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# RD Sharma solutions for Class 12 Mathematics chapter 12 - Higher Order Derivatives

## Chapter 12: Higher Order Derivatives

Ex. 12.1Others

#### Chapter 12: Higher Order Derivatives Exercise 12.1 solutions [Pages 16 - 18]

Ex. 12.1 | Q 1.1 | Page 16

Find the second order derivatives of the following function  x3 + tan x ?

Ex. 12.1 | Q 1.2 | Page 16

Find the second order derivatives of the following function sin (log x) ?

Ex. 12.1 | Q 1.3 | Page 16

Find the second order derivatives of the following function  log (sin x) ?

Ex. 12.1 | Q 1.4 | Page 16

Find the second order derivatives of the following function ex sin 5x  ?

Ex. 12.1 | Q 1.5 | Page 16

Find the second order derivatives of the following function e6x cos 3x  ?

Ex. 12.1 | Q 1.6 | Page 16

Find the second order derivatives of the following function x3 log ?

Ex. 12.1 | Q 1.7 | Page 16

Find the second order derivatives of the following function tan−1 x ?

Ex. 12.1 | Q 1.8 | Page 16

Find the second order derivatives of the following function x cos x ?

Ex. 12.1 | Q 1.9 | Page 16

Find the second order derivatives of the following function  log (log x)  ?

Ex. 12.1 | Q 2 | Page 16

If y = ex cos x, show that $\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x$ ?

Ex. 12.1 | Q 3 | Page 16

If y = x + tan x, show that  $\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0$ ?

Ex. 12.1 | Q 4 | Page 16

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

Ex. 12.1 | Q 5 | Page 16

If y = log (sin x), prove that $\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x$ ?

Ex. 12.1 | Q 6 | Page 16

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

Ex. 12.1 | Q 7 | Page 16

If $y = \frac{\log x}{x}$ show that $\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}$ ?

Ex. 12.1 | Q 8 | Page 16

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

Ex. 12.1 | Q 9 | Page 16

If x=a (cos t +t sint )and y= a(sint-cos t )  Prove that Sec^3 t/(at),0<t< pi/2

Ex. 12.1 | Q 10 | Page 16

If y = ex cos x, prove that $\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)$ ?

Ex. 12.1 | Q 11 | Page 16

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

Ex. 12.1 | Q 12 | Page 16

If x = a (1 − cos3 θ), y = a sin3 θ, prove that $\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}$ ?

Ex. 12.1 | Q 13 | Page 16

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

Ex. 12.1 | Q 14 | Page 16

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

Ex. 12.1 | Q 15 | Page 16

If x = a(1 − cos θ), y = a(θ + sin θ), prove that $\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}$ ?

Ex. 12.1 | Q 16 | Page 17

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

Ex. 12.1 | Q 17 | Page 17

If x = cos θ, y = sin3 θ, prove that $y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)$ ?

Ex. 12.1 | Q 18 | Page 17

If y = sin (sin x), prove that $\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0$ ?

Ex. 12.1 | Q 19 | Page 17

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

Ex. 12.1 | Q 20 | Page 17

If y = (sin−1 x)2, prove that (1 − x2)

$\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0$ ?

Ex. 12.1 | Q 21 | Page 17

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

Ex. 12.1 | Q 22 | Page 17

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

Ex. 12.1 | Q 23 | Page 17

If $y = e^{2x} \left( ax + b \right)$  show that  $y_2 - 4 y_1 + 4y = 0$ ?

Ex. 12.1 | Q 24 | Page 17

If $x = \sin \left( \frac{1}{a}\log y \right)$ show that (1 − x2)y2 − xy1 − a2y = 0 ?

Ex. 12.1 | Q 25 | Page 17

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

Ex. 12.1 | Q 26 | Page 17

If y = tan−1 x, show that $\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0$ ?

Ex. 12.1 | Q 27 | Page 17

If $y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2$ show that $\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2$ ?

