MATHEMATICS for CLASS 12 - deepak sir13) √ 3− 2−7 +8 , 1 √ 2− ë,√𝑙 𝑔{sin[ ë 2 3...

MATHEMATICS for CLASS 12 ( CBSE NEW CCE patten )www.cbsehelp.weebly/deepaksirweebly.com

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SHRI SAI MASTERS TITION CENTER, ARJUN NAGAR, SJE. NEW DELHI-10029. , DEEPAK SIR 9811291604

Find dy/dx of following

1) 𝑥2 sin 𝑥

2) 𝑥2𝑒𝑥

3) sin 𝑥 𝑒𝑥

4) sin 𝑥 log 𝑥

5) 𝑥2𝑎𝑥

6) sin 𝑥 log 𝑥 𝑒𝑥

7) 𝑥2 log 𝑥 𝑒𝑥

8) 𝑥2 sin 𝑥 𝑒𝑥

9) 𝑥2 sin 𝑥 log 𝑥

10) 𝑥2 + sin 𝑥

log 𝑥

11) 𝑥2 sin 𝑥

log 𝑥

12) 𝑥2+5𝑥+6

𝑥3+𝑥2+5𝑥−8

13) (𝑥2+5𝑥+6) sin 𝑥

log 𝑥 𝑒𝑥


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1) 𝑥11 , 𝑥22 , 𝑥34 , 𝑥100 , 𝑥20

2) 𝑥1

2 , 𝑥3

4 , 𝑥1

34 , 𝑥1

400 , 𝑥3

20

3) 1

𝑥2 ,

1

𝑥3 ,

1

𝑥5

1

𝑥22 ,

1

𝑥37 ,

1

𝑥42 ,

1

𝑥12

,1

𝑥12

4) (5𝑥 + 7)11 , ( 𝑥2 − 7𝑥 + 8)2 , ( 𝑥3−𝑥2 − 7𝑥 + 8)24

5) 𝑒5𝑥+7 , 𝑒 𝑥2−7𝑥+8 , 𝑒 𝑥3−𝑥2−7𝑥+8 , 𝑒𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥 , 𝑒𝑥 𝑠𝑖𝑛𝑥

6) 𝑒𝑥 + 𝑒−𝑥

𝑒𝑥 − 𝑒−𝑥

7) log(5𝑥 + 7) , log 𝑥2 − 7𝑥 + 6 , log 𝑥3−𝑥2 − 7𝑥 + 8 , log 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥

8) log [tan [𝜋

4+

𝑥

2]], log[x + √𝑎2 + 𝑥2], log [

𝑎+𝑏 𝑠𝑖𝑛𝑥

𝑎−𝑏 𝑠𝑖𝑛𝑥], log √

1+ 𝑠𝑖𝑛𝑥

1− 𝑠𝑖𝑛𝑥

𝑦 = 𝑥𝑛 ⇒ 𝑦′ = 𝑛 𝑥𝑛−1

𝑦 = ⎕𝑛 ⇒ 𝑦′ = 𝑛 ⎕𝑛−1

𝑦 = 𝑒𝑥 ⇒ 𝑦′ = 𝑒𝑥

𝑦 = 𝑒⎕ ⇒ 𝑦′ = 𝑒⎕ ⎕′

𝑦 = log 𝑥 ⇒ 𝑦′ = 1/𝑥

𝑦 = log ⎕ ⇒ 𝑦′ = 1/⎕ ⎕′

𝑦 = sin 𝑥 ⇒ 𝑦′ = cos 𝑥

𝑦 = sin ⎕ ⇒ 𝑦′ = cos ⎕


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9) sin(5𝑥 + 7) , sin(𝑥2 − 7𝑥 + 6) , sin( 𝑥3−𝑥2 − 7𝑥 + 8) , sin( 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 )

