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Edited By Ramraj Saini | Updated on Dec 03, 2023 05:24 PM IST | #CBSE Class 12th

**NCERT Solutions for Exercise 5.1 Class 12 Maths Chapter 5 Continuity and Differentiability **are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the **latest syllabus and pattern of CBSE 2023-24. **In the previous class, you have already learned about limits, derivatives, limits, and derivatives of trigonometric functions. In this article, you will get NCERT solutions for Class 12 Maths chapter 5 exercise 5.1. NCERT book Class 12 Maths chapter 5 exercise 5.1 consists of questions related to finding whether a function is continuous or not.

Continuity of functions can't be learned without fundamental knowledge of limit which you have learned already. It is a fundamental concept of calculus that you must know to understand more concepts of calculus. Solving exercise 5.1 Class 12 Maths questions are very important to get conceptual clarity about continuity. There are different theorems to check the continuity of different types of functions mentioned in the NCERT syllabus of Class 12 Maths. **12th class Maths exercise 5.1 **answers** **are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

**Also, see**

- Continuity and Differentiability Exercise 5.2
- Continuity and Differentiability Exercise 5.3
- Continuity and Differentiability Exercise 5.4
- Continuity and Differentiability Exercise 5.5
- Continuity and Differentiability Exercise 5.6
- Continuity and Differentiability Exercise 5.7
- Continuity and Differentiability Exercise 5.8
- Continuity and Differentiability Miscellaneous Exercise

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**Question:1**. Prove that the function is continuous at and at

**Answer:**

Given function is

Hence, function is continous at x = 0

Hence, function is continous at x = -3

Hence, function is continuous at x = 5

**Question:2**.Examine the continuity of the function

**Answer:**

Given function is

at x = 3

Hence, function is continous at x = 3

**Question:3 **Examine the following functions for continuity.

**Answer:**

Given function is

Our function is defined for every real number say k

and value at x = k ,

and also,

Hence, the function is continuous at every real number

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

**Answer:**

Given function is

For every real number k ,

We get,

Hence, function continuous for every real value of x,

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

**Answer:**

Given function is

For every real number k ,

We gwt,

Hence, function continuous for every real value of x ,

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

**Answer:**

Given function is

for x > 5 , f(x) = x - 5

for x < 5 , f(x) = 5 - x

SO, different cases are their

case(i) x > 5

for every real number k > 5 , f(x) = x - 5 is defined

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

case (ii) x < 5

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

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

case(iii) x = 5

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

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

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

**Question:4**. Prove that the function is continuous at x = n, where n is a positive integer

**Answer:**

GIven function is

the function is defined for all positive integer, n

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

**Question:5.** Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?

**Answer:**

Given function is

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

Hence , given function** is continous at **x = 0

given function is defined for x = 1

Now, for x = 1 Right-hand limit and left-hand limit are not equal

R.H.L L.H.L.

