RD SHARMA Solutions for Class 12-science Maths Chapter 12 - Higher Order Derivatives

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Chapter 12 - Higher Order Derivatives Exercise MCQ

Question 27

If   then   is equal to

a. 25y

b. 5y

c. -25y

d. 15y

Solution 27

Given:

Differentiating the above equation w.r.t x, we get

  

Differentiating the equation (i) w.r.t x, we get

  

Question 28

If   then   equals

a.   

b.   

c.   

d.   

Solution 28

Given:

Differentiating the above equation w.r.t x, we get

  

Again differentiating w.r.t x, we get

  

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Correct option: (d)

Question 6

If y = a + bx2, a,b arbitarary constansts, then

begin mathsize 12px style left parenthesis straight a right parenthesis space fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 xy
left parenthesis straight b right parenthesis space straight x fraction numerator straight d squared straight y over denominator dx squared end fraction equals y subscript 1
left parenthesis straight c right parenthesis space straight x fraction numerator straight d squared straight y over denominator dx squared end fraction minus dy over dx plus straight y equals 0
left parenthesis straight d right parenthesis space straight x fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 xy end style

Solution 6

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

Solution 7

  

Question 8

Solution 8

Question 9

If f"(x)=begin mathsize 12px style fraction numerator sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction comma end style then (1-x2) f"(x) - x f"(x)=

(a) 1

(b) -1

(c) 0

(d) none of these

Solution 9

Question 10

Solution 10

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)

Solution 11

 

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

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

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

(a) m2y

(b) my

(c) -m2y

(d) none of these

Solution 15

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

Solution 16

Question 17

If y = e tanx, then (cos2 x) y2 =

(a) (1 - sin 2x) y1

(b) - 1 (1 + sin 2x) y1

(c) (1+ sin 2x) y1

(d) none of these

Solution 17

Question 18

begin mathsize 12px style If space straight y equals fraction numerator 2 over denominator square root of straight a squared minus straight b squared end root end fraction tan to the power of negative 1 end exponent open parentheses square root of fraction numerator straight a minus straight b over denominator straight a plus straight b end fraction end root tan straight x over 2 close parentheses comma space straight a greater than straight b greater than 0 comma space then
left parenthesis straight a right parenthesis space straight y subscript 1 equals fraction numerator negative 1 over denominator straight a plus straight b space cosx end fraction
left parenthesis straight b right parenthesis space straight y subscript 2 space equals fraction numerator straight b space sinx over denominator open parentheses straight a plus straight b space cosx close parentheses squared end fraction
left parenthesis straight c right parenthesis space straight y subscript 1 equals fraction numerator 1 over denominator straight a minus bcosx end fraction
left parenthesis straight d right parenthesis space straight y subscript 2 equals fraction numerator negative straight b space sinx over denominator open parentheses straight a minus bcosx close parentheses squared end fraction end style

Solution 18

 Question is incorrect.

   

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

If y1/n + y-1/n = 2x, then (x2 - 1) y2 + xy1 =

(a) -n2y

(b) n2y

(c) 0

(d) none of these

Solution 22

Question 23

Solution 23

Question 24

If y = xn-1 log x then x2y2+(3 -2n) xy1 is equal to

(a) -3

(b) 1

(c) 3

(d) none of these

Solution 24

Question 25

If xy - log e y = 1 satisfies the equation x (yy2 +  ) - y2 + λyy1 = 0, then λ =

(a) -3

(b) 1

(c) 3

(d) none of these

Solution 25

Question 26

If y2 = ax2 + bx + c, then

(a) a constant

(b) a function of x only

(c) a function of y only

(d) a function of x and y

Solution 26

Chapter 12 - Higher Order Derivatives Exercise Ex. 12.1

Question 1(i)
Solution 1(i)
Question 1(ii)
Solution 1(ii)
Question 1(iii)

Find the second order derivative of log(sinx).