10) sin [log(5𝑥 + 7)] , 𝑠𝑖𝑛[𝑒 𝑥2−7𝑥+8] , 𝑠𝑖𝑛2(5𝑥 + 7)11

11) √𝑎𝑥 + 𝑏 , √log(5𝑥 + 7) , √sin( 𝑥3−𝑥2 − 7𝑥 + 8) , √𝑒𝑠𝑖𝑛𝑥+𝑐𝑜𝑠𝑥

12) √𝑠𝑖𝑛√𝑥 , √𝑙𝑜𝑔𝑠𝑖𝑛𝑥 , √𝑠𝑒𝑐𝑙𝑜𝑔(𝑥) , √𝑥 log 𝑥 cos 𝑥

13) √ 𝑥3−𝑥2 − 7𝑥 + 8 ,1

√𝑎2−𝑥2 , √𝑙𝑜𝑔 {sin [

𝑥2

3− 1]} 𝑥, √𝑥

Prove the following or find 𝑑𝑦

𝑑𝑥

14) y=[x + √𝑎2 + 𝑥2],𝑛then prove that 𝑑𝑦

𝑑𝑥 =

𝑛𝑦

√𝑎2+𝑥2

15) y= 𝑥 sin−1 𝑥

√1−𝑥2 + log[ √1 − 𝑥2], then prove that

𝑑𝑦

𝑑𝑥 =

sin−1 𝑥

[1−𝑥2]3/2

16) y= log [ √1+tan 𝑥

1−tan 𝑥], then prove that

𝑑𝑦

𝑑𝑥 = sec 2x. (2011)

17) y= log [ √1−𝑐𝑜𝑠𝑥

1+cos 𝑥], then prove that

𝑑𝑦

𝑑𝑥 = 2cosec 2x.

18) y= log [ √𝑥 + 1

√𝑥], then prove that

𝑑𝑦

𝑑𝑥 =

𝑥−1

2𝑥(𝑥+1)

19) y= √𝑥 + 1

√𝑥 then prove that 2x

𝑑𝑦

𝑑𝑥 = √𝑥 −

1

√𝑥

20) y= √𝑎2 + 𝑥2 , prove that y 𝑑𝑦

𝑑𝑥 -x = 0

21) 𝑥

2√𝑎2 − 𝑥2 +

𝑎2

2 sin−1 𝑥

𝑎, 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡

𝑑𝑦

𝑑𝑥 = √𝑎2 − 𝑥2

𝑦 = √𝑥 ⇒ 𝑦′ =1

2√𝑥

𝑦 = √⎕ ⇒ 𝑦′ =1

2√⎕ ⎕′


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22) sin−1 (sin 𝑥)

23) cos−1(cos 𝑥)

24) tan−1(tan 𝑥)

25) sin−1(2𝑥 √1 − 𝑥2 )

26) sin−1 ( 3𝑥 − 4𝑥3)

27) cos−1(4𝑥3 − 3𝑥)

28) cos−1(2𝑥2 − 1)

29) cos−1(1 − 2𝑥2)

30) tan−1(2𝑥

1−𝑥2)

31) tan−1(3𝑥−𝑥3

1−3𝑥2)

32) sin−1(2𝑥

1+𝑥2 )

33) cos−1(1−𝑥2

1+𝑥2 )

34) tan−1 1−cos 𝑥

sin 𝑥

35) tan−1 √1−cos 𝑥

1+cos 𝑥

36) tan−1 √1+cos 𝑥

1−cos 𝑥

37) tan−1 cos 𝑥

1+ sin 𝑥

38) tan−1 √1−cos 𝑥

1+cos 𝑥

39) tan−1 √1+sin 𝑥

1−sin 𝑥

40) tan−1( 𝑠𝑒𝑐𝑥 + tan 𝑥 )

41) sin−1( 2𝑥√1 − 𝑥2 )

42) cos−1( 2𝑥√1 − 𝑥2 )

43) tan−1{√ 1 + 𝑥2 + 𝑥}

44) tan−1{√ 1 + 𝑥2 − 𝑥}

45) tan−1 {√ 1+𝑥2 −1

𝑥}


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46) tan−1 {√ 1+𝑥2 + 1

𝑥}

47) tan−1{√ 1 + 𝑥2 + 𝑥}

48) tan−1 {√1+sin 𝑥 +√1−sin 𝑥

√1+sin 𝑥−√1−sin 𝑥 }

49) tan−1 {𝑎+𝑥

1−𝑎𝑥}

50) tan−1 {a cos 𝑥 −𝑏 𝑠𝑖𝑛𝑥

a cos 𝑥 + 𝑏 𝑠𝑖𝑛𝑥}

51) tan−1 {3𝑥𝑎2− 𝑥3

𝑎3− 3𝑎𝑥2}

52) tan−1 √ 𝑎−𝑥

𝑎+𝑥

53) tan−1 {√ 1+𝑥2 +√ 1−𝑥2

√ 1+𝑥2 −√ 1−𝑥2 }

54) sin−1(2𝑥

1+𝑥2 ) +cos−1(

1−𝑥2

1+𝑥2 )