Therefore, given function is **not continous **at x =1

Given function is defined for x = 2 and its value at x = 2 is 5

Hence, given function** is continous** at x = 2

**Question:6.** Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 2

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

case(ii) k < 2

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

case(iii) x = 2

Right hand limit at x= 2 Left hand limit at x = 2

Therefore, x = 2 is the point of discontinuity

**Question:7.**** **Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

GIven function is defined for every real number k

Different cases are their

case (i) k < -3

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

case(ii) k = -3

Hence, given function is continous for x = -3

case(iii) -3 < k < 3

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

case(iv) k = 3

Hence**. x = 3 is the point of discontinuity**

case(v) k > 3

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

**Question:8.** Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

if x > 0 ,

if x < 0 ,

given function is defined for every real number k

Now,

case(i) k < 0

Hence, given function is continuous for every value of k < 0

case(ii) k > 0

Hence, given function is continuous for every value of k > 0

case(iii) x = 0

Hence,** 0 is the only point of discontinuity**

**Question:9. **Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

if x < 0 ,

Now, for any value of x, the value of our function is -1

Therefore, the given function is continuous for each and every value of x

Hence, no point of discontinuity

**Question:10.**** **Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 1

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

case(ii) k < 1

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

case(iii) x = 1

Hence, at x = 2 given function is continuous

Therefore, no point of discontinuity

**Question:11.** Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 2

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

case(ii) k < 2

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

case(iii) x = 2

Hence, given function is continuous at x = 2

There, no point of discontinuity

**Question:12.** Find all points of discontinuity of f, where f is defined by

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 1

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

case(ii) k < 1

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

case(iii) x = 1

Hence, x = 1 is the point of discontinuity

**Question:13.** Is the function defined by

a continuous function?

**Answer:**

Given function is

given function is defined for every real number k

There are different cases for the given function

case(i) k > 1

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

case(ii) k < 1

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

case(iii) x = 1

Hence, x = 1 is the point of discontinuity

**Question:14.** Discuss the continuity of the function f, where f is defined by

**Answer:**

Given function is

GIven function is defined for every real number k

Different cases are their

case (i) k < 1

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

case(ii) k = 1

Hence, given function is discontinous at x = 1

Therefore, x = 1 is he point od discontinuity

case(iii) 1 < k < 3

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

case(iv) k = 3

Hence. x = 3 is the point of discontinuity

case(v) k > 3

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

case(vi) when k < 3

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

**Question:15 Discuss the continuity of the function f, where f is defined by**

**Answer:**

Given function is

Given function is satisfies for the all real values of x

case (i) k < 0

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

case (ii) x = 0

L.H.L at x= 0

R.H.L. at x = 0

L.H.L. = R.H.L. = f(0)

Hence, function is continuous at x = 0

case (iii) k > 0

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

case (iv) k < 1

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

case (v) k > 1

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

case (vi) x = 1

Hence, function is not continuous at x = 1

**Question:16.** Discuss the continuity of the function f, where f is defined by

**Answer:**

Given function is

GIven function is defined for every real number k

Different cases are their

case (i) k < -1

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

case(ii) k = -1

Hence, given function is continous at x = -1

case(iii) k > -1

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

case(vi) -1 < k < 1

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

case(v) k = 1

Hence.at x =1 function is continous

case(vi) k > 1

Hence, given function is continous for each and every value of k > 1

case(vii) when k < 1

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

Therefore, continuous at all points

**Question:17.** Find the relationship between a and b so that the function f defined by

is continuous at x = 3.

**Answer:**

Given function is

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

For the function to be continuous

**Question:18.** For what value of l is the function defined by

continuous at x = 0? What about continuity at x = 1?

**Answer:**

Given function is

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

For the function to be continuous

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

For x = 1

Hence, given function is continuous at x =1

**Answer:**

Given function is

Given is defined for all real numbers k

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

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

**Answer:**

Given function is

Clearly, Given function is defined at x =

Hence, the function defined by continuous at x =

**Question:21. **Discuss the continuity of the following functions:

a)

**Answer:**

Given function is

Given function is defined for all real number

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

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

f(x) = g(x) + h(x) = sin x + cos x is also a continuous function

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

**Answer:**

Given function is

Given function is defined for all real number

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

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

f(x) = g(x) - h(x) = sin x - cos x is also a continuous function

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

**Answer:**

Given function is

Given function is defined for all real number

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

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

f(x) = g(x).h(x) = sin x .cos x is also a continuous function

**Question:22.** Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

**Answer:**

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

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, the function is a continuous function

We proved independently that sin x and cos x is a continous function

So, we can say that

cosec x = is also continuous except at

sec x = is also continuous except at

cot x = is also continuous except at

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

**Answer:**

Given function is

Hence, the function is continuous

Therefore, no point of discontinuity

**Question:24. **Determine if f defined by

is a continuous function?