Solution 1(iii)

L e t space y equals log left parenthesis sin x right parenthesis
D i f f e r e n t i a t i n g space w i t h space r e p e c t space t o space x comma space w e space g e t comma
fraction numerator d y over denominator d x end fraction equals fraction numerator cos x over denominator sin x end fraction
A g a i n space d i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space x comma space w e space g e t comma
fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus sin x cross times sin x minus cos x cross times cos x over denominator sin squared x end fraction
rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus sin squared x minus cos squared x over denominator sin squared x end fraction
rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus open parentheses sin squared x plus cos squared x close parentheses over denominator sin squared x end fraction
rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator minus 1 over denominator sin squared x end fraction
rightwards double arrow fraction numerator d squared y over denominator d x squared end fraction equals minus cos e c squared x

Question 1(iv)
Solution 1(iv)

Question 1(v)
Solution 1(v)
Question 1(vi)
Solution 1(vi)
Question 1(vii)
Solution 1(vii)
Question 1(viii)

Solution 1(viii)

              Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 1(ix)
Solution 1(ix)
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Syntax error from line 1 column 49 to line 1 column 73. Unexpected '<mstyle '.

Question 6

Solution 6

Question 7
Solution 7
Question 8
Solution 8
Question 10

Solution 10

Question 11
Solution 11

Question 12
Solution 12
Question 13

Solution 13

Question 14

If x equals a open parentheses theta minus sin theta close parentheses comma space y equals a open parentheses 1 plus cos theta close parentheses, find fraction numerator d squared y over denominator d x squared end fraction

Solution 14

x equals a open parentheses theta minus sin theta close parentheses ; space y equals a open parentheses 1 plus cos theta close parentheses
D i i f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n s space w i t h space r e s p e c t space t o space theta comma space w e space g e t comma
fraction numerator d x over denominator d theta end fraction equals a open parentheses 1 minus cos theta close parentheses space space space... left parenthesis 1 right parenthesis
fraction numerator d y over denominator d theta end fraction equals a open parentheses minus sin theta close parentheses space space space space space space space... left parenthesis 2 right parenthesis
D i v i d i n g space e q u a t i o n space left parenthesis 2 right parenthesis space b y space left parenthesis 1 right parenthesis comma space w e space h a v e comma
fraction numerator d y over denominator d x end fraction equals fraction numerator a open parentheses minus sin theta close parentheses over denominator a open parentheses 1 minus cos theta close parentheses end fraction space equals fraction numerator minus sin theta over denominator 1 minus cos theta end fraction
D i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space theta comma space w e space h a v e comma
fraction numerator d open parentheses fraction numerator d y over denominator d x end fraction close parentheses over denominator d theta end fraction equals fraction numerator open parentheses 1 minus cos theta close parentheses open parentheses minus cos theta close parentheses plus sin theta open parentheses sin theta close parentheses over denominator open parentheses 1 minus cos theta close parentheses squared end fraction
equals fraction numerator minus cos theta plus cos squared theta plus sin squared theta over denominator open parentheses 1 minus cos theta close parentheses squared end fraction
equals fraction numerator 1 minus cos theta over denominator open parentheses 1 minus cos theta close parentheses squared end fraction
fraction numerator d open parentheses fraction numerator d y over denominator d x end fraction close parentheses over denominator d theta end fraction equals fraction numerator 1 over denominator 1 minus cos theta end fraction... left parenthesis 3 right parenthesis
D i v i d i n g space e q u a t i o n space left parenthesis 3 right parenthesis space b y space left parenthesis 1 right parenthesis comma space w e space h a v e comma
fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator 1 over denominator 1 minus cos theta end fraction cross times fraction numerator 1 over denominator a open parentheses 1 minus cos theta close parentheses end fraction
equals fraction numerator 1 over denominator a open parentheses 1 minus cos theta close parentheses squared end fraction
equals fraction numerator 1 over denominator a open parentheses 2 sin squared begin display style theta over 2 end style close parentheses squared end fraction
equals fraction numerator 1 over denominator 4 a sin to the power of 4 open parentheses theta over 2 close parentheses end fraction
equals fraction numerator 1 over denominator 4 a end fraction cos e c to the power of 4 open parentheses theta over 2 close parentheses

Question 15

Solution 15

Question 16

Solution 16

Question 17
Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

begin mathsize 12px style If space straight y equals straight e to the power of acos to the power of negative 1 end exponent straight x end exponent comma space show space that space open parentheses 1 minus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx minus straight a squared straight y equals 0 end style