55) tan−1(2𝑥

1−𝑥2) + cos−1(

1−𝑥2

1+𝑥2 )

56) cos−1(cos 𝑥+𝑠𝑖𝑛 𝑥

√2 )

57) sin−1 {𝑥+ √1−𝑥2

√2}

58) tan−1(4𝑥

1−4𝑥2)

59) tan−1(2𝑥+1

1−4𝑥)

60) tan−1(2𝑎𝑥

1−𝑎2𝑥) xy = k

61) tan−1(√𝑥+√𝑎

1−√𝑎𝑥)

62) tan−1(5𝑥

1−6𝑥2)

63) tan−1(𝑥

1+6𝑥2)

64) sin−1 2𝑥+1

1+4𝑥

65) tan−1 𝑥1/3+𝑎1/3

1− (𝑎𝑥)1/3

66) sin−1( 3𝑥 2𝑥+1

1+(36)𝑥)


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67) xy = k

68) 𝑥1/3 + 𝑎1/3 = 𝑦1/3

69) {𝑥 + 𝑦}2 = 2axy

70) {𝑥2 + 𝑦2}2 = 2axy

71) 𝑒𝑥−𝑦 = log𝑥

𝑦

72) sin 𝑥𝑦 + cos(𝑥 + 𝑦) = 1

73) log (𝑥2 + 𝑦2 ) = 2 tan−1 {𝑦

𝑥} , show that

dy

dx=

𝑥+𝑦

𝑥−𝑦

74) x√1 + 𝑦 + 𝑦 √1 + 𝑥 = 0 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 dy

dx=

−1

{𝑥+1}2

75) sin y = x sin ( a+ y ) , show that dy

dx=

{sin( 𝑎+𝑦)}2

sin 𝑎 ( 09, 11 , 12)

76) cos y = x cos ( a+ y ) , show that dy

dx=

{cos( 𝑎+𝑦)}2

sin 𝑎

77) ex +ey =ex+y show that dy

dx=

−ey(ey−1)

ex(ex−1)

78) √1 − 𝑥6 + √1 − 𝑦2 = 𝑎(𝑥3 − 𝑦3) , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 dy

dx=

𝑥2

𝑦2√

1−𝑦6

1−𝑥6

𝑝𝑢𝑡 𝑥3 = sin 𝐴 𝑎𝑛𝑑 𝑦3= sin B


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79) 𝑥𝑥 , 𝑥𝑠𝑖𝑛𝑥 , 𝑥cos−1 𝑥 , 𝑥cot 𝑥 , 𝑥𝑙𝑜𝑔𝑥 , 𝑥1/𝑥 , 𝑥√𝑥

80) {sin 𝑥 }𝑥 , {sin 𝑥 }𝑙𝑜𝑔𝑥 , {sin 𝑥 }𝑐𝑜𝑠𝑥 , , {sin 𝑥 }cos−1 𝑥 , {sin 𝑥 }𝑥𝑡𝑎𝑛𝑥

81) {log 𝑥}𝑥 , {log 𝑥}𝑠𝑖𝑛𝑥 , {log 𝑥}𝑙𝑜𝑔𝑥 , {log 𝑥}𝑐𝑜𝑠𝑥 {log 𝑥}1

𝑥

82) 𝑥𝑥𝑥 , {𝑥𝑥}𝑥

83) {sin 𝑥 }𝑡𝑎𝑛𝑥 + {cos 𝑥 }𝑠𝑒𝑐𝑥

84) Y = {log 𝑥}𝑥 + 𝑥𝑙𝑜𝑔𝑥

85) 𝑥𝑐𝑜𝑡𝑥 +2𝑥2−3

𝑥2+𝑥+2

86) cos(𝑥𝑥) , sin (𝑥𝑥)

87) Log (𝑥𝑥 + 𝑐𝑜𝑠𝑒𝑐2 𝑥 )

88) 𝑥𝑦 = 𝑒𝑥−𝑦 , prove that dy

dx=

log 𝑥

( 1+log x )2 ( 02 , 10 , 11 , 13 )

