NCERT Solutions class 12 Maths Exercise 5.1 Part-2

NCERT 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

 

 

 

 

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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.

 

 

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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.

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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

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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.

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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.

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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.

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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

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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

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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),

 

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27.  at  

Ans. Here,

  and

 

 

Now,   when , we have

Therefore,  is continuous at  when .

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28.  at  

Ans. Here,

And

Also

Since the given function is continuous at

 

 

 

 

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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)

 

 

 

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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

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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.

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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.

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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.

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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

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