NCERT Solutions Class 12

NCERT Solutions Class 12

Continuity and Differentiability : NCERT Solutions – Class 12 Maths (Ex 9)

Miscellaneous

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.

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