Ex. 12.1 | Q 28 | Page 17

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

Ex. 12.1 | Q 29 | Page 17

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

Ex. 12.1 | Q 30 | Page 17

Find $\frac{d^2 y}{d x^2}$ where $y = \log \left( \frac{x^2}{e^2} \right)$ ?

Ex. 12.1 | Q 31 | Page 17

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

Ex. 12.1 | Q 32 | Page 17

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

Ex. 12.1 | Q 33 | Page 17

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

Ex. 12.1 | Q 34 | Page 17

If  $y = e^{a \cos^{- 1}} x$ ,prove that $\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0$ ?

Ex. 12.1 | Q 35 | Page 17

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

Ex. 12.1 | Q 36 | Page 17

If x = 2 cos t − cos 2ty = 2 sin t − sin 2t, find $\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}$ ?

Ex. 12.1 | Q 37 | Page 17

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

Ex. 12.1 | Q 38 | Page 17

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

Ex. 12.1 | Q 39 | Page 17

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

Ex. 12.1 | Q 40 | Page 18

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

Ex. 12.1 | Q 41 | Page 18

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

Ex. 12.1 | Q 42 | Page 18

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

Ex. 12.1 | Q 43 | Page 18

$\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4}$ ?

Ex. 12.1 | Q 44 | Page 18

$\text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2}$ ?

Ex. 12.1 | Q 45 | Page 18

$\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4}$ ?

Ex. 12.1 | Q 46 | Page 18

$\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3}$ ?

Ex. 12.1 | Q 47 | Page 18

$\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2}$ ?

Ex. 12.1 | Q 48 | Page 18

If $x = 3 \ cot - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,$ find $\frac{d^2 y}{d x^2}$ ?

Ex. 12.1 | Q 49 | Page 18

$\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3}$ ?

Ex. 12.1 | Q 50 | Page 18

$\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }$

$\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x$ ?

Ex. 12.1 | Q 51 | Page 18

$\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2$ ?

Ex. 12.1 | Q 52 | Page 18

$\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0$ Disclaimer: There is a misprint in the question. It must be

$x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0$ instead of 1

$x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0$ ?

Ex. 12.1 | Q 53 | Page 18

$\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0$

Disclaimer: There is a misprint in the question,

$\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0$ must be written instead of

$\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0$ ?

#### Chapter 12: Higher Order Derivatives solutions [Page 22]

Q 1 | Page 22

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

Q 2 | Page 22

If x = a cos nt − b sin nt and $\frac{d^2 x}{dt} = \lambda x$  then find the value of λ ?

Q 3 | Page 22

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

Q 4 | Page 22

If x = 2aty = at2, where a is a constant, then find $\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}$ ?

Q 5 | Page 22

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

Q 6 | Page 22

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 ?

Q 7 | Page 22

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

Q 8 | Page 22

If y = |x − x2|, then find $\frac{d^2 y}{d x^2}$ ?

Q 9 | Page 22

If $y = \left| \log_e x \right|$ find$\frac{d^2 y}{d x^2}$ ?

#### Chapter 12: Higher Order Derivatives solutions [Pages 22 - 24]

Q 1 | Page 22

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

• n2 x

• −n2 x

• −nx

• nx

Q 2 | Page 22

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

• $- \frac{1}{t^2}$

• $\frac{1}{2 \ at^3}$

• $- \frac{1}{t^3}$

• $- \frac{1}{ 2 \ at^3}$

Q 3 | Page 23

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

• n (n − 1)y

• n (n − 1)y

•  ny

•  n2y

Q 4 | Page 23

$\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =$

• 220 (cos 2 x − 220 cos 4 x)

• 220 (cos 2 x + 220 cos 4 x)

• 220 (sin 2 x + 220 sin 4 x)

• 220 (sin 2 x − 220 sin 4 x)

