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Install NowNCERT Solutions class 12 Maths Exercise 5.1 Part-2 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Continuity and Differentiability as PDF.**

## NCERT Solutions class 12 Continuity & Differentiability

**7. Find the relationship between **** and **** so that the function **** defined by**** is continuous at **** **

**Ans. **Given:

Continuity at

Also

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

**18. For what value of **** is the function defined by ****continuous at **** What about continuity at **** **

**Ans. **Since is continuous at

And

Here, therefore should be L.H.L. = R.H.L.

0 = 1, which is not possible.

Again Since is continuous at

And

Here, therefore should be L.H.L. = R.H.L.

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

**19. Show that the function defined by **** is discontinuous at all integral points. Here **** denotes the greatest integer less than or equal to **** **

**Ans. **For any real number we use the symbol to denote the fractional part or decimal part of For example,

[3.45] = 0.45

[–7.25] = 0.25

[3] = 0

[–7] = 0

The function : R R defined by is called the fractional part function. It is observed that the domain of the fractional part function is the set R of all real numbers and the range of the set [0, 1).

Hence given function is discontinuous function.

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

**20. Is the function **** continuous at **** ?**

**Ans. **Given:

L.H.L. =

R.H.L. =

And

Since L.H.L. = R.H.L. =

Therefore, is continuous at

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

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

**(a) **

**(b) **** **

**(c) **** **

**Ans. (a)** Let be an arbitrary real number then

=

=

=

Similarly, we have

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

**(b) **Let be an arbitrary real number then

=

=

=

Similarly, we have

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

**(c)** Let be an arbitrary real number then

=

=

=

Similarly, we have

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

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

**Ans. **(a) Let be an arbitrary real number then

=

= =

for all R

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

**(b)** and domain I

=

=

=

=

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

**(c) ** and domain I

=

=

= =

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

**(d)** and domain I

= =

= =

Therefore, is continuous at

Since, is an arbitrary real number, therefore, is continuous.

**23. Find all points of discontinuity of **** where ****.**

**Ans. **Given:

At L.H.L. =

R.H.L. =

is continuous at

When and are continuous, then is also continuous.

When is a polynomial, then is continuous.

Therefore, is continuous at any point.

**24. Determine if **** defined by **** is a continuous function.**

**Ans. **Here, = 0 x a finite quantity = 0

Also

Since, therefore, the function is continuous at

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

**25. Examine the continuity of **** where **** is defined by ****.**

**Ans. **At L.H.L. =

=

R.H.L. =

=

And

Therefore, is discontinuous at

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

**Find the values of **** so that the function **** is continuous at the indicated point in Exercise 26 to 29.**

**26. **** at **** **

**Ans. **Here,

Putting where

= =

= =

= ……….(i)

And ……….(ii)

when [Given]

Because is continuous at

From eq. (i) and (ii),

**27. **** at **** **

**Ans. **Here,

and

Now, when , we have

Therefore, is continuous at when .

**28. **** at **** **

**Ans. **Here,

And

Also

Since the given function is continuous at

**29. **** at **** **

**Ans. **When we have which being a polynomial is continuous at each point

And, when we have which being a polynomial is continuous at each point

Now

……….(i)

=

…….(i)

Since function is continuous, therefore, eq. (i) = eq. (ii)

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

**30. Find the values of **** and **** such that the function defined by **** is a continuous function.**

**Ans. **For function is constant, therefore it is continuous.

For function polynomial, therefore, it is continuous.

For function is constant, therefore it is continuous.

For continuity at

……….(i)

For continuity at

……….(ii)

Solving eq. (i) and eq. (ii), we get

and

### NCERT Solutions class 12 Maths Exercise 5.1 Part-2

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

**Ans. **Let and , then

Now and being continuous it follows that their composite is continuous.

Hence is continuous function.

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

**Ans. **Given: ….(i)

has a real and finite value for all R.

Domain of is R.

Let and

Since and being cosine function and modulus function are continuous for all real

Now, being the composite function of two continuous functions is continuous, but not equal to

Again, [Using eq. (i)]

Therefore, being the composite function of two continuous functions is continuous.

**33. Examine that **** is a continuous function.**

**Ans. **Let and , then

Now, and being continuous, it follows that their composite, is continuous.

Therefore, is continuous.

**34. Find all points of discontinuity of **** defined by **** **

**Ans. **Given:

When =

When

When

At L.H.L. =

R.H.L. =

And

Therefore, at is continuous.

At L.H.L. =

R.H.L. =

And

Therefore, at is continuous.

Hence, there is no point of discontinuity.

## NCERT Solutions class 12 Maths Exercise 5.1 Part-2

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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Exercise :- 5.1, In solution of question no.18 is wrong because we can’t get the exact value ‘?’.