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#### Question 1:

Find the second order derivatives of each of the following functions:

(i) x3 + tan x
(ii) sin (log x)
(iii) log (sin x)
(iv) ex sin 5x
(v) e6x cos 3x
(vi) x3 log x
(vii) tan−1 x
(viii) x cos x
(ix) log (log x)

(i) We have,

(ii) We have,

(iii) We have,

(iv) We have,

(v) We have,

(vi) We have,

(vii) We have,

(viii) We have,

(ix) We have,

#### Question 2:

If y = ex cos x, show that .

Here,

Hence proved.

#### Question 3:

If y = x + tan x, show that .

Here,

Hence proved.

#### Question 4:

If y = x3 log x, prove that $\frac{{d}^{4}y}{d{x}^{4}}=\frac{6}{x}$.

Here,

Hence proved.

#### Question 5:

If y = log (sin x), prove that .

Here,

Hence proved.

#### Question 6:

If y = 2 sin x + 3 cos x, show that $\frac{{d}^{2}y}{d{x}^{2}}+y=0$.

Here,

Hence proved.

If , show that .

Here,

Hence proved.

#### Question 8:

If x = a sec θ, y = b tan θ, prove that $\frac{{d}^{2}y}{d{x}^{2}}=-\frac{{b}^{4}}{{a}^{2}{y}^{3}}$.

Here,

Hence proved.

#### Question 9:

If x = a (cos θ + θ sin θ), y = a (sin θ – θ cos θ), prove that

We have, $x=a\left(\mathrm{cos}\theta +\theta \mathrm{sin}\theta \right)$

From (i) and (ii), we have
$\frac{dy}{dx}=\frac{dy}{d\theta }}{dx}{d\theta }}=\frac{a\theta \mathrm{sin}\theta }{a\theta \mathrm{cos}\theta }=\mathrm{tan}\theta \phantom{\rule{0ex}{0ex}}\frac{{d}^{2}y}{d{x}^{2}}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d\left(\mathrm{tan}\theta \right)}{dx}={\mathrm{sec}}^{2}\theta \frac{d\theta }{dx}\phantom{\rule{0ex}{0ex}}⇒\frac{{d}^{2}y}{d{x}^{2}}=\left({\mathrm{sec}}^{2}\theta \right)\left(\frac{1}{a\theta \mathrm{cos}\theta }\right)\phantom{\rule{0ex}{0ex}}⇒\frac{{d}^{2}y}{d{x}^{2}}=\frac{{\mathrm{sec}}^{3}\theta }{a\theta }$
Hence proved.

#### Question 10:

If y = ex cos x, prove that .

Here,

Hence proved.

#### Question 11:

If x = a cos θ, y = b sin θ, show that $\frac{{d}^{2}y}{d{x}^{2}}=-\frac{{b}^{4}}{{a}^{2}{y}^{3}}$.

Here,

Hence proved.

#### Question 12:

If x = a (1 − cos3 θ), y = a sin3 θ, prove that .

Here,

#### Question 13:

If x = a (θ + sin θ), y = a (1 + cos θ), prove that $\frac{{d}^{2}y}{d{x}^{2}}=-\frac{a}{{y}^{2}}$.

Here,

Hence proved.

#### Question 14:

If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find $\frac{{d}^{2}y}{d{x}^{2}}$.

Here,

#### Question 15:

If x = a(1 − cos θ), y = a(θ + sin θ), prove that .

Here,

Hence proved.

#### Question 16:

If x = a (1 + cos θ), y = a(θ + sin θ), prove that .

Here,

#### Question 17:

If x = cos θ, y = sin3 θ, prove that .

Here,

Hence proved.

#### Question 18:

If y = sin (sin x), prove that .

Here,

Hence proved.

#### Question 19:

If x = sin t, y = sin pt, prove that $\left(1-{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{p}^{2}y=0$.

Here,
.

Hence proved.

#### Question 20:

If y = (sin−1 x)2, prove that (1 − x2) $\frac{{d}^{2}y}{d{x}^{2}}-x\frac{dy}{dx}+{p}^{2}y=0$.

Here,

Hence proved.

#### Question 21:

If $y={e}^{{\mathrm{tan}}^{-1}x}$, prove that (1 + x2)y2 + (2x − 1)y1 = 0.

Here,

Hence proved.

#### Question 22:

If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0.