**Answer:**

Given function is

Given function is defined for all real numbers k

when x = 0

Hence, function is continuous at x = 0

when

Hence, the given function is continuous for all points

**Question:25**. Examine the continuity of f, where f is defined by

**Answer:**

Given function is

Given function is defined for all real number

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

Lets take g(x) = sin x and h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

We proved independently that sin x and cos x is continous function

So, we can say that

f(x) = g(x) - h(x) = sin x - cos x is also a continuous function

When x = 0

Hence, function is also continuous at x = 0

**Question:26.** Find the values of k so that the function f is continuous at the indicated point in Exercises

**Answer:**

Given function is

When

For the function to be continuous

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

**Question:27**. Find the values of k so that the function f is continuous at the indicated point in Exercises

**Answer:**

Given function is

When x = 2

For the function to be continuous

f(2) = R.H.L. = LH.L.

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

**Question:28**. Find the values of k so that the function f is continuous at the indicated point in Exercises

**Answer:**

Given function is

When x =

For the function to be continuous

f() = R.H.L. = LH.L.

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

**Question:29** Find the values of k so that the function f is continuous at the indicated point in Exercises

**Answer:**

Given function is

When x = 5

For the function to be continuous

f(5) = R.H.L. = LH.L.

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

**Question:30** Find the values of a and b such that the function defined by

is a continuous function.

**Answer:**

Given continuous function is

The function is continuous so

By solving equation (i) and (ii)

a = 2 and b = 1

Hence, values of a and b such that the function defined by is a continuous function is 2 and 1 respectively

**Question:31.** Show that the function defined by is a continuous function.

**Answer:**

Given function is

given function is defined for all real values of x

Let x = k + h

if

Hence, the function is a continuous function

**Question:32.** Show that the function defined by is a continuous function.

**Answer:**

Given function is

given function is defined for all values of x

f = g o h , g(x) = |x| and h(x) = cos x

Now,

g(x) is defined for all real numbers k

case(i) k < 0

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

case (ii) k > 0

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

case (iii) k = 0

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

Therefore, g(x) = |x| is continuous for all real values of x

Now,

h(x) = cos x

Let suppose x = c + h

if

Hence, function is a continuous function

g(x) is continuous , h(x) is continuous

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

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

**Answer:**

Given function is

f(x) = sin |x|

f(x) = h o g , h(x) = sin x and g(x) = |x|

Now,

g(x) is defined for all real numbers k

case(i) k < 0

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

case (ii) k > 0

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

case (iii) k = 0

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

Therefore, g(x) = |x| is continuous for all real values of x

Now,

h(x) = sin x

Let suppose x = c + h

if

Hence, function is a continuous function

g(x) is continuous , h(x) is continuous

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

**Question:34. **Find all the points of discontinuity of f defined by

**Answer:**

Given function is

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

Now,

g(x) is defined for all real numbers k

case(i) k < 0

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

case (ii) k > 0

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

case (iii) k = 0

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

Therefore, g(x) = |x| is continuous for all real values of x

Now,

g(x) is defined for all real numbers k

case(i) k < -1

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

case (ii) k > -1

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

case (iii) k = -1

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

Therefore, h(x) = |x+1| is continuous for all real values of x

g(x) is continuous and h(x) is continuous

Therefore, f(x) = g(x) - h(x) = |x| - |x+1| is also continuous

In Class 12th Maths chapter 5 exercise 5.1 there are 34 long answer types questions only checking your knowledge of continuity. Many questions in Class 12 Maths ch 5 ex 5.1 are related to checking the continuity of trigonometric functions. There are 20 examples and some important theorems given before this exercise in the NCERT textbook. Solving these examples is a must to do before going to the Class 12th Maths chapter 5 exercise 5.1 questions because it will help you in solving NCERT problems.

**Also Read|** Continuity and Differentiability Class 12th Chapter 5 Notes

- Class 12 Maths chapter 5 exercise 5.1 solutions are described in a detailed manner, so you will get the concept of continuity easily.
- In Class 12 Maths chapter 5 exercise 5.1 solutions , you will get different ways to approach the problem.
- There are some properties of continuous functions which make it easy to check the continuity of the given functions.
- In Class 12 Maths chapter 5 exercise 5.1 solutions, you will learn about the algebra of continuous functions.

