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# NCERT Solutions class 12 Maths Miscellaneous

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## NCERT Solutions class 12 Continuity & Differentiability

Differentiate with respect to  the functions in Exercises 1 to 5.

1.

Ans. Let

=

2.

Ans. Let  =

=

3.

Ans. Let ……….(i)

4.

Ans. Let  =

=

=

=

5.

Ans. Let

[By Quotient Rule]

=

=

Differentiate with respect to  the functions in Exercises 6 to 11.

6.

Ans. Let   ……….(i)

Now,

=  =

And

=  =

From eq. (i),

=

=

7.

Ans. Let  ……….(i)

=

=

=

8.  for some constants  and

Ans. Let  for some constants  and

9.

Ans. Let  ……….(i)

=

10.  for some fixed  and

Ans. Let

=   …….(i)

Now taking ,  let  ……….(ii)

=

=

From eq. (ii),

From eq. (i),

11.  for

Ans. Let  for

Putting  and

……….(i)

Now

=

=

……….(ii)

Again

=

=

……….(iii)

Putting the values from eq. (ii) and (iii) in eq. (i),

12. Find  if  and

Ans. Given:      and

and

=

=

=

13. Find  if

Ans. Given:

=

=  = 0

14. If  for  prove that

Ans. Given:

=

=      Proved.

### 15. If  for some  prove that

Ans. Given: ……….(i)

……….(ii)

Again

[From eq. (ii)

=

=  ……….(iii)

Putting values of  and  in the given expression,

=

=  =  =

which is a constant and is independent of  and

### 16. If  with  prove that

Ans. Given:

=

=

[Taking reciprocal]

### 17. If  and  find

Ans. Given:  and

Differentiating both sides with respect to

and

and

and

and

Now

Again   =

=  =

=

### 18. If  show that  exists for all real  and find it.

Ans. Given: =

Now, L.H.D. at

=

=

L.H.D. at  = R.H.D. at

Therefore,  is differentiable at .

Now, L.H.D. at

=  =

And R.H.D. at

=

=

Again L.H.D. at  = R.H.D. at

### 19. Using mathematical induction, prove that  for all positive integers

Ans. Let  be the given statement in the problem.

…..(i)

= , which is true as

Now we suppose  is true.

…….(ii)

To establish the truth of  we prove,

=

=

Therefore,  is true if  is true but  is true.

By Principal of Induction  is true for all  N.

### 20. Using the fact that  and the differentiation, obtain the sum formula for cosines.

Ans. Given:

Consider A and B as function of  and differentiating both sides w.r.t.

### 21. Does there exist a function which is continuous everywhere but not differentiable at exactly two points?

Ans. Let us consider the function

is continuous everywhere but it is not differentiable at  and

22. If  prove that

Ans. Given:

=

### .23. If  show that

Ans. Given:

=

=

Differentiating both sides with respect to

Proved.

## NCERT Solutions class 12 Maths Miscellaneous

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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1. Very nice