Here,

Hence proved.

#### Question 23:

If $y={e}^{2x}\left(ax+b\right)$, show that ${y}_{2}-4{y}_{1}+4y=0$.

Given,

$y={e}^{2x}\left(ax+b\right)$

To prove: ${y}_{2}-4{y}_{1}+4y=0$

Proof:

We have,

$y={e}^{2x}\left(ax+b\right)$         ...(i)

#### Question 24:

If , show that (1 − x2)y2xy1a2y = 0.

Here,

#### Question 25:

If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0

Here,

Hence proved.

#### Question 26:

If y = tan−1 x, show that .

Here,

Hence proved.

#### Question 27:

If , show that $\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}=2$.

Here,

#### Question 28:

If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2.

Here,

Hence proved.

#### Question 29:

If y = cot x show that $\frac{{d}^{2}y}{d{x}^{2}}+2y\frac{dy}{dx}=0$.

Here,

Hence proved.

#### Question 30:

Find $\frac{{d}^{2}y}{d{x}^{2}}$, where .

Here,

#### Question 31:

If y = ae2x + bex, show that, $\frac{{d}^{2}y}{d{x}^{2}}-\frac{dy}{dx}-2y=0$.

Here,

Hence proved.

#### Question 32:

If y = ex (sin x + cos x) prove that $\frac{{d}^{2}y}{d{x}^{2}}-2\frac{dy}{dx}+2y=0$.

Here,

Hence proved.

#### Question 33:

If y = cos−1 x, find $\frac{{d}^{2}y}{d{x}^{2}}$ in terms of y alone.

Here,

#### Question 34:

If , prove that .

Here,

Hence proved.

#### Question 35:

If y = 500 e7x + 600 e−7x, show that $\frac{{d}^{2}y}{d{x}^{2}}=49y$.

Here,

#### Question 36:

If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find .

Here,

#### Question 37:

If x = 4z2 + 5, y = 6z2 + 7z + 3, find $\frac{{d}^{2}y}{d{x}^{2}}$.

Here,

#### Question 38:

If y log (1 + cos x), prove that $\frac{{d}^{3}y}{d{x}^{3}}+\frac{{d}^{2}y}{d{x}^{2}}·\frac{dy}{dx}=0$

Here,

#### Question 39:

If y = sin (log x), prove that ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}+y=0$.

Here,

#### Question 40:

If y = 3 e2x + 2 e3x, prove that $\frac{{d}^{2}y}{d{x}^{2}}-5\frac{dy}{dx}+6y=0$

Here,

#### Question 41:

If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2.

Here,

Hence proved.

#### Question 42:

If y = cosec−1 x, x >1, then show that $x\left({x}^{2}-1\right)\frac{{d}^{2}y}{d{x}^{2}}+\left(2{x}^{2}-1\right)\frac{dy}{dx}=0$.

Here,

Hence proved.

#### Question 48:

If  find $\frac{{d}^{2}y}{d{x}^{2}}.$

We have,

Also,

Now,

So,

#### Question 52:

Disclaimer: There is a misprint in the question. It must be ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+\left(1-2n\right)x\frac{dy}{dx}+\left(1+{n}^{2}\right)y=0$ instead of ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}+\left(1-2n\right)\frac{dy}{dx}+\left(1+{n}^{2}\right)y=0$.

#### Question 53:

Disclaimer: There is a misprint in the question, $\left({x}^{2}+1\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}-{n}^{2}y=0$ must be written instead of $\left({x}^{2}-1\right)\frac{{d}^{2}y}{d{x}^{2}}+x\frac{dy}{dx}-{n}^{2}y=0.$

#### Question 1:

If x = a cos nt b sin nt, then $\frac{{d}^{2}x}{d{t}^{2}}$is

(a) n2 x
(b) −n2 x
(c) −nx
(d) nx

(b) −n2x

Here,

#### Question 2:

If x = at2, y = 2 at, then $\frac{{d}^{2}y}{d{x}^{2}}=$

(a) $-\frac{1}{{t}^{2}}$
(b)
(c) $-\frac{1}{{t}^{3}}$
(d) $-\frac{1}{2a{t}^{3}}$

(d) $-\frac{1}{2a{t}^{3}}$

Here,

#### Question 3:

If y = axn+1 + bxn, then ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}=$

