NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 Continuity and Differentiability
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 are created by our experts to help students to understand the concepts of the chapter better. Solve these questions to ace your examinations.
Krati Saraswat18 Jan, 2024
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NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 (Continuity and Differentiability)
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 Continuity and Differentiability is prepared by the academic team of Physics Wallah. We have prepared NCERT Solutions for all exercises of Chapter 5. Given below is step by step solutions of all questions given in NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 Continuity and Differentiability.NCERT Solutions for Class 12 Maths Chapter 5 Miscellaneous Exercise
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 Overview
NCERT Solutions for Class 12 Maths Chapter 5 contains all the important topics for the exams. Our experts created these questions for the students to ace the examination. This article contains all the important questions and their easy to understand answers for the better understanding. Students are advised to go through these questions to clarify their concepts better. These questions are created to help students in the better understanding of the NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1.NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1
Solve The Following Questions NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 Continuity and Differentiability:
Question 1. Prove that the function f(x) = 5x - 3 is continuous at x = 0, at x = - 3 and x = 5 Solution :
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.2
Question 2. Examine the continuity of the function f(x) = 2x 2 - 1 at x = 3 Solution :
Thus, f is continuous at x = 3
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3
Question 3. Examine the following functions for continuity. (a)
(c)
Solution :
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.4
Question 4. Prove that the function f(x) = x n is continuous at x = n, where n is a positive integer. Solution : 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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.5
Question 5. Is the function f defined by
continuous at x = 0? At x = 1? At x = 2?
Solution :
The given function f is
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.
Therefore, f is continuous at x = 0
At x = 1,
f is defined at 1 and its value at 1 is 1.
The left hand limit of f at x = 1 is,
The right hand limit of f at x = 1 is,
Therefore, f is not continuous at x = 1
At x = 2,
f is defined at 2 and its value at 2 is 5.
Therefore, f is continuous at x = 2
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.6
Question 6. Find all points of discontinuity of f, where f is defined by
Solution :
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7
Question 7. Find all points of discontinuity of f, where f is defined by
Solution :
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:
Therefore, f is continuous at x = −3
Case III:
Therefore, f is continuous in (−3, 3).
Case IV:
If c = 3, then the left hand limit of f at x = 3 is,
The right hand limit of f at x = 3 is,
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.8
Question 8. Find all points of discontinuity of f, where f is defined by
Solution :
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3
Question 9. Find all points of discontinuity of f, where f is defined by
Solution :
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.4
Question 10. Find all points of discontinuity of f, where f is defined by
Solution :
Therefore,
f
is continuous at all points
x
, such that
x
> 1
Hence, the given function
f
has no point of discontinuity.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.5
Question 11. Find all points of discontinuity of f, where f is defined by
Solution :
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.6
Question 12. Find all points of discontinuity of f, where f is defined by
Solution :
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.
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7
Question 13. Is the function defined by
a continuous function?
Solution :
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.8
Question 14. Discuss the continuity of the function f, where f is defined by f =
Solution :
The given function is f =
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 (3, 10].
Hence, f is not continuous at x = 1 and x = 3
Question
15. Discuss the continuity of the function f, where f is defined by
Solution :
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:
Question
16. Discuss the continuity of the function f, where f is defined by
Solution :
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 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
is continuous at x = 3.
Solution :
The given function f is
If f is continuous at x = 3, then
Question 18. For what value of λ is the function defined by
continuous at x = 0?
What about continuity at x = 1?
Solution :
The given function f is
If f is continuous at x = 0, then
Therefore, for any values of λ, f is continuous at x = 1
Question
19. Show that the function defined by
is discontinuous at all integral point. Here [denotes the greatest integer less than or equal to x.
Solution :
The given function is
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.
Question
20. Is the function defined by
continuous at x = π ?
Solution :
The given function is
It is evident that f is defined at x = π
Therefore, the given function f is continuous at x = 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
Solution :
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
Question
22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
Solution :
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
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 (x) = cos x is continuous function.
It can be concluded that,
Question
23. Find the points of discontinuity of f, where
Solution :
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 all points of the real line.
Thus, f has no point of discontinuity.
Question
24. Determine if f defined by
is a continuous function?
Solution :
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.
Question
25. Examine the continuity of f, where f is defined by
Solution :
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.
Question
26. Find the values of k so that the function f is continuous at the indicated point.
Solution :
The given function f is
The given function f is continuous at x = π/2 , if f is defined at x = π/2 and if the value of the f at x = π/2 equals the limit of f at x = π/2 .
It is evident that f is defined at x = π/2 and f( π/2) = 3
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.
Solution :
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
Therefore, the required value of k is 3/4.
Question
28. Find the values of k so that the function f is continuous at the indicated point.
Solution :
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 f is defined at x = p and f(π) = kπ + 1
Therefore, the required value of k is -2/π
Question
29. Find the values of k so that the function f is continuous at the indicated point.
Solution :
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
It is evident that f is defined at x = 5 and f(5) = kx + 1 = 5k + 1
Therefore, the required value of k is 9/5
Question
30. Find the values of a and b such that the function defined by
is a continuous function.
Solution :
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.
Question
31. Show that the function defined by f (x) = cos (x
2
) is a continuous function.
Solution :
The given function is f (x) = cos (x
2
)
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) = x
2
It has to be first proved that g (x) = cos x and h (x) = x
2
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
Therefore, g (x) = cos x is continuous function.
h (x) = x
2
Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k
2
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, h is a continuous function.
Question
32. Show that the function defined by f(x) = |cos x| is a continuous function.
Solution :
The given function is f(x) = |cos x|
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) = |x| and h(x) = cos x
It has to be first proved that g(x) = |x| and h(x) = cos x are continuous functions.
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
Case II:
Therefore, g is continuous at all points x, such that x > 0
Case III:
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) = 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 (x) = cos x 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.
Question
33. Examine that sin|x| is a continuous function.
Solution :
Let, f(x) = sin|x|
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) = |x| and h (x) = sin x
It has to be proved first that g (x) = |x| and h (x) = sin x are continuous functions.
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
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Therefore, g is continuous at all points x, such that x < 0
Case II:
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Therefore, g is continuous at all points x, such that x > 0
Case III:
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.
Therefore,
is a continuous function.
Question
34. Find all the points of discontinuity of f defined by
f(x) = |x| - |x + 1|.
Solution :
The given function is f(x) = |x| - |x + 1|
The two functions, g and h, are defined as
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 a continuous function.
Therefore, f has no point of discontinuity.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 FAQs
What are the conditions for continuity and differentiability?
If f is differentiable at x=a, then f is continuous at x=a. Equivalently, if f fails to be continuous at x=a, then f will not be differentiable at x=a. A function can be continuous at a point, but not be differentiable there.
What are the 3 conditions of continuity?
The three conditions of continuity are as follows: The function is expressed at x = a. The limit of the function as the approaching of x takes place, a exists. The limit of the function as the approaching of x takes place, a is equal to the function value f(a).
What is the SI unit of equation of continuity?
the continuity equation can be written as, A × v = constant. In this equation A indicates the area of cross section of the fluid and v indicates the speed. SI unit of A is m² and SI unit of v is m/s.
What is the concept of continuity?
A function is said to be continuous in a given interval if there is no break in the graph of the function in the entire interval range.
Is continuity necessary for integration?
Every continuous function is integrable, but there are integrable functions which are not continuous.
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