## NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Miscellaneous Exercise

#### Page No 191:

#### Question 1:

#### Answer:

Using chain rule, we obtain

#### Question 2:

#### Answer:

#### Question 3:

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to

*x*, we obtain#### Question 4:

#### Answer:

Using chain rule, we obtain

#### Question 5:

#### Answer:

#### Question 6:

#### Answer:

Therefore, equation (1) becomes

#### Question 7:

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to

*x*, we obtain#### Question 8:

**,**for some constant

*a*and

*b*.

#### Answer:

By using chain rule, we obtain

#### Question 9:

#### Answer:

Taking logarithm on both the sides, we obtain

Differentiating both sides with respect to

*x*, we obtain#### Question 10:

, for some fixed and

#### Answer:

Differentiating both sides with respect to

*x*, we obtain
Differentiating both sides with respect to

*x*, we obtain*s*=

*a*

^{a}

Since

*a*is constant,*a*^{a}is also a constant.
∴

From (1), (2), (3), (4), and (5), we obtain

#### Question 11:

, for

#### Answer:

Differentiating both sides with respect to

*x,*we obtain
Differentiating with respect to

*x*, we obtain
Also,

Differentiating both sides with respect to

*x*, we obtain
Substituting the expressions of in equation (1), we obtain

#### Question 12:

Find, if

#### Answer:

#### Question 13:

Find, if

#### Answer:

#### Question 14:

If, for, −1 <

*x*<1 font="" prove="" that="">#### Answer:

_{It is given that,}

_{Differentiating both sides with respect to }

_{x}

_{, we obtain}

_{Hence, proved.}

#### Question 15:

If, for some prove that

is a constant independent of

*a*and*b*.#### Answer:

It is given that,

Differentiating both sides with respect to

*x*, we obtain
Hence, proved.

#### Page No 192:

#### Question 16:

If with prove that

#### Answer:

Then, equation (1) reduces to

⇒sina+y-ydydx=cos2a+y⇒dydx=cos2a+ysina

Hence, proved.

#### Question 17:

If and, find

#### Answer:

#### Question 18:

If, show that exists for all real

*x*, and find it.#### Answer:

It is known that,

Therefore, when

*x*≥ 0,
In this case, and hence,

When

*x*< 0,
In this case, and hence,

Thus, for, exists for all real

*x*and is given by,#### Question 19:

Using mathematical induction prove that for all positive integers

*n*.#### Answer:

_{For }

_{n}

_{ = 1,}

∴P(

*n*) is true for*n*= 1
Let P(

*k*) is true for some positive integer*k*.
That is,

It has to be proved that P(

*k*+ 1) is also true.
Thus, P(

*k*+ 1) is true whenever P (*k*) is true.
Therefore, by the principle of mathematical induction, the statement P(

*n*) is true for every positive integer*n*._{Hence, proved.}

#### Question 20:

Using the fact that sin (

*A*+*B*) = sin*A*cos*B*+ cos*A*sin*B*and the differentiation, obtain the sum formula for cosines.#### Answer:

Differentiating both sides with respect to

*x*, we obtain#### Question 21:

Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

#### Answer:

y=x -∞ It can be seen from the above graph that, the given function is continuos everywhere but not differentiable at exactly two points which are 0 and 1.

#### Question 22:

If, prove that

#### Answer:

Thus,

#### Question 23:

If, show that

#### Answer:

It is given that,

_{}

^{}