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

I f space y equals cos e c to the power of minus 1 end exponent x comma space x greater than 1 comma space t h e n space s h o w space t h a t space x open parentheses x squared minus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction plus open parentheses 2 x squared minus 1 close parentheses fraction numerator d y over denominator d x end fraction equals 0

Solution 42

W e space k n o w space t h a t comma space fraction numerator d over denominator d x end fraction open parentheses cos e c to the power of minus 1 end exponent x close parentheses equals fraction numerator minus 1 over denominator open vertical bar x close vertical bar square root of x squared minus 1 end root end fraction
L e t space y equals cos e c to the power of minus 1 end exponent x
fraction numerator d y over denominator d x end fraction equals fraction numerator minus 1 over denominator open vertical bar x close vertical bar square root of x squared minus 1 end root end fraction
S i n c e space x greater than 1 comma space open vertical bar x close vertical bar equals x
T h u s comma
fraction numerator d y over denominator d x end fraction equals fraction numerator minus 1 over denominator x square root of x squared minus 1 end root end fraction... left parenthesis 1 right parenthesis
D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma
fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator x begin display style fraction numerator 2 x over denominator 2 square root of x squared minus 1 end root end fraction end style plus square root of x squared minus 1 end root over denominator x squared open parentheses x squared minus 1 close parentheses end fraction
equals fraction numerator begin display style fraction numerator x squared over denominator square root of x squared minus 1 end root end fraction end style plus square root of x squared minus 1 end root over denominator x squared open parentheses x squared minus 1 close parentheses end fraction
equals fraction numerator x squared plus x squared minus 1 over denominator x squared open parentheses x squared minus 1 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction
equals fraction numerator 2 x squared minus 1 over denominator x squared open parentheses x squared minus 1 close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction
T h u s comma space x open parentheses x squared minus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator 2 x squared minus 1 over denominator x square root of x squared minus 1 end root end fraction... left parenthesis 2 right parenthesis
S i m i l a r l y comma space f r o m space left parenthesis 1 right parenthesis comma space w e space h a v e
open parentheses 2 x squared minus 1 close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator minus 2 x squared plus 1 over denominator x square root of x squared minus 1 end root end fraction... left parenthesis 3 right parenthesis
T h u s comma space f r o m space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space h a v e comma
space x open parentheses x squared minus 1 close parentheses fraction numerator d squared y over denominator d x squared end fraction plus open parentheses 2 x squared minus 1 close parentheses fraction numerator d y over denominator d x end fraction equals fraction numerator 2 x squared minus 1 over denominator x square root of x squared minus 1 end root end fraction plus open parentheses fraction numerator minus 2 x squared plus 1 over denominator x square root of x squared minus 1 end root end fraction close parentheses equals 0
H e n c e space p r o v e d.

Question 43

I f space x equals cos t plus log tan t over 2 comma space y equals sin t comma space t h e n space f i n d space t h e space v a l u e space o f space fraction numerator d squared y over denominator d t squared end fraction space a n d space fraction numerator d squared y over denominator d x squared end fraction space a t space t equals straight pi over 4.

Solution 43

G i v e n space t h a t comma space x equals cos t plus log tan t over 2 comma space y equals sin t
D i f f e r e n t i a t i n g space w i t h space r e s p e c t space t o space t comma space w e space h a v e comma
fraction numerator d x over denominator d t end fraction equals minus sin t plus fraction numerator space 1 over denominator tan begin display style t over 2 end style end fraction cross times s e c squared t over 2 cross times 1 half
equals minus sin t plus fraction numerator space 1 over denominator begin display style fraction numerator sin begin display style t over 2 end style over denominator cos t over 2 end fraction end style end fraction cross times fraction numerator 1 over denominator cos squared begin display style t over 2 end style end fraction cross times 1 half
equals minus sin t plus fraction numerator space 1 over denominator begin display style fraction numerator sin begin display style t over 2 end style over denominator cos t over 2 end fraction end style end fraction cross times fraction numerator 1 over denominator cos squared begin display style t over 2 end style end fraction cross times 1 half
equals minus sin t plus fraction numerator space 1 over denominator 2 sin t over 2 cos t over 2 end fraction
equals minus sin t plus fraction numerator space 1 over denominator sin t end fraction
equals fraction numerator 1 minus sin squared t over denominator sin t end fraction
equals fraction numerator cos squared t over denominator sin t end fraction
equals cos t cross times c o t t

