NCERT Solutions class 12 Maths Miscellaneous

NCERT Solutions class 12 Maths Miscellaneous 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

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

1.  

Ans. Let

 

  =

 

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2.  

Ans. Let  =

 

  =

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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

  

 

 

 

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9.  

Ans. Let  ……….(i)

 

=

 

 

  

 

 

 

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10.  for some fixed  and  

Ans. Let

 

=   …….(i)

Now taking ,  let  ……….(ii)

  =

 

 

 

=

 

 From eq. (ii),

 From eq. (i),

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11.  for  

Ans. Let  for

Putting  and

  ……….(i)

Now

  =

 

=

 

 

  ……….(ii)

Again 

 

=

 

=

 

 

  ……….(iii)

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

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12. Find  if  and  

Ans. Given:      and

  and

 

=

=

=

---

13. Find  if  

Ans. Given:    

 

 

 

=

 

=  = 0

---

14. If  for  prove that  

Ans. Given:

 

 

 

 

 

 

 

 

 

 

=

=      Proved.

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NCERT Solutions class 12 Maths Miscellaneous

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

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16. If  with  prove that  

Ans. Given:

 

 

 

 

=

 

=

   [Taking reciprocal]

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NCERT Solutions class 12 Maths Miscellaneous

17. If  and  find  

Ans. Given:  and

Differentiating both sides with respect to

 and

  and

        and

  and 

Now   

Again   =

=  =

=

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NCERT Solutions class 12 Maths Miscellaneous

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

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NCERT Solutions class 12 Maths Miscellaneous

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.

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NCERT Solutions class 12 Maths Miscellaneous

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.

  

 

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NCERT Solutions class 12 Maths Miscellaneous

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

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22. If  prove that

Ans. Given:

 

=

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NCERT Solutions class 12 Maths Miscellaneous

.23. If  show that  

Ans. Given:    

 

=

=

 

 

Differentiating both sides with respect to

 

       Proved.

NCERT Solutions class 12 Maths Miscellaneous

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