89) 𝑥𝑦 = 𝑦𝑥

90) 𝑥𝑦 + 𝑦𝑥 = 2

91) (𝑐𝑜𝑠𝑥 )𝑦 = (sin 𝑥)𝑥 , prove that dy

dx=

log sin 𝑦+𝑦 tan 𝑥

log cos 𝑥−𝑥 𝑐𝑜𝑡𝑦

92) 𝑒𝑥 + 𝑒𝑦 = 𝑒𝑥+𝑦 prove that dy

dx + 𝑒𝑥−𝑦 = 0

93) 𝑥 sin( 𝑎 + 𝑦 ) + sin 𝑎 cos( 𝑎 + 𝑦) = 0 𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 dy

dx =

(sin(𝑎+𝑦)𝑥

sina

94) {sin 𝑥 }𝑥 + sin−1 √𝑥 (09 , 13 )

95) 𝑥𝑠𝑖𝑛𝑥 + {sin 𝑥 }tan 𝑥

96) 𝑥𝑥𝑐𝑜𝑠𝑥 +𝑥2−1

𝑥2−1 (2011)

97) (𝑥𝑐𝑜𝑠𝑥)𝑥 + (𝑥𝑠𝑖𝑛 𝑥)1/𝑥

98) 𝑥𝑠𝑖𝑛𝑥−𝑐𝑜𝑠𝑥 +𝑥2−1

𝑥2−1

99) {𝑥 + 1

𝑥}

𝑥 + {𝑥}{𝑥+

1

𝑥}

100) 𝑥𝑚𝑦𝑛 = {𝑥 + 𝑦}𝑚+𝑛 , prove that dy

dx=

y

𝑥

101) (𝑥2 + 1)( 𝑥4 + 1)( 𝑥6 + 1)( 𝑥8 + 1)

102) sinx sin2x sin4x sin8x

103) y=√1−𝑥2 (2𝑥−3)1/2

{𝑥2+2}2/3

104) √𝑥 (𝑥+4)3/2

{4𝑥 −3}4/3


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105) x= 𝑎 {cos 𝑡 +1

2log tan 𝑡𝑎𝑛2 𝑡

2} and y = a sin t

106) x= 𝑎{t − sin t } and y = 𝑎{1 − cost } at t=π/3 , dy

dx= − √3 (2011)

107) x= a cos3t , y = sin 3t

108) x= 𝑎{cos 𝑡 + t sin t } and y = 𝑎{sin 𝑡 − t cos 𝑡 }

109) x= 2 cos t – cos 2t and y=2 sin t – sin 2t , prove that dy

dx = tan {

3𝑡

2}

110) x =√𝑎sin−1 𝑡 , y =√𝑎cos−1 𝑡 show that dy

dx=

−y

𝑥

111) x= {sin 𝑡}3

√cos 2𝑡 and x=

{cos 𝑡}3

√cos 2𝑡 show that

dy

dx= − cot 3𝑡

DIFFERENTIATION OF FUNCTION WITH RESPECT TO OTHER FUNCTION

112) differentiate log sin x w.r.t √cos 𝑥

113) differentiate sin−1 {2𝑥

1+ 𝑥2} 𝑤. 𝑟. 𝑡 tan−1 𝑥


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114) ax3 + 4x + 5 , log x + x4 ,

115) y= x + tan x , show that {cos 𝑥}2 𝑦′′ − 2𝑦 + 2𝑥 = 0

116) x= 𝑎{cos 𝑡 + t sin t } and y = 𝑎{sin 𝑡 − t cos 𝑡 } , 𝑦′′ = {sec 𝑡}3/𝑎𝑡

117) y = 3 cos ( log x) + 4 sin ( log x), prove that 𝑥2𝑦′′ + 𝑥𝑦′ + 𝑦 = 0

118) x ={𝑠𝑖𝑛1

𝑎log 𝑦} show that (1-𝑥2) 𝑦′′ − 𝑥𝑦′ − 𝑎2 = 0 (2010)

119) y={𝑙𝑜𝑔⟨𝑥 + √𝑥2 + 1⟩}2

show that (1+𝑥2) 𝑦′′ + 𝑥𝑦′ = 2 (2008)

120) y={tan−1 𝑥}2 show that (1 + 𝑥2)2 𝑦′′ + 2𝑥(1 + 𝑥2)𝑦′ = 2 (2012)

