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Edited By Kuldeep Maurya | Updated on Jan 27, 2022 02:13 PM IST

RD Sharma is one of the most well-known books in the country. RD Sharma's books are detailed, informative, and also contain step-by-step solutions for their problems. Class 12 RD Sharma chapter 8 exercise 8.2 solution deals with the chapter 'Continuity.' The book has plenty of examples that the students can practice to develop their skills. Still, when it comes to solving a complex chapter like Continuity, students also need to exercise problems. This is where RD Sharma class 12th exercise 8.2 solution comes to light.

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- Chapter 8 - Continuity -Ex-8.1
- Chapter 8 - Continuity -Ex-FBQ
- Chapter 8 - Continuity -Ex-MCQ
- Chapter 8 - Continuity -Ex-VSA

Continuity exercise 8.2 question 1

f(x) is everywhere continuous.

A function is everywhere continuous when it is continuous at every

Now, at x = 0

And

As,

Hence, f(x) is everywhere continuous.

Note: sine function, identity function, polynomial functions are everywhere continuous.

Continuity exercise 8.2 question 2

Discontinuous at x = 0.

If a function is not continuous at one point then it is discontinuous. As at that point

and this is the definition of discontinuity.

Now consider at x = 0

So

Function is discontinuous at X = 0

x = 1

Show LHS RHS or LHL the value of function at given point or RHL the value of function at given point.

Now consider the point x = 1

As

f(x) is discontinuous at x =1

x = 2

To check the continuity of such type of function we check at the breaking point.

At point x =2

Therefore, f(x) is discontinuous at x = 2

Continuity exercise 8.2 question 3 (iii)

x = 0

At x = 0

f is discontinuous at x = 0

Continuity exercise 8.2 question 3 (iv)

x = 0

At x = 0

f is not continuous at x = 0.

Continuity exercise 8.2 question 3 (v)

x = 0

At x = 0

As

f is not continuous at x = 0

Continuity exercise 8.2 question 3 (vi)

x = 0

Use L-Hospital Rule when we find 0/0 form

at x = 0

Hence, f(x) is discontinuous at x = 0.

Continuity exercise 8.2 question 3 (vii)

x = 0

Use L-Hospital Rule when we find 0/0 form

At x = 0

f(x) is not continuous at x = 0

Continuity exercise 8.2 question 3 (viii)

f(x) is nowhere discontinuous

Now, we check the limit at x = 1 and x = 3

At x = 3

As

Hence, f(x) is continuous at x = 3

Now, at x = 1

As

f(x) is continuous at x = 1

As polynomial function is everywhere continuous & greater integer function is continuous except its end point &

is continuous.

Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 3 (ix)

at x = 3

at x = -3

Now at x = 3

As

f(x) is not continuous at x = 3.

Continuity exercise 8.2 question 3 (x)

x = 1

At x = 1

As

f(x) is not continuous at x = 1

Continuity exercise 8.2 question 3 (xi)

x = 1

Check at end points.

At x = 0

As (1)=(2)=(3)

Hence, f(x) is continuous at x = 0

At x = 1

As

Hence, f(X) is discontinuous at x = 1

Continuity exercise 8.2 question 3 (xii)

f(x) is continuous everywhere.

At x = 0

And

Constant function,

Sine and cosine function is everywhere continuous and f(x) is continuous at x = 0

f(x) is everywhere continuous.

Continuity exercise 8.2 question 3 (xiii)

everywhere continuous.

Constant and identity function are everywhere continuous. So, we only have to check at x = -1, x = 1 (end points) -(1)

At x = -1

As

At x = 1

As

So from (1), (2) & (3)

f(x) is everywhere continuous.

f(x) is continuous so it is also continuous at end points. Put LHL= RHL

At x = 0

As f(X) is continuous at x = 0 when

Continuity exercise 8.2 question 4 (ii)

Put LHL = RHL = f(2)

LHL = RHL = f(2) ....(1)

Using (1) we get

Continuity exercise 8.2 question 4 (iii)

No value of k can make +

Put L.H.L = R.H.L at x = 0

At x = 0

L.H.L = R.H.L = f(0) ....(1)

Hence no value of k can make f continuous.

Continuity exercise 8.2 question 4 (iv)

Put LHL = RHL at x = 3 , 5

At x = 3

Now, f(x) is continuous at x = 3

If 3a + b =2 .....(A)

Now, at x = 5

Now, f(x) is continuous at x = 5

If 5a + b = 9 .....(B)

on solving A and B we get,

Continuity exercise 8.2 question 4 (v)

a = 3, b = 1

Put LHL = RHL at x = 3 , 5

At x = -1

Now at x = 0

Hence, a = 3, b = 1

Put LHL = RHL at x = 0

At x = 0

L.H.L

As f(x) is continuous at x = 0

Hence, p = -1/2

Continuity exercise 8.2 question 4 (vii)

a = 2, b = 1

Put LHL = RHL at x = 2 &x = 10

At x = 2

As, f(x) is continuous at x = 2 when 5 + 2a +b - (1)

At x = 10

As, f(x) is continuous at x = 10

10a + b =21

From (1) & (2) we have

8a = 16

a = 2

Put in (1)

b = 1

Hence, a = 2 & b = 1

Continuity exercise 8.2 question 5

p at LHL = RHL at

Using (A)

So using (B) we get

Hence,

Continuity exercise 8.2 question 6

Put at LHL = RHL at

At x = /4

Now,

L.H.L

Using (A)

Now

Using (C)

From (B)

Hence

Continuity exercise 8.2 question 6

Put at LHL = RHL at

At x = /4

Now,

L.H.L

Using (A)

Now

Using (C)

From (B)

Hence,

Continuity exercise 8.2 question 7

a = 3, b = -2

Put at LHL = RHL at x = 2, x = 4

At x = 2

from (B)

From (C)

Solving (B) & (D) we get

Hence, a = 3, b = -2

Continuity exercise 8.2 question 8

Apply L-Hospital Rule when you get 0/0 from.