(a) n (n − 1)y
(b) n (n + 1)y
(c) ny
(d) n2y

(b) n(n+1)y

Here,

#### Question 4:

(a) 220 (cos 2 x − 220 cos 4 x)
(b) 220 (cos 2 x + 220 cos 4 x)
(c) 220 (sin 2 x + 220 sin 4 x)
(d) 220 (sin 2 x − 220 sin 4 x)

(b) 220(cos2x + 220cos4x)

Here,

#### Question 5:

If x = t2, y = t3, then $\frac{{d}^{2}y}{d{x}^{2}}=$

(a) 3/2
(b) 3/4t
(c) 3/2t
(d) 3t/2

(b) 3/4t

Here,

#### Question 6:

If y = a + bx2, a, b arbitrary constants, then

(a)
(b) $x\frac{{d}^{2}y}{d{x}^{2}}={y}_{1}$
(c) $x\frac{{d}^{2}y}{d{x}^{2}}-\frac{dy}{dx}+y=0$
(d)

(b) $x\frac{{d}^{2}y}{d{x}^{2}}={y}_{1}$

Here,

#### Question 7:

If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to

(a) $\frac{n\left(n+1\right)}{2}$
(b) ${\left\{\frac{n\left(n+1\right)}{2}\right\}}^{2}$
(c) $-{\left\{\frac{n\left(n+1\right)}{2}\right\}}^{2}$
(d) none of these

(c) $-{\left\{\frac{n\left(n+1\right)}{2}\right\}}^{2}$

Here,

#### Question 8:

If y = a sin mx + b cos mx, then $\frac{{d}^{2}y}{d{x}^{2}}$ is equal to

(a) −m2y
(b) m2y
(c) −my
(d) my

(a) −m2y

Here,

#### Question 9:

If $f\left(x\right)=\frac{{\mathrm{sin}}^{-1}x}{\sqrt{1-{x}^{2}}}$, then (1 − x)2 f '' (x) − xf(x) =

(a) 1
(b) −1
(c) 0
(d) none of these

(a) 1

Here,

DISCLAIMER : In the question instead of (1 − x)2 f '' (x) − xf(x)
it should be (1 − x)2 f ' (x) − xf(x)

#### Question 10:

If , then $\frac{{d}^{2}y}{d{x}^{2}}=$

(a) 2
(b) 1
(c) 0
(d) −1

(c) 0

#### Question 11:

Let f(x) be a polynomial. Then, the second order derivative of f(ex) is

(a) f'' (ex) e2x + f'(ex) ex
(b) f'' (ex) ex + f' (ex)
(c) f'' (ex) e2x + f'' (ex) ex
(d) f'' (ex)

(a) f''(ex)e2x + f'(ex)ex

Since f(x) is a polynomial,

#### Question 12:

If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =

(a) 0
(b) y
(c) −y
(d) none of these

(c) −y

Here,

#### Question 13:

If x = 2 at, y = at2, where a is a constant, then is

(a) 1/2a
(b) 1
(c) 2a
(d) none of these

(a) 1/2a

Here,

#### Question 14:

If x = f(t) and y = g(t), then $\frac{{d}^{2}y}{d{x}^{2}}$is equal to

(a)
(b)
(c) $\frac{g\text{'}\text{'}}{f\text{'}\text{'}}$
(d)

(a)

Here,
x = f(t) and y = g(t)

#### Question 15:

If y = sin (m sin−1 x), then (1 − x2) y2xy1 is equal to

(a) m2y
(b) my
(c) −m2y
(d) none of these

(c)−m2y

Here,

#### Question 16:

If y = (sin−1 x)2, then (1 − x2)y2 is equal to

(a) xy1 + 2
(b) xy1 − 2
(c) −xy1+2
(d) none of these

(a) xy1 + 2

Here,

#### Question 17:

If y = etan x, then (cos2 x)y2 =

(a) (1 − sin 2x) y1
(b) −(1 + sin 2x)y1
(c) (1 + sin 2x)y1
(d) none of these

(c) (1 + sin 2x)y1
Here,

#### Question 18:

If , then

(a)
(b)
(c)
(d)

Disclaimer: The question given in the book is wrong.