Q 5 | Page 23

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

• 3/2

• 3/4t

• 3/2t

• 3t/2

Q 6 | Page 23

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

• $\frac{d^2 y}{d x^2} = 2xy$

• $x\frac{d^2 y}{d x^2} = y_1$

• $x\frac{d^2 y}{d x^2} - \frac{dy}{dx} + y = 0$

• $x\frac{d^2 y}{d x^2} = 2 xy$

Q 7 | Page 23

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

• $\frac{n\left( n + 1 \right)}{2}$

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

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

• none of these

Q 8 | Page 23

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

• −m2y

• m2y

• −my

• my

Q 9 | Page 23

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

• 1

• −1

• 0

• none of these

Q 10 | Page 23

If $y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)$, then $\frac{d^2 y}{d x^2} =$

• 2

• 1

• 0

• −1

Q 11 | Page 23

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

• f'' (ex) e2x + f'(ex) ex

•  f'' (ex) ex + f' (ex)

• f'' (ex) e2x + f'' (ex) ex

•  f'' (ex)

Q 12 | Page 23

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

• 0

• y

• y

• none of these

Q 13 | Page 23

If x = 2 at, y = at2, where a is a constant, then $\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}$ is

• 1/2a

• 1

• 2a

• none of these

Q 14 | Page 23

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

• $\frac{f' g'' - g'f''}{\left( f' \right)^3}$

• $\frac{f' g'' - g'f''}{\left( f' \right)^2}$

• $\frac{g''}{f''}$

• $\frac{f'' g' - g'' f'}{\left( g' \right)^3}$

Q 15 | Page 24

If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to

• m2y

• my

• −m2y

• none of these

Q 16 | Page 24

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

• xy1 + 2

•  xy1 − 2

• xy1+2

• none of these

Q 17 | Page 24

If y = etan x, then (cos2 x)y2 =

• (1 − sin 2xy1

• −(1 + sin 2x)y1

• (1 + sin 2x)y1

• none of these

Q 19 | Page 24

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

• 3(xy2 + y1)y2

• 3(xy1 + y2)y2

• 3(xy2 + y1)y1

• none of these

Q 20 | Page 24

If $y = \log_e \left( \frac{x}{a + bx} \right)^x$ then x3 y2 =

• (xy1 − y)2

• (1 + y)2

• $\left( \frac{y - x y_1}{y_1} \right)^2$

• none of these

Q 21 | Page 24

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

•  f(t) − f''(t)

• {f(t) − f'' (t)}2

• {f(t) + f''(t)}2

• none of these

Q 22 | Page 24

If $y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =$ ?

• -n^2y

• my

• n^2y

• None of these

Q 23 | Page 24

If $\frac{d}{dx}\left[ x^n - a_1 x^{n - 1} + a_2 x^{n - 2} + . . . + \left( - 1 \right)^n a_n \right] e^x = x^n e^x$ then the value of ar, 0 < r ≤ n, is equal to

• $\frac{n!}{r!}$

• $\frac{\left( n - r \right)!}{r!}$

• $\frac{n!}{\left( n - r \right)!}$

• none of these

Q 24 | Page 24

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

• −(n − 1)2 y

• (n − 1)2y

• −n2y

•  n2y

Q 25 | Page 24

If xy − loge y = 1 satisfies the equation $x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0$

• −3

• 1

• 3

• none of these

Q 26 | Page 24

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

• a constant

• a function of x only

• a function of y  only

• a function of x and y

Ex. 12.1Others

## RD Sharma solutions for Class 12 Mathematics chapter 12 - Higher Order Derivatives

RD Sharma solutions for Class 12 Maths chapter 12 (Higher Order Derivatives) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 12 Mathematics chapter 12 Higher Order Derivatives are Maximum and Minimum Values of a Function in a Closed Interval, Maxima and Minima, Simple Problems on Applications of Derivatives, Graph of Maxima and Minima, Approximations, Tangents and Normals, Increasing and Decreasing Functions, Rate of Change of Bodies Or Quantities, Introduction to Applications of Derivatives.

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