentiation.
Answer:
Let NCERT Solutions class 12 maths chapter 5 ex 5.5 q 18(a)

By applying product rule, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 18(b)

By taking logarithm on both sides of the equation y=u,v,w, we obtain
log y= log u+ log v+ log w
Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.5 q 18(c)

NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.6 Derivatives of Implicit Functions in Parametric Forms

Question 1:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 1 NCERT Solutions class 12 maths chapter 5 ex 5.6 q 1(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 1(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 1(b)


Question 2:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 2x = a cos θy = b cos θ
Answer:
The given equations are x = a cos θ and y = b cos θ
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 2(a)



Question 3:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 3 x = sin ty = cos 2t
Answer:
The given equations are x = sin t and y = cos 2t

NCERT Solutions class 12 maths chapter 5 ex 5.6 q 3(a)


Question 4:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 4NCERT Solutions class 12 maths chapter 5 ex 5.6 q 4(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 4(a)

NCERT Solutions class 12 maths chapter 5 ex 5.6 q 4(b)


Question 5:
If x and y are connected parametrically by the equation, without eliminating the parameter, findNCERT Solutions class 12 maths chapter 5 ex 5.6 q 5 NCERT Solutions class 12 maths chapter 5 ex 5.6 q 5(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 5(a)
NCRET Solutions class 12 maths chapter 5 ex 5.6 q 5(a)


Question 6:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 6 NCERT Solutions class 12 maths chapter 5 ex 5.6 q 6(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 6(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 6(b)


Question 7:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 7
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 7(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 7(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 7(b)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 7(c)


Question 8:
If x and y are connected parametrically by the equation,  without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 8NCERT Solutions class 12 maths chapter 5 ex 5.6 q 8(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 8(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 8(b)


Question 9:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 9
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 9(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 9(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 9(b)


Question 10:
If x and y are connected parametrically by the equation, without eliminating the parameter, find NCERT Solutions class 12 maths chapter 5 ex 5.6 q 10
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 10(a)
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 10(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 10(b)


Question 11:
If NCERT Solutions class 12 maths chapter 5 ex 5.6 q 11
Answer:
The given equations are NCERT Solutions class 12 maths chapter 5 ex 5.6 q 11(a)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 11(b)
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 11(c)
Hence, proved


NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 Derivatives of Inverse Trigonometric Functions

Question 1:
Find the second order derivatives of the function. NCERT Solutions class 12 maths chapter 5 ex 5.7 q 1
Answer:
Let y=NCERT Solutions class 12 maths chapter 5 ex 5.7 q 1
Then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 1(a)


Question 2:
Find the second order derivatives of the function. NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 2
Answer:
Let y= NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 2
Then,
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 2(a)


Question 3:
Find the second order derivatives of the function.
x. cos x
Answer:
Let y= x. cos x
Then,
NCERT Solutions class 12 maths chapter 5 ex 5.6 q 3(b)


Question 4:

Find the second order derivatives of the function. log x
Answer:
Let y= log x
Then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 4


Question 5:
Find the second order derivatives of the function. NCERT Solutions class 12 maths chapter 5 ex 5.7 q 5
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.7 q 5
then,
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 5(a)


Question 6:
Find the second order derivatives of the function.
NCERT Solutions class 12 maths chapter 5ex 5.7 q 6
Answer:
Let y= NCERT Solutions class 12 maths chapter 5ex 5.7 q 6
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 6(a)


Question 7:
Find the second order derivatives of the function.
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 7
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.7 q 7
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 7(a)


Question 8:
Find the second order derivatives of the function.
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 8
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ex 5.7 q 8
then,
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 8(a)


Question 9:
Find the second order derivatives of the function.
log(log x)
Answer:
Let y= log(log x)
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 9


Question 10:
Find the second order derivatives of the function.
sin(log x)
Answer:
Let y= sin(log x)
then,
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 10


Question 11:
If y= 5 cos x-3 sin x, prove that NCERT Solutions class 12 maths chapter 5 ex 5.7 q 11
Answer:
It is given that, y= 5 cos x-3 sin x
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 11(a)
Hence, proved.


Question 12:
If NCERT Solutions class 12 maths chapter 5 ex 5.7 q 12find, NCERT Solutions class 12 maths chapter 5 ex 5.7 q 12(a)in terms of y alone.
Answer:
It is given that, NCERT Solutions class 12 maths chapter 5 ex 5.7 q 12
then,
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 12(b)


Question 13:
If y= 3cos(log x)+4sin(log x), show that NCERT Solutions class 12 maths chapter 5 ex 5.7 q 13
Answer:
It is given that, y= 3cos(log x)+4sin(log x)
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 13(a)

Hence, proved.