N o w space f i n d space t h e space v a l u e space o f space fraction numerator d y over denominator d t end fraction :
fraction numerator d y over denominator d t end fraction equals cos t
T h u s comma space fraction numerator d y over denominator d x end fraction equals fraction numerator d y over denominator d t end fraction cross times fraction numerator d t over denominator d x end fraction equals cos t cross times fraction numerator 1 over denominator cos t cross times c o t t end fraction
rightwards double arrow fraction numerator d y over denominator d x end fraction equals tan t
S i n c e space fraction numerator d y over denominator d t end fraction equals cos t comma space w e space h a v e space fraction numerator d squared y over denominator d t squared end fraction equals minus sin t
A t space t equals straight pi over 4 comma space open parentheses fraction numerator d squared y over denominator d t squared end fraction close parentheses subscript t equals straight pi over 4 end subscript equals minus sin open parentheses straight pi over 4 close parentheses equals fraction numerator minus 1 over denominator square root of 2 end fraction

 

fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator begin display style fraction numerator d over denominator d t end fraction end style open parentheses begin display style fraction numerator d y over denominator d x end fraction end style close parentheses over denominator begin display style fraction numerator d x over denominator d t end fraction end style end fraction
equals fraction numerator begin display style fraction numerator d over denominator d t end fraction end style open parentheses begin display style tan t end style close parentheses over denominator begin display style cos t cross times c o t t end style end fraction
equals fraction numerator s e c squared t over denominator begin display style cos t cross times c o t t end style end fraction
equals fraction numerator s e c squared t over denominator begin display style cos t cross times fraction numerator begin display style cos t end style over denominator sin t end fraction end style end fraction
equals fraction numerator s e c squared t over denominator begin display style cos squared t end style end fraction cross times sin t
equals s e c to the power of 4 t cross times sin t
T h u s comma space open parentheses fraction numerator d squared y over denominator d x squared end fraction close parentheses subscript t equals straight pi over 4 end subscript equals s e c to the power of 4 open parentheses straight pi over 4 close parentheses cross times sin straight pi over 4 equals 2


Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 49

Solution 49

Question 50

Solution 50


Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 9

If   prove that   and   

Solution 9

Given:

Differentiating 'x' w.r.t   we get

  

Differentiating 'y' w.r.t   we get

  

Dividing (ii) by (i), we get

  … (iii)

Differentiating above equation w.r.t x, we get

  

Hence,   

Question 48

If   find   

Solution 48

Given:

Differentiating 'x' w.r.t t, we get

  

Differentiating 'y' w.r.t t, we get

  

Dividing (ii) by (i), we get

  

Differentiating above equation w.r.t x, we get

  

Hence,   

Chapter 12 - Higher Order Derivatives Exercise Ex. 12VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

y equals x plus e to the power of x
D i f f e r e n t i a t i n g space t h e space a b o v e space f u n c t i o n space w i t h space r e s p e c t space t o space x comma space w e space h a v e comma
fraction numerator d y over denominator d x end fraction equals 1 plus e to the power of x
T h u s comma space fraction numerator d x over denominator d y end fraction equals fraction numerator 1 over denominator 1 plus e to the power of x end fraction
D i f f e r e n t i a t i n g space t h e space a b o v e space e q u a t i o n space w i t h space r e s p e c t space t o space y comma space w e space h a v e comma
fraction numerator d squared x over denominator d y squared end fraction equals fraction numerator minus e to the power of x over denominator open parentheses 1 plus e to the power of x close parentheses squared end fraction cross times fraction numerator d x over denominator d y end fraction
equals fraction numerator minus e to the power of x over denominator open parentheses 1 plus e to the power of x close parentheses cubed end fraction

Question 8

Solution 8

Question 9

Solution 9