#### Question 19:

If $y=\frac{ax+b}{{x}^{2}+c}$, then (2xy1 + y)y3 =

(a) 3(xy2 + y1)y2
(b) 3(xy1 + y2)y2
(c) 3(xy2 + y1)y1
(d) none of these

(a) 3(xy2 + y1)y2

Here,

#### Question 20:

If $y={\mathrm{log}}_{e}{\left(\frac{x}{a+bx}\right)}^{x}$, then x3 y2 =

(a) (xy1y)2
(b) (1 + y)2
(c) ${\left(\frac{y-x{y}_{1}}{{y}_{1}}\right)}^{2}$
(d) none of these

(a) (xy1y)2

Here,

#### Question 21:

If x = f(t) cos tf' (t) sin t and y = f(t) sin t + f'(t) cos t, then ${\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}=$

(a) f(t) − f''(t)
(b) {f(t) − f'' (t)}2
(c) {f(t) + f''(t)}2
(d) none of these

(c){f(t) + f''(t)}2

Here,

If

#### Question 23:

If , then the value of ar, 0 < rn, is equal to

(a) $\frac{n!}{r!}$
(b) $\frac{\left(n-r\right)!}{r!}$
(c) $\frac{n!}{\left(n-r\right)!}$
(d) none of these

(c) $\frac{n!}{\left(n-r\right)!}$

According to the given equation,

#### Question 24:

If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to

(a) −(n − 1)2 y
(b) (n − 1)2y
(c) −n2y
(d) n2y

(a) −(n − 1)2 y

Here,

#### Question 25:

If xy − loge y = 1 satisfies the equation , then λ =

(a) −3
(b) 1
(c) 3
(d) none of these

(c) 3

Here,

#### Question 26:

If y2 = ax2 + bx + c, then ${y}^{3}\frac{{d}^{2}y}{d{x}^{2}}$is

(a) a constant
(b) a function of x only
(c) a function of y  only
(d) a function of x and y

(a) a constant

Here,

#### Question 1:

If y = t10 + 1 and x = t8 + 1, then $\frac{{d}^{2}y}{d{x}^{2}}$ = ___________________.

Given, y = t10 + 1 and x = t8 + 1.

$y={t}^{10}+1$

Differentiating both sides with respect to t, we get

$\frac{dy}{dt}=10{t}^{9}$

$x={t}^{8}+1$

Differentiating both sides with respect to t, we get

$\frac{dx}{dt}=8{t}^{7}$

$\therefore \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx}=\frac{10{t}^{9}}{8{t}^{7}}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dx}=\frac{5{t}^{2}}{4}$

Differentiating both sides with respect to x, we get

$\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(\frac{5{t}^{2}}{4}\right)$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=\frac{5}{4}×2t\frac{dt}{dx}$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=\frac{5}{16{t}^{6}}$

If y = t10 + 1 and x = t8 + 1, then $\frac{{d}^{2}y}{d{x}^{2}}$ = .

#### Question 2:

If x = a sin θ and y = b cos θ, then $\frac{{d}^{2}y}{d{x}^{2}}$ = ______________________.

Given, $x=a\mathrm{sin}\theta$ and $y=b\mathrm{cos}\theta$.

$x=a\mathrm{sin}\theta$

Differentiating both sides with respect to θ, we get

$\frac{dx}{d\theta }=a\mathrm{cos}\theta$

$y=b\mathrm{cos}\theta$

Differentiating both sides with respect to θ, we get

$\frac{dy}{d\theta }=-b\mathrm{sin}\theta$

$\therefore \frac{dy}{dx}=\frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$

$⇒\frac{dy}{dx}=\frac{-b\mathrm{sin}\theta }{a\mathrm{cos}\theta }$

$⇒\frac{dy}{dx}=-\frac{b\mathrm{tan}\theta }{a}$

Differentiating both sides with respect to x, we get

$\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-\frac{b\mathrm{tan}\theta }{a}\right)$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=-\frac{b}{a}{\mathrm{sec}}^{2}\theta \frac{d\theta }{dx}$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=-\frac{b}{{a}^{2}}{\mathrm{sec}}^{3}\theta$

If x = a sin θ and y = b cos θ, then $\frac{{d}^{2}y}{d{x}^{2}}$ = .

#### Question 3:

If y = x + ex, then $\frac{{d}^{2}y}{d{x}^{2}}$ = _____________________.

