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# NCERT Solutions class 12 Maths Exercise 5.1 Part-2

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## 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

### 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.

### 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.

### 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

### 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.

### 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

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

Ans. At  L.H.L. =

=

R.H.L. =

=

And

Therefore,  is discontinuous at

### 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)

### 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

### 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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### 6 thoughts on “NCERT Solutions class 12 Maths Exercise 5.1 Part-2”

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