Question 14:
If NCERT Solutions class 12 maths chapter 5 ex 5.7 q 14show that, NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 14(a)
Answer:
It is given that, NCERT Solutions class 12 maths chapter 5 ex 5.7 q 14
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 14(b)


Question 15:
If NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 15show that, NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 15(a)
Answer:
It is given that, NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 15
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 15(b)

Hence, proved.


Question 16:
If NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 16show that, NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 16(a)
Answer:
The given relationship is NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 16
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 16(b)

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ex 5.7 q 16(c)

Differentiating this relationship with respect to x, we obtain
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 16(d)

Hence, proved.


Question 17:
If NCERT Solutins class 12 maths cahpter 5 ex 5.7 q 17show that, NCERT Solutions class 12 maths chapter 5 ex 5.7 q 17(a)
Answer:
The given relationship is NCERT Solutins class 12 maths cahpter 5 ex 5.7 q 17
then,
NCERT Solutions class 12 maths cahpter 5 ex 5.7 q 17(b)


NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.8 Mean Value theorem

 Question 1:
Verify Rolle’s Theorem for the function NCERT Solutions class 12 maths chapter 5 ex 5.8 q 1,x∈[4-2] Answer:
The given function, NCERT Solutions class 12 maths chapter 5 ex 5.8 q 1 being a polynomial function, is continuous in [−4, 2] and is differentiable in (−4, 2).
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 1(a)
∴ f (−4) = f (2) = 0⇒ The value of f (x) at −4 and 2 coincides.Rolle’s Theorem states that there is a point c ∈ (−4, 2) such that
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 1(b)

Hence, Rolle’s Theorem is verified for the given function.


Question 2:
Examine if Rolle’s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle’s Theorem from these examples?
(i)NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2
(ii)NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(a)
(iii)NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(b)
Answer:
By Rolle’s Theorem, for a function NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(c)if

(a) f is continuous on [ab]

(b) f is differentiable on (ab)

(c) (a) = f (b)

then, there exists some c ∈ (ab) such that NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(d)
Therefore, Rolle’s Theorem is not applicable to those functions that do not satisfy any of the three conditions of the hypothesis
(i) f(x)=[x] for x∈[5,9]

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at = 5 and = 9

⇒ f (x) is not continuous in [5, 9].
also f(5)=[5]=5 and f(9)=[9] ∴f(5)≠f(9)

The differentiability of f in (5, 9) is checked as follows.

Let be an integer such that n ∈ (5, 9).
the left hand limit of f at x=n is
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(e)

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

is not differentiable in (5, 9).

It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s Theorem.

Hence, Rolle’s Theorem is not applicable for
f(x)=[x]for x∈[5,9] (ii) f(x)=[x] for x∈[-2,2]

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at = −2 and = 2

⇒ f (x) is not continuous in [−2, 2].
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(f)

The differentiability of f in (−2, 2) is checked as follows.

Let be an integer such that n ∈ (−2, 2).
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(g)

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

is not differentiable in (−2, 2).

It is observed that f does not satisfy all the conditions of the hypothesis of Rolle’s Theorem.

Hence, Rolle’s Theorem is not applicable for
f(x)=[x] for x∈[-2,2] (iii) NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(b)
It is evident that f, being a polynomial function, is continuous in [1, 2] and is differentiable in (1, 2).
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(h)

(1) ≠ f (2)

It is observed that f does not satisfy a condition of the hypothesis of Rolle’s Theorem.

Hence, Rolle’s Theorem is not applicable forNCERT Solutions class 12 maths cahpter 5 ex 5.8 q 2(b)


Question 3:
If f:[-5,5]→R is a differentiable function and if NCERT Solutions class 12 maths chapter 5 ex 5.8 q 3 does not vanish anywhere, then prove that NCERT Solutions class 12 maths chapter 5 ex 5.8 q 3(a)
Answer:
It is given that f:[-5,5]→R

is a differentiable function.

Since every differentiable function is a continuous function, we obtain

(a) f is continuous on [−5, 5].

(b) is differentiable on (−5, 5).

Therefore, by the Mean Value Theorem, there exists c ∈ (−5, 5) such that
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 3(b)
It is also given that NCERT Solutions class 12 maths chapter 5 ex 5.8 q 3 does not vanish anywhere.
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 3(c)

Hence, proved.


Question 4:
Verify Mean Value Theorem, if NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 4in the interval [a,b], where a=1 and b=4
Answer:
The given function is NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 4
f, being a polynomial function, is continuous in [1, 4] and is differentiable in (1, 4) whose derivative is 2x − 4.
NCERT Solutions class 12 maths cahpter 5 ex 5.8 q 4(a)
Mean Value Theorem states that there is a point c ∈ (1, 4) such that NCERT Solutions class 12 maths chapter 5 ex 5.8 q 4(b)
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 4(c)

Hence, Mean Value Theorem is verified for the given function.