At x = /4

Hence F (x) will be continious on

If

Continuity exercise 8.2 question 9

Everywhere continuous

Polynomial & identity functions are everywhere continuous.

As polynomial & identity function are everywhere continuous.

So, we only have to check at end point i.e. x = 2

f(x) is continuous at x = 2 and everywhere

Continuity exercise 8.2 question 10

everywhere continuous.

sine function is everywhere continuous.

As sine function is everywhere continuous &

Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 11

Everywhere continuous

sine function, polynomial function and identity are everywhere continuous.

At x = 0

So, f(x) is everywhere continuous.

Continuity exercise 8.2 question 12

Discontinuous at all integral points.

It is defined at all integral points

let n be an integer.

Then,

Hence g is discontinious at all integral points.

Continuity exercise 8.2 question 13 (i)

everywhere continuous.

sine and cosine function are everywhere continuous.

As sine and cosine function are everywhere continuous and addition of two continuous functions is again continuous.

Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 13 (ii)

everywhere continuous.

As sine and cosine function are everywhere continuous and subtraction of two continuous functions is continuous.

Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 13 (iii)

sine and cosine function are everywhere continuous.

As sine and cosine function are everywhere continuous and product of two continuous functions is continuous.

Hence, f(x) is everywhere continuous.

Continuity exercise 8.2 question 14

cos

f(x) = cos x

f(x) = cos x

Let a be any real number then,

L.H.L

R.H.L

Also, L.H.L=R.H.L=f(a)

f(x) is continuous everywhere

Continuity exercise 8.2 question 15

|cos x| is continuous.

Continuous of two continuous functions is continuous.

f(x) = |cos x|

f(x) = |cos x|

Let a be any real number

L.H.L

R.H.L

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

f(x) is continuous everywhere

Continuity exercise 8.2 question 16

No point of discontinuity.

Continuity at x = -1

L.H.L

R.H.L

And

F(x) is continuous at x=-1

Continuous at x=0

L.H.L

R.H.L

And

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

F(x) is continuous at x=0, hence continuous everywhere

Continuity exercise 8.2 question 17

f(x) is continuous.

f is defined at all points of the real line

Let c be a real number

Case 1:

Case 2:

Similarly,

Therefore f is continuous at x=0

From the above observations.

It can be conclude that f is continuous at every point of the real line.

Thus f is a continuous function.

Continuity exercise 8.2 question 18

Clearly,

is discontinuous at

X = -2

Also, it is not defined at x = -2

For x ≠ -2

We observe that,

f(f(x)) is discontinuous and not defined a

Hence

f(f(x)) is not continuous at x = -2 and

Continuity exercise 8.2 question 19

Discontinuous at x = 1/2 , 1 , 2

f(x)/g(x) is continuous at every point hen f(x) & g(x) are continuous except

Hence F is discontinuous at

The RD Sharma class 12 solution of Continuity exercise 8.2 consists of 40 questions that cover up almost the majority of all the topics of the chapter continuity. The concept covered in this chapter are-

Continuous function.

Absolute continuous function.

Absolute Continuity of a measure concerning another measure.

Continuous probability distribution.

The questions in this exercise are divided into two parts: level 1 and level 2. With such a vast number of topics to cover, it gets easier for students to study in two parts, understanding the level of difficulty of the question and practicing accordingly.

The solutions in RD Sharma class 12th exercise 8.2 are created by experts all around the country who are excellent in the field of academics, thus providing helpful tips to students which help them solve questions more efficiently. The format of the RD Sharma class 12 solutions chapter 8 exercise 8.2 solution corresponds with the syllabus of NCERT, which makes it more useful for students to prepare for public exams as well.

A few reasons are listed below why the RD Sharma class 12th exercise 8.2 are helpful in the preparation of exams:-

The questions are designed to cover up almost all the topics that have the possible chances to be asked in the exams.

The RD Sharma class 12 chapter 8 exercise 8.2 consists of questions that are frequently asked in board exams and also provides tricks and tips to solve the questions in an alternate and easy way.

The solution also helps solve the homework, as teachers take the help of the same book to assign a task.

The RD Sharma class 12th exercise 8.2 is trusted by a thousand students across the country. By practicing the questions from it has made them set a benchmark in the maths subject and score high.

The best part about RD Sharma class 12th exercise 8.2 is that it can be downloaded free from the Career360 website; it doesn't cost a single penny to own this book

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1. Do the RD Sharma solutions match up to the recent syllabus?

Yes, the RD Sharma books are revised every year to correspond to the syllabus of NCERT.

2. What are the charges of the RD Sharma solution class 12 chapter 8?

RD Sharma class 12 chapter 8 ex 8.2, is in fact, free of cost to own. You can download the E-book and pdf through the Career360 website.

3. How many questions does the RD Sharma class 12 chapter 8 solution consist of?

The solution consists of only 40 questions that cover up the entire syllabus of the chapter, making it efficient enough to practice

4. Can the RD Sharma class 12 chapter 8 ex 8.2 help solve homework?

Yes, it is completely helpful to solve homework as teachers always use the same book to assign homework. So, solving homework with the help of RD Sharma solutions can help save time as well.

5. From which source can I download the RD Sharma class 12 chapter 8 ex 8.2?

You can download the E-book and PDF from the official website of Career360, and that is also free of cost.

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