Rd Sharma XII Vol 1 2018 Solutions for Class 12 Science Math Chapter 12 Higher Order Derivatives are provided here with simple step-by-step explanations. These solutions for Higher Order Derivatives are extremely popular among Class 12 Science students for Math Higher Order Derivatives Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma XII Vol 1 2018 Book of Class 12 Science Math Chapter 12 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma XII Vol 1 2018 Solutions. All Rd Sharma XII Vol 1 2018 Solutions for class Class 12 Science Math are prepared by experts and are 100% accurate.

#### 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 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 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 $\frac{{d}^{2}y}{d{x}^{2}}$.

Here,

#### 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