Question 5:
Verify Mean Value Theorem, if NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5in the interval [ab], where a = 1 and b = 3. Find all c∈(1,3)for which NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5(a)
Answer:
The given function f is NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5
f, being a polynomial function, is continuous in [1, 3] and is differentiable in (1, 3) whose derivative is 3x2 − 10x − 3.
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5(b)
Mean Value Theorem states that there exist a point c ∈ (1, 3) such that NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5(c)
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5(d)
Hence, Mean Value Theorem is verified for the given function and NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5(e)is the only point for which NCERT Solutions class 12 maths chapter 5 ex 5.8 q 5(c)


Question 6:
Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
Answer:
Mean Value Theorem states that for a function f:[a, b]→R, if

(a) f is continuous on [ab]

(b) f is differentiable on (ab)
then, there exists some c ∈ (ab) such that NCERT Solutions class 12 maths chapter 5 ex 5.8 q 6
Therefore, Mean Value Theorem is not applicable to those functions that do not satisfy any of the two conditions of the hypothesis.
(i) f(x)=[x] for x∈[5,9]

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at = 5 and = 9

⇒ f (x) is not continuous in [5, 9].

The differentiability of f in (5, 9) is checked as follows.

Let be an integer such that n ∈ (5, 9).
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 6(a)

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

is not differentiable in (5, 9).

It is observed that f does not satisfy all the conditions of the hypothesis of Mean Value Theorem.

Hence, Mean Value Theorem is not applicable for
f(x)=[x] for x∈[5,9] (ii) f(x)=[x] for x∈[-2,2]

It is evident that the given function f (x) is not continuous at every integral point.

In particular, f(x) is not continuous at = −2 and = 2

⇒ f (x) is not continuous in [−2, 2].

The differentiability of f in (−2, 2) is checked as follows.

Let be an integer such that n ∈ (−2, 2).
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 6(b)

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

is not differentiable in (−2, 2).

It is observed that f does not satisfy all the conditions of the hypothesis of Mean Value Theorem.

Hence, Mean Value Theorem is not applicable for
f(x)=[x]for x∈[-2,2] (iii) NCERT Solutions class 12 maths chapter 5 ex 5.8 q 6(c)

It is evident that f, being a polynomial function, is continuous in [1, 2] and is differentiable in (1, 2).

It is observed that f satisfies all the conditions of the hypothesis of Mean Value Theorem.

Hence, Mean Value Theorem is applicable for
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 6(d)

It can be proved as follows.
NCERT Solutions class 12 maths chapter 5 ex 5.8 q 6(e)

NCERT Solutions For Class 12 Maths Chapter 5 Continuity and Differentiability Miscellaneous Solutions

Question 1:
NCERT Solutions class 12 maths chapter 5 ms q 1(a)
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ms q 1(a)

Using chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 1(b)


Question 2:
NCERT Solutions class 12 maths chapter 5 ms q 2
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 2(a)



Question 3:
NCERT Solutions class 12 maths chapter 5 ms q 3
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ms q 3

Taking logarithm on both the sides, we obtain
log y=3 cos 2x log 5x

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 3(a)


Question 4:
NCERT Solutions class 12 maths chapter 5 ms q 4
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ms q 4(a)

Using chain rule, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 4(b)


Question 5:
NCERT Solutions class 12 maths chapter 5 ms q 5
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 5(a)



Question 6:
NCERT Solutions class 12 maths chapter 5 ms q 6
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 6(a)

Therefore, equation (1) becomes
NCERT Solutions class 12 maths chapter 5 ms q 6(b)


Question 7:
NCERT Solutions class 12 maths chapter 5 ms q 7(a)
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 7(b)

Taking logarithm on both the sides, we obtain
log y=log x.log(log x)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 7(c)


Question 8:
NCERT Solutions class 12 maths chapter 5 ms q 8
for some constant a and b
Answer:
Let y= NCERT Solutions class 12 maths chapter 5 ms q 8

By using chain rule, we obtain
NCERT Solutions class 12 maths chapter 5ms q 8(a)


Question 9:
NCERT Solutions class 12 maths chapter 5 ms q 9
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 9(a)

Taking logarithm on both the sides, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 9(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 9(c)


Question 10:
NCERT Solutions class 12 maths chapter 5 ms q 10
for some fixed a>0 and x>0
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 10(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 10(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 10(c)

s = aa

Since a is constant, aa is also a constant.
NCERT Solutions class 12 maths chapter 5 ms q 10(d)

From (1), (2), (3), (4), and (5), we obtain
NCERT Solutions class 12 maths chapter 5 ms q 10(e)


Question 11:
NCERT Solutions class 12 maths chapter 5 ms q 11
for x>3
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 11(a)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 11(b)