121) y={cos−1 𝑥} show that 𝑦′′ = − cot 𝑦 𝑐𝑜𝑠𝑒𝑐2𝑦

122) y= e𝑎 cos−1 𝑥 prove that (1-𝑥2) 𝑦′′ − 𝑥𝑦′ − 𝑎2𝑦 = 0

123) y= 500 e7𝑥 + 600 e−7𝑥prove that 𝑦′′ = 49 𝑦

124) y= 3 e2𝑥 + 2 e3𝑥prove that 𝑦′′ − 5𝑦′ + 6𝑦 = 0

125) y= csc−1 𝑥 show that (1-𝑥2) 𝑦′′ + (2𝑥2 − 1)𝑦′ = 0

126) x= cos t + log tan𝑡

2 and y = sin t ,show that y’’=2√2 at t = π/4

127) x= a sint and y= a{cos 𝑡 + log tan𝑡

2} show that 𝑦′′ =

−1

𝑎 𝑠𝑖𝑛2𝑡 𝑐𝑜𝑠𝑡

128) y= ex(sin 𝑥 + cos 𝑥) , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑦′′ − 2𝑦′ + 2𝑦 = 0

129) cos𝑥

2cos

𝑥

4cos

𝑥

8cos

𝑥

16… … … … . . =

sin 𝑥

𝑥 show that

1

22{sec

𝑥

2}

2

+ 1

24{sec

𝑥

4}

2

+1

28{sec

𝑥

8}

2

… … … … . . = 𝑐𝑜𝑠𝑒𝑐 2𝑥 −1

𝑥2


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1) A balloon, which is always spherical, has variable radius. Find the rate at which its volume is

increasing with respect to its radius when radius is 7 cm.

2) Find the rate of change of area of circle with respect to its radius. How fast area is changing

when radius is 3 cm.

3) The cost function of a firm is given by C = 4x2 - x + 70. Find the marginal cost, when x = 3.

4) The cost function of a firm is given by C = 3x2 + 2x - 3. Find the marginal cost, when x = 3.

5) The total revenue received from the sale of x units of a product is given by

R(x) = 3X2 + 36x + 5. The marginal revenue, when x = 15.

6) The total cost C(x) of producing x units of an article is given by

C(x) = 0.005x3 - 0.02x2 + 3 x+ 5000 Find the marginal cost when 3 units are produced.

7) The total revenue received from the sale of x units of a product is given by R(x)

= 3x2 + 36x + 5. Find the marginal revenue when x = 5.

RATES RELATED TO EACH OTHER

8) The radius of a spherical air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the

volume of the bubble increasing when its radius is 1 cm?

9) The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of

increase of its surface area when radius is 4 cm.

10) An edge of a variable cube is increasing at the rate of 5 cm/sec. How fast is the volume of cube

increasing when edge is 10 cm long?

11) A balloon which always remains spherical is being inflated by pumping in gas at the rate of

900 cm3 sec. Find the rate at which the radius of the balloon is increasing when the radius of

the balloon is 15 cm. [Delhi 2003]

12) The volume of a spherical balloon is increasing at the rate of 25 cm3/sec. Find the rate of

change of its surface area at the instant when its radius is 5 cm.

13) The surface area of a spherical bubble is increasing at the rate of 2 cm /sec. Find the rate at

which the volume of the bubble is increasing at the instant if its radius is 6 cm.

14) Radius of a variable circle is changing at the rate of 5 cm/sec. What is the radius of the circle at

a time when its area is changing at the rate of 100 cm2/sec?

15) Gas is escaping from a spherical balloon at the rate of 900 cm3/sec how fast is the surface area,

radius of balloon shrinking when the radius of the balloon is 30 cm?

16) What is the radius of a variable sphere when its volume and surface area are changing at the

rate of 200 cm3/sec and 10 cm2/sec respectively? Also calculate the rate of change of radius at

that moment.

17) Side of an equilateral triangle is increasing at the rate of 2 cm/sec. At what rate is the area

increasing when side is 15 cm? ( 15√3 )

18) A particle moves along the curve 6y = x3 + 2. Find the points on the curve at which the y-

coordinate is changing 8 times as fast as x-coordinate.

19) A plane is ascending at the rate of 100 km/hr. If the radius of the earth is r km, how fast is the

area of earth, visible from the plane, increasing at three minutes after it started

ascending.[Visible area A at height h is given by A = [ h / (r + h)] ].