$y=x+{e}^{x}$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(x+{e}^{x}\right)$

$⇒\frac{dy}{dx}=1+{e}^{x}$

Again differentiating both sides with respect to x, we get

$\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(1+{e}^{x}\right)$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=0+{e}^{x}={e}^{x}$

$\therefore \frac{{d}^{2}y}{d{x}^{2}}={e}^{x}$

If y = x + ex, then $\frac{{d}^{2}y}{d{x}^{2}}$ = .

#### Question 4:

If

$y=1-x+\frac{{x}^{2}}{2!}-\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}-...$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(1-x+\frac{{x}^{2}}{2!}-\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}-...\right)$

$⇒\frac{dy}{dx}=\frac{d}{dx}\left(1\right)-\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(\frac{{x}^{2}}{2!}\right)-\frac{d}{dx}\left(\frac{{x}^{3}}{3!}\right)+\frac{d}{dx}\left(\frac{{x}^{4}}{4!}\right)-...$

$⇒\frac{dy}{dx}=0-1+\frac{2x}{2!}-\frac{3{x}^{2}}{3!}+\frac{4{x}^{3}}{4!}-...$

$⇒\frac{dy}{dx}=-1+\frac{x}{1!}-\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}-...$

Again differentiating both sides with respect to x, we get

$\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}\left(-1+\frac{x}{1!}-\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}-...\right)$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=\frac{d}{dx}\left(-1\right)+\frac{d}{dx}\left(\frac{x}{1!}\right)-\frac{d}{dx}\left(\frac{{x}^{2}}{2!}\right)+\frac{d}{dx}\left(\frac{{x}^{3}}{3!}\right)-...$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=0+1-\frac{2x}{2!}+\frac{3{x}^{2}}{3!}-...$

$⇒\frac{{d}^{2}y}{d{x}^{2}}=1-x+\frac{{x}^{2}}{2!}-\frac{{x}^{3}}{3!}+...$

If

#### Question 5:

If y = x + ex , then $\frac{{d}^{2}x}{d{y}^{2}}$ = ______________.

$y=x+{e}^{x}$

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=\frac{d}{dx}\left(x+{e}^{x}\right)$

$⇒\frac{dy}{dx}=1+{e}^{x}$

Differentiating both sides with respect to y, we get

$\frac{d}{dy}\left(\frac{dx}{dy}\right)=\frac{d}{dy}\left(\frac{1}{1+{e}^{x}}\right)$

$⇒\frac{{d}^{2}x}{d{y}^{2}}=\frac{0-1×\frac{d}{dy}\left(1+{e}^{x}\right)}{{\left(1+{e}^{x}\right)}^{2}}$

$⇒\frac{{d}^{2}x}{d{y}^{2}}=-\frac{0+{e}^{x}\frac{dx}{dy}}{{\left(1+{e}^{x}\right)}^{2}}$

$⇒\frac{{d}^{2}x}{d{y}^{2}}=-\frac{{e}^{x}}{{\left(1+{e}^{x}\right)}^{3}}$

If y = x + ex , then $\frac{{d}^{2}x}{d{y}^{2}}$ = .

#### Question 1:

If y = a xn + 1 + bxn and ${x}^{2}\frac{{d}^{2}y}{d{x}^{2}}=\mathrm{\lambda }y$, then write the value of λ.

Here,

#### Question 2:

If x = a cos ntb sin nt and $\frac{{d}^{2}x}{dt}=\mathrm{\lambda }x$, then find the value of λ.

Here,

#### Question 3:

If x = t2 and y = t3, find$\frac{{d}^{2}y}{d{x}^{2}}$.

Here,

#### Question 4:

If x = 2at, y = at2, where a is a constant, then find .

Here,

#### Question 5:

If x = f(t) and y = g(t), then write the value of $\frac{{d}^{2}y}{d{x}^{2}}$.

Here.
x = f(t) and y = g(t)

#### Question 6:

If $y=1-x+\frac{{x}^{2}}{2!}-\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}$.....to ∞, then write $\frac{{d}^{2}y}{d{x}^{2}}$in terms of y.

Here,

#### Question 7:

If y = x + ex, find $\frac{{d}^{2}x}{d{y}^{2}}$.

Here,

#### Question 8:

If y = |xx2|, then find $\frac{{d}^{2}y}{d{x}^{2}}$.

Here,

#### Question 9:

If , find $\frac{{d}^{2}y}{d{x}^{2}}$.