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Download EBook**Comprehensive Coverage:**The solutions encompass all the topics covered in ex 5.1 class 12, ensuring a thorough understanding of the concepts.**Step-by-Step Solutions:**In this class 12 maths ex 5.1, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.**Accuracy and Clarity:**Solutions for class 12 ex 5.1 are presented accurately and concisely, using simple language to help students grasp the concepts easily.**Conceptual Clarity:**In this 12th class maths exercise 5.1 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.**Inclusive Approach:**Solutions for ex 5.1 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.**Relevance to Curriculum:**The solutions for class 12 maths ex 5.1 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

**Also see-**

- NCERT Solutions for Class 12 Maths
- NCERT Solutions for Class 12 Physics
- NCERT Solutions for Class 12 Chemistry
- NCERT Solutions for Class 12 Biology

- NCERT Exemplar Solutions for Class 12th Maths
- NCERT Exemplar Solutions for Class 12th Physics
- NCERT Exemplar Solutions for Class 12th Chemistry
- NCERT Exemplar Solutions for Class 12th Biology

**Happy learning!!!**

1. Check whether function f(x) = x + 3 is continuous at x = 1 ?

f(1^+)= 1+3 = 4

f(1^-)= 1+3 = 4 = f(1)

Hence f(x) is continuous at x=1.

2. If f(x), g(x) are continuous functions then what about continuity of f(x) + g(x) ?

If f(x), g(x) are continuous functions then f(x) + g(x) is also a continuous function.

3. If f(x), g(x) are continuous functions then what about continuity of f(x) - g(x) ?

If f(x), g(x) are continuous functions then f(x) - g(x) is also a continuous function.

4. If two functions are continuous then check continuity of product of the given two functions.

If two functions are continuous then check the product of the given two functions is also a continuous function.

5. Do every continuous function is a differential function ?

No, every continuous function need not to be a differential function.

6. Do every differential function is a continuous function ?

Yes, every differential function is a continuous function.

7. How many questions are there in the exercise 5.1 Class 12 Maths ?

NCERT book Exercise 5.1 Class 12 Maths is consists of 34 long answer questions related to checking the continuity of the functions. For more questions students can refer to NCERT exemplar problems.

8. How many exercises are in there in the chapter 5 Class 12 Maths ?

There are eight main exercises and one miscellaneous exercise given in the chapter 5 Class 12 Maths.

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6 minHave a question related to CBSE Class 12th ?

You can use them people also used problem

Hi,

The Medhavi National Scholarship Program, under the Human Resources & Development Mission (HRDM), offers financial assistance to meritorious students through a scholarship exam. To be eligible, candidates must be between 16 and 40 years old as of the last date of registration and have at least passed the 10th grade from a recognized board. Higher qualifications, such as 11th/12th grade, graduation, post-graduation, or a diploma, are also acceptable.

To apply, download the Medhavi App from the Google Play Store, sign up, and read the detailed notification about the scholarship exam. Complete the registration within the app, take the exam from home using the app, and receive your results within two days. Following this, upload the necessary documents and bank account details for verification. Upon successful verification, the scholarship amount will be directly transferred to your bank account.

The scholarships are categorized based on the marks obtained in the exam: Type A for those scoring 60% or above, Type B for scores between 50% and 60%, and Type C for scores between 40% and 50%. The cash scholarships range from Rs. 2,000 to Rs. 18,000 per month, depending on the exam and the marks obtained.

Since you already have a 12th-grade qualification with 84%, you meet the eligibility criteria and can apply for the Medhavi Scholarship exam. Preparing well for the exam can increase your chances of receiving a higher scholarship.

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

**
Thank you and wishing you all the best for your bright future.
**

Hello student,

**
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
**

- No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
- You have to appear for the 2025 12th board exams.
- Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
- Aim to register before late October to avoid extra fees.
- Schools might not offer classes for private students, so focus on self-study or coaching.

**
Remember
**
, these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

**
Good luck with your studies!
**

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