NCERT Solutions class 12 Maths Exercise 5.7 (Ex 5.7) Chapter 5 Continuity and Differentiability

NCERT Solutions for Class 12 Maths Exercise 5.7 Chapter 5 Continuity and Differentiability – FREE PDF Download

Free PDF download of NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.7 (Ex 5.7) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.7 Questions with Solutions to help you to revise complete Syllabus and Score More marks.

NCERT Solutions for Class 12 Maths Chapter 5 Continuity and Differentiability Exercise 5.7 (Ex 5.7)

Find the second order derivatives of the functions given in Exercises 1 to 5.

1.

Ans. Let

2.

Ans. Let

3.

Ans. Let

=

=

=

=

=

=

4.

Ans. Let

=

5.

Ans. Let

=

=

=

=

=

=

=

=

=

Find the second order derivatives of the functions given in Exercises 6 to 10.

6.

Ans. Let

=

=

$= e^{x} [5 (- s i n 5 x) \times 5 + (c o s 5 x) \times 5] + (5 c o s 5 x + s i n 5 x) e^{x}$

=

=

=

7.

Ans. Let

=

=

= $e^{6 x} (- 3 \sin 3 x + 6 \cos 3 x)$

=

=

=

8.

Ans. Let

=

= =

9.

Ans. Let

=

=

= $\frac{- [x . \frac{1}{x} + \log x \times 1]}{{(x \log x)}^{2}}$

= $\frac{- [1 + \log x]}{{(x \log x)}^{2}}$

10.

Ans. Let

= =

=

=

=

11. If prove that

Ans. Let

…….(i)

= = [From eq. (i)]

12. If Find in terms of alone.

Ans. Given:

……….(i)

= [From eq. (i)]

= ……….(ii)

=

=

=

[From eq. (ii)]

13. If show that

Ans. Given:

….(i)

=

Now

[From eq. (i)]

Hence proved.

14. If show that

Ans. Given:

….(i)

To prove:

….(ii)

Again
= ….(iii)

Now, L.H.S.=
= $A m^{2} e^{m x} + B n^{2} e^{n x} - (m + n) (A m e^{m x} + B n e^{n x}) + m n (A e^{m x} + B e^{n x})$

= $A m^{2} e^{m x} + B n^{2} e^{n x} - A m^{2} e^{m x} - B m n e^{n x} - A m n e^{m x} - B n^{2} e^{n x} + A m n e^{m x} + B m n e^{n x}$

= 0

= R.H.S. Hence proved.

15. If show that

Ans. Given:

……….(i)

= $500 (7) e^{7 x} - 600 (7) e^{- 7 x}$

$\frac{d^{2} y}{d x^{2}} = 500 (7) e^{7 x} (7) - 600 (7) e^{- 7 x} (- 7)$ = $500 (49) e^{7 x} + 600 (49) e^{- 7 x}$

$\frac{d^{2} y}{d x^{2}} = 49 [500 e^{7 x} + 600 e^{- 7 x}]$

= [From eq. (i)]

= Hence proved.

16. If show that

Ans. Given:

Taking log on both sides,

=

And

Now, L.H.S. =

And R.H.S. =

L.H.S. = R.H.S. Hence proved.

17. If show that

Ans. Given:

……….(i)

And

=

Again differentiating both sides w.r.t.

Hence proved.