20) A man 160 cm tall walks away from a source of light situated at the top of the pole 6 m high at

the rate of 1.1 m/sec. How fast is the length of the shadow increasing when he is 1 m away

from the pole?


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21) A man 1.6 m tall walks at the rate of 0.5 m/sec away from a lamp post, 8 meters high. Find the

rate at which his shadow is increasing' and the rate with which the tip of shadow is moving

away from the pole.

22) A ladder 13m long is leaning against a vertical wall. The bottom of the ladder is dragged away

from the wall along the ground at the rate of 2 cm/sec. How fast is its height on the wall

decreasing when the foot of the ladder is 5 m away from the wall?

23) The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm

per second. How fast is the area decreasing when the two equal sides are equal to the base?

24) The length x of a rectangle is decreasing at the rate of 3 cm/minute and the width is increasing

at the rate of 2 cm/minute. When x = 10 cm and y = 6 cm, find the rate of change of (I) the

perimeter (ii) the area of the rectangle. [NCERT]

25) For the function y = x3 + 21, find the values of x when y increases 75 times as fast as x.

( 𝑑𝑦

𝑑𝑡 = 75

𝑑𝑥

𝑑𝑡 )

26) A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm per second.

At the instant when the radius of the circular wave is 10 cm, how fast is the enclosed area

increasing?

27) The volume of a cube is increasing at the rate of 8 cm3/sec. How fast is the surface area

increasing when the length of an edge is 12 cm? [NCERT]

28) A balloon which always remains spherical has a variable diameter(2x + 1)3. Find the rate of

change of its volume with respect to x. [NCERT]


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1) If y= x4 -10 and if x changes from 2 to 1.99 what is the approximate change in y.

2) A circular plate expends under heating so that its radius increases by 2%. Find the approximate

change (increase) in area of plate it radius before heating is 10 cm.

3) Find the approximate value of f (3.02) when f(x) =3x2 + 5x +3.

4) Find the approximate value of f (2.01) when f(x) =4x2 + 5x +2.

5) Find the approximate value of f (5.001) when f(x) =x3 -7x2 +15.

6) Using differentials to approximate following

a. √25.2

b) √0.037

c) √26

d) √37

e) √36.6

f) √1273

g) √663

h) √293

i) √253

j) √804

k) √824

l) √17

81

4


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Role’s theorem

Let a f be a real value function defined on the close interval [a, b] such that

I. It is continuous on close interval [a, b].

II. It is differentiable on open interval (a, b).

III. f(a) =f( b)

Then, there exist a real number c ϵ (a, b), such that f’(c) = 0

Geometrically: it give a (at least one) unique point where tangent is parallel to x-axis.

Lagrange’s mean value theorem

Let a f be a function defined on the close interval [a, b] such that

I. It is continuous on close interval [a, b].

II. It is differentiable on open interval (a, b).

Then, there exist a real number c ϵ (a, b), such that

f’(c) =𝑓(𝑏)−𝑓( 𝑏)

𝑏−𝑎

Geometrically: it give a (at least one) unique point , except at the end point, where tangent is parallel to chord

joining the end points of the curve .

Important point .

Sno. Function name

Continuous or discontinuous

Differentiable domain examples

1 polynomial Continuous Differentiable R X2 + x+ 3 , X3 + x+ 3 etc.

2 ex ,sin 𝑥 cos 𝑥 Continuous Differentiable Every where

3 Log x Continuous Differentiable (0 , ∞ )

4 Tanx Discontinuous at ±π/2 , ±3π/2, ±5π/2

Differentiable

5 IxI Continuous Non Differentiable Every where

6 All polynomic fn with fractional power

Continuous or discontinuous

Non Differentiable Every where F(x) =√2𝑥 − 1 at x=1/2 ,f ⇒∞ so non differentiable

7 Sum Difference Multiplication quotient

Continuous (of continuous functions)

Differentiable ( of differentiable functions)

Every where Sin x + cos x Sin x - cos x Sin x cos x Sin x / cos x


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VERIFY ROLLE’S THEORM for following functions

1) f(x)=IxI on [-1 , 1] not applicable

2) f(x)=[x] on [-1 , 1] not applicable

3) f(x)=3+ (x-2)2/3 on [1, 3] not applicable

4) f(x)= (x)2/3 on [-1 , 1] not applicable

5) f(x)= (x)1/3 on [1, 3] not applicable

6) f(x)=(2x-3)2/3 on [1, 3] not applicable

7) f(x)=Tanx on [0,π] not applicable

8) f(x)=x2-5x+6 on [2,3] 2.5

9) f(x)=x2-8x+12 on [2,6]

