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

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

Question 1.

x2 + 3x + 2
Answer:
Let y = x2 + 3x + 2
Differentiating w.r.t. x
$\frac{dy}{dx}$ = 2x + 3
Again, differentiating w.r.t. x
$\frac{d^{2}y}{d{x}^2}$ = $\frac{d}{dx}(2x+3)$ = 2
Thus, $\frac{d^{2}y}{d{x}^2}$ = 2

Question 2.
x20
Answer:
Let y = x20
Differentiating w.r.t. x

Question 3.
x cos x
Answer:
Let y = x cos x
Differentiating w.r.t x

Question 4.
log x
Answer:
Let y = log x
Differentiating w.r.t. x
$\frac{dy}{dx}$ = $\frac{1}{x}$ = x-1
Again, differentiating w.r.t. x
$\frac{d^{2}y}{d{x}^2}$ = (- 1)x-1-1 = (- 1)x-2
∴ $\frac{d^{2}y}{d{x}^2}=-\frac{1}{x^2}$

Question 5.
x3 log x
Answer:
Let y = x3 log x
Differentiating w.r.t. x

Question 6.
ex sin 5x
Answer:
Let y = ex sin 5x
Differentiating w.r.t. x

Question 7.
e6x cos 3x
Answer:
Let y = e6x cos 3x
Differentiating w.r.t. x

Question 8.
tan-1 x
Answer:
Let y = tan-1 x
Differentiating w.r.t. x

Question 9.
log (log x)
Answer:
Let y = log(log x)
Differentiating w.r.t. x

Question 10.
sin (log x)
Answer:
Let y = sin (log x)
Differentiating w.r.t. x

Question 11.
If y = 5 cos x - 3 sin x then show that $\frac{d^{2}y}{d{x}^2}$ + y = 0.
Answer:
Given y = 5 cos x - 3 sin x
Differentiating w.r.t. x
$\frac{dy}{dx}$ = 5 $\frac{d}{dx}$ cos x - 3 $\frac{d}{dx}$ sin x
⇒ $\frac{dy}{dx}$ = 5(- sin x) - 3 cos x
⇒ $\frac{dy}{dx}$ = - 5 sin x - 3 cos x
Again, differentiating w.r.t. x
$\frac{d^{2}y}{d{x}^2}$ = - 5 cos x - 3(- sin x)
= -5 cos x + 3 sin x
= - (5 cos x - 3 sin x) = - y [From (1)]
Thus, $\frac{d^{2}y}{d{x}^2}$ + y = 0
Hence proved.

Question 12.
If y = cos-1 x then find $\frac{d^{2}y}{d{x}^2}$ only in terms of y.
Answer:
Differentiating y = cos-1x w.r.t. x

Question 13.
If y = 3 cos (log x) + 4 sin (log x) then show that x2y2 + xy1 + y = 0.
Answer:
y = 3 cos(log x) + 4 sin(log x)
Differentiating both sides w.r.t. x

Question 14.
If y = Aemx + Benx then show that
$\frac{d^{2}y}{d{x}^2}$ - (m + n)$\frac{dy}{dx}$ + mny = 0
Answer:
Differentiating y = Aemx + Benx w.r.t. x
$\frac{dy}{dx}$ = mAemx + nBenx
Again, differentiating w.r.t. x
$\frac{d^{2}y}{d{x}^2}$ = m2Aemx + n2Benx
Now L.H.S. = $\frac{d^{2}y}{d{x}^2}$ - (m + n)$\frac{dy}{dx}$ + mny
= m2Aemx + n2Benx - (m + n)(mAemx + nBenx)
= m2Aemx + n2Benx - m2Aemx - mnBenx - mnAemx - n2Benx + mnAemx+ mnBenx)
= 0 = R.HS.
Thus, $\frac{d^{2}y}{d{x}^2}$ -(m + n)$\frac{dy}{dx}$ + mny = 0
Hence Proved.

Question 15.
If y = 500e7x + 600e-7x then show that $\frac{d^{2}y}{d{x}^2}$ = 49y
Answer:
Given y = 500e7x + 600e-7x
Differentiating both sides w.r.t. x

Question 16.
If ey (x + 1) = 1 then show that $\frac{d^{2}y}{d{x}^2}={\left(\frac{dy}{dx}\right)}^2$
Answer:
Given ey (x + 1) = 1 ⇒ (x + 1)ey = 1
Differentiating both sides w.r.t. x

Question 17.
If y = (tan-1 x)2 then show that
(x2 + 1)2 y2 + 2x(x2 + 1)y1 = 2
Answer:
Given, y = (tan-1 x)2 Differentiating both sides w.r.t. x

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