GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Gujarat Board GSEB Textbook Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 Textbook Questions and Answers.

Gujarat Board Textbook Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Question 1.
x² + 3x + 2
Solution:
∴ \(\frac { dy }{ dx }\) = 2x + 3
So, \(\frac{d^{2} y}{d x^{2}}\) = \(\frac { d }{ dx }\)(2x + 3) = 2.

Question 2.
x20
Solution:
∴ \(\frac { dy }{ dx }\) = x20
So, \(\frac{d^{2} y}{d x^{2}}\) = 20 \(\frac { d }{ dx }\)(x19) = 20 x 19 x18 = 380 x18.

GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Question 3.
x cos x
Solution:
Let y = x cos x.
∴ \(\frac { dy }{ dx }\) = x \(\frac { d }{ dx }\) (cos x) + cos x \(\frac { d }{ dx }\)(x).
= x(- sin x) + cos x . 1
= – x sin x + cos x.
So, \(\frac{d^{2} y}{d x^{2}}\) = – \(\frac { d }{ dx }\)(x sin x) + \(\frac { d }{ dx }\) (cos x)
= – (x cos x + sin x . 1) – sin x
= – x cos x – 2 sin x.

Question 4.
log x
Solution:
Let y = log x
∴ \(\frac { dy }{ dx }\) = \(\frac { 1 }{ x }\) = x-1
So, \(\frac{d^{2} y}{d x^{2}}\) = \(\frac { d }{ dx }\)(x-1) = – 1. x-2 = – \(\frac{1}{x^{2}}\)

Question 5.
x³ log x
Solution:
Let y = x³ log x
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 1

Question 6.
ex sin 5x
Solution:
Let y = ex sin 5x
∴ \(\frac { dy }{ dx }\) = ex. \(\frac { d }{ dx }\)(sin 5x) + sin 5x\(\frac { d }{ dx }\)ex
= ex.cos 5x . 5 + sin 5x . ex
= ex(5 cos 5x + sin 5x)
So, \(\frac{d^{2} y}{d x^{2}}\) = ex . \(\frac { d }{ dx }\)(5 cos 5x + sin 5x) + (5 cos 5x + sin 5x) . \(\frac { d }{ dx }\)(ex)
= ex(- 25 sin 5x + 5 cos 5x) + (5 cos 5x + sin 5x) ex
= ex(- 24 sin 5x + 10 cos 5x)
= 2ex (5 cos 5x – 12 sin 5x).

GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Question 7.
e6x cos 3x
Solution:
Let y = e6x cos 3x
∴ \(\frac { dy }{ dx }\) = e6x. \(\frac { d }{ dx }\)(cos 5x) + cos 3x \(\frac { d }{ dx }\)e6x
= e6x(- 3 sin 3x (6 e6x)
= e6x(- 3 sin 3x + 6 cos 3x)
So, \(\frac{d^{2} y}{d x^{2}}\) = e6x \(\frac { d }{ dx }\)(- 3 sin 3x + 6 cos 3x) + (- 3 sin 3x + 6 cos 3x) \(\frac { d }{ dx }\)e6x
= e6x(- 9 cos 3x – 18 sin 3x) + (- 3 sin 3x + 6 cos 3x) e6x . 6
= 9 e6x(3 cos 3x – 4 sin 3x).

Question 8.
tan-1 x
Solution:
∴ \(\frac { dy }{ dx }\) = \(\frac{1}{1+x^{2}}\) = (1 + x²)-1.
So, \(\frac{d^{2} y}{d x^{2}}\) = \(\frac { dy }{ dx }\) (1 + x²)-1.
= (-1) . (1 + x²)-2 . 2x = \(\frac{-2 x}{\left(1+x^{2}\right)^{2}}\).

Question 9.
log (log x)
Solution:
Let y = log (log x)
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 2

Question 10.
sin (log x)
Solution:
Let y = sin (log x).
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 3

Question 11.
If y = 5 cos x – 3 sin x, prove that \(\frac{d^{2} y}{d x^{2}}\) + y = 0.
Solution:
We have: y = 5 cos x – 3 sin x.
∴ \(\frac { dy }{ dx }\) = – 5 sin x – 3 cos x.
So, \(\frac{d^{2} y}{d x^{2}}\) = – 5 cos x + 3 sin x = – (5 cos x – 3 sin x) = – y.
⇒ \(\frac{d^{2} y}{d x^{2}}\) + y = 0.

GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Question 12.
If y = cos-1x, find \(\frac{d^{2} y}{d x^{2}}\) in terms of y alone.
Solution:
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 4

Question 13.
If y = 3 cos (log x) + 4 sin (log x), show that x²y2 + xy1 + y = 0.
Solution:
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 5

Question 14.
If y = Aemx + Benx, show that \(\frac{d^{2} y}{d x^{2}}\) – (m + n)\(\frac { dy }{ dx }\) + mny = 0.
Solution:
We have : y = Aemx + Benx.
∴ \(\frac { dy }{ dx }\) = A . memx + B . nenx
So, \(\frac{d^{2} y}{d x^{2}}\) = A . m²emx + B . n² enx.
∴ \(\frac{d^{2} y}{d x^{2}}\) – (m+n) \(\frac { dy }{ dx }\) + mny
= A m² emx + B n² enx – (m + n) x (A memx + B nenx) + mn (A emx + B enx)
= A m² emx + B n² enx – A m² emx – B mnenx – A mnemx – B n² enx + A mnemx + B mnenx
= 0.

GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

Question 15.
If y = 500 e7x + 600 e-7x, show that \(\frac{d^{2} y}{d x^{2}}\) = 49y.
Solution:
We have:
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 6

Question 16.
If ey (x + 1) = 1, show that \(\frac{d^{2} y}{d x^{2}}\) = (\(\frac { dy }{ dx }\))².
Solution:
GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7 7

Question 17.
If y = (tan-1 x)², show that (x² + 1)² y2 + 2x (x² + 1) y1 = 2.
Solution:
We have : y = (tan-1 x)².
∴ y1 = 2 tan-1x \(\frac { d }{ dx }\)(tan-1x),
⇒ y1 = \(\frac{2 \tan ^{-1} x}{1+x^{2}}\)
⇒ (1 + x²)y1 = 2 tan-1x.
Differentiating again w.r.t. x, we have :
(1 + x²)y2 + y1(2x) = \(\frac{2}{1+x^{2}}\)
⇒ (1 + x²)y2 + 2x(1 + x²)y1 = 2.

GSEB Solutions Class 12 Maths Chapter 5 Continuity and Differentiability Ex 5.7

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