10) f(x)=(x-1)(x-2)2 on [1,3]

11) f(x)= x(x-2)2 on 0≤x≤3

12) f(x)=x3-6 x2+11x-6 on [1,3] 2±√1

3

13) f(x)=𝑠𝑖𝑛2𝑥 on [0, π]

14) f(x)=sin x +cos x -1 on [0, π/2]

15) f(x)=ex(sin x -cos x ) on [π/4, 5π/4] (note ex ≠ 0)

16) f(x)= sin 2x on[0, π]

17) if f:[-5 ,5]→R is differnentiable and if f’(x) does not vanish any where , then prove that f(-5)≠f(5).

{let f(-5)=f(5), then by rolles there must be a number such that f’(c)=0, but given that f’(x)≠0 never be

for any xϵ(-5 ,5) . hance it contradict our assumption so f(-5) ≠ f(5).

18) Using Rolle’s , find the point on the curve y= 16-x2 ,xϵ[-1, 1]. Where tangent is parallel to x-axis.

19) Given that f(x)=x3-6x2+ax+b on [1,3], Rolle’s hold with c=2±√1

3 . Find the value of ‘a’ and ‘b’, if f(1) =

f(3)=0.


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Verfy lagerange’s mean value theorm for the following functions also ,find a point c in given interval.

1) F(x) = x( x-2) on [1,3] 2

2) F(x) = x(x-1 )(x-2) on [0, ½ ] 1±√21

6

3) F( x) = √𝑥 − 2 on [2,3] ½

4) f(x)=x2-3x+2 on [-1,2]

5) f(x)=2x2-3x+1 on [1,3]

6) f(x)=x2-2x+4 on [1,5]

7) f(x)=2x -x2 on [0, 1]

8) f(x)= (x-1 )(x-2)(x-3) on [0, 4]

9) F( x) = √25 − 𝑥2 on [-3, 4] ½

10) f(x)= 𝑥 + 1

𝑥 on [1,3]

11) F( x) = √𝑥2 − 4 on [2,4 ]

12) f(x)=x3-5 x2-3x on [1,3]

13) Discusses the applicability of Lagrange’s mean value theorem for f(x) =IxI no [-1, 1].

14) Discusses the applicability of Lagrange’s mean value theorem for f(x) =1

𝑥 no [-1, 1].

15) Discusses the applicability of lagrange’s mean value theorem for f(x)=1

4𝑥−1 no [-1,4].

16) Find the point on the function y= (𝑥 − 4)2, where tangent is parallel to chord joing (4,0) and(5,1).

17) Find the point on the function y= (𝑥 − 3)2, where tangent is parallel to chord joing (0,0) and(1,2).

18) Find the point on the function y= 𝑥2 + 𝑥 , where tangent is parallel to chord joing (3,0) and(4,1).

19) Find the point on the function y= 𝑥3 + 3𝑥 , where tangent is parallel to chord joing (1,-2) and(2,2).

20) Find the point on the function y= 𝑥3 + 1, where tangent is parallel to chord joing (1,2) and(3,28).

Answers.

5. 2 6.3 7. ½ 8.2±2

√3 9. ±

1

√2 10. √3 11. √6 12.7/3

16. (9/2,1/4 )

17. (1/2 , ¾ )

18.(7/2,1/4 )

19±√7

3

20. √13

3


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Slope : the tangent of a angle that a line make with the positive direaction of x-axis in anti-clockwise sense is

called the slope of line .

Thus

.

When φ = 0 , then slope of line tan 0 = 0 , that means line is parallel to x-axis .

When φ = 90 , then slope of line tan 90 = ∞, that means line is perpendeculer to x-axis .

When φ = 45 or 135 , then slope of line tan 45 = 1 or tan 135=-1, that means line is equally inclined axis .

Slope of a line when equation of line ( ax + by + c =0) is given = −𝒂

𝒃

Angle between to line : tan φ =±{𝒎𝟏− 𝒎𝟐

𝟏+ 𝒎𝟏𝒎𝟐}

Condition for two parallel line : 𝒎𝟏 = 𝒎