Differentiating with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 11(c)
also,
NCERT Solutions class 12 maths chapter 5 ms q 11(d)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 11(e)
Substituting the expressions of NCERT Solutions class 12 maths chapter 5 ms q 11(f)in equation (1), we obtain
NCERT Solutions class 12 maths chapter 5 ms q 11(g)


Question 12:
Find NCERT Solutions class 12 maths chapter 5 ms q 12if NCERT Solutions class 12 maths chapter 5 ms q 12(a)
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 12(b)



Question 13:
Find NCERT Solutions class 12 maths chapter 5 ms q 12if NCERT Solutions class 12 maths chapter 5 ms q 13
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 13(a)



Question 14:
If NCERT Solutions class 12 maths cahpter 5 ms q 14 for, −1 < x <1, prove that
NCERT Solutions class 12 maths chapter 5 ms q 14(a)
Answer:

It is given that,
NCERT Solutions class 12 maths chapter 5 ms q 14(b)

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 14(c)

Hence, proved.


Question 15:
If NCERT Solutions class 12 maths chapter 5 ms q 15 for some c>0. prove that

NCERT Solutions class 12 maths chapter 5 ms q 15(a)is a constant independent of a and A
Answer:
It is given that, NCERT Solutions class 12 maths chapter 5 ms q 15
Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 15(b)
NCERT Solutions class 12 maths chapter 5 ms q 15(c)

Hence, proved.


Question 16:
If cos y = x cos(a + y), with cos a≠±1, prove that NCERT Solutions class 12 maths chapter 5 ms q 16
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 16(a)

Then, equation (1) reduces to
NCERT Solutions class 12 maths cahpter 5 ms q 16(b)
⇒sina+y-ydydx=cos2a+y⇒dydx=cos2a+ysinaHence, proved.


Question 17:
If NCERT Solutions class 12 maths chapter 5 ms q 17and NCERT Solutions class 12 maths chapter 5 ms q 17(a)find NCERT Solutions class 12 maths chapter 5 ms q 17(b)
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 17(c)



Question 18:
If NCERT Solutions class 12 maths chapter 5 ms q 18show that NCERT Solutions class 12 maths chapter 5 ms q 18(a)

exists for all real x, and find it.
Answer:
It is known that, NCERT Solutions class 12 maths chapter 5 ms q 18(b)
Therefore, when x ≥ 0 NCERT Solutions class 12 maths cahpter 5 ms q 18(c)
In this case, NCERT Solutions class 12 maths chapter 5 ms q 18(d)and hence, NCERT Solutions class 12 maths chapter 5 ms q 18(e)
When x < 0, NCERT Solutions class 12 maths chapter 5 ms q 18(f)
In this case, NCERT Solutions class 12 maths chapter 5 ms q 18(g)and hence NCERT Solutions class 12 maths chapter 5 ms q 18(h)
Thus for NCERT Solutions class 12 maths chapter 5 ms q 18,NCERT Solutions class 12 maths chapter 5 ms q 18(a)exists for all real x and is given by,
NCERT Solutions class 12 maths chapter 5 ms q 18(i)


Question 19:
Using mathematical induction prove that NCERT Solutions class 12 maths chapter 5 ms q 19for all positive integers n.
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 19(a)
for n=1
NCERT Solutions class 12 maths chapter 5 ms q 19(c)
∴P(n) is true for n = 1

Let P(k) is true for some positive integer k.
that is, NCERT Solutions class 12 maths chapter 5 ms q 19)d)

It has to be proved that P(k + 1) is also true.
NCERT Solutoins class 12 maths chapter 5 ms q 19(e)

Thus, P(k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, the statement P(n) is true for every positive integer n.

Hence, proved.



Question 20:
Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.
Answer:
sin(A+B)=sin A cos B+ cos A sin B

Differentiating both sides with respect to x, we obtain
NCERT Solutions class 12 maths chapter 5 ms q 20


Question 21:
Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?
Answer:
y=x           -∞<x≤1    2-x         1≤x≤∞
NCERT Solutions class 12 maths chapter 5 ms q 21(a)
It can be seen from the above graph that, the given function is continuos everywhere but not differentiable at exactly two points which are 0 and 1.


Question 22:
If NCERT Solutions class 12 maths chapter 5 ms q 22prove that NCERT Solutions class 12 maths chapter 5 ms q 22(a)
Answer:
NCERT Solutions class 12 maths chapter 5 ms q 22(b)
thus,
NCERT Solutions class 12 maths chapter 5 ms q 22(c)


Question 23:
If NCERT Solutions class 12 maths chapter 5 ms q 23-1≤x≤1 show that NCERT Solutions class 12 maths chapter 5 ms q 23(a)
Answer:
It is given that, NCERT Solutions class 12 maths chapter 5 ms q 23
NCERT Solutions class 12 maths chapter 5 ms q 23(b)
Hence proved.