NCERT Solutions Class 12

Miscellaneous

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)

 

 

  

 

 

 

 

 

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4.  

Ans. Let  = 

 

= 

= 

= 

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5.  

Ans. Let 

     [By Quotient Rule]

 

  = 

  = 

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

6.  

Ans. Let   ……….(i)

Now, 

=  = 

And 

=  = 

 From eq. (i),  

= 

  = 

 

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7.  

Ans. Let  ……….(i)

  = 

 

 

 

= 

 

= 

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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 

 

= 

= 

= 

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13. Find  if  

Ans. Given:     

 

 

 

= 

 

=  = 0

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14. If  for  prove that  

Ans. Given: 

 

 

 

 

 

 

 

 

 

 

= 

=      Proved.

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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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17. If  and  find  

Ans. Given:  and 

Differentiating both sides with respect to 

 and 

  and 

        and 

  and  

Now    

Again   = 

=  = 

= 

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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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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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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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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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.23. If  show that  

Ans. Given:     

 

= 

= 

 

 

Differentiating both sides with respect to 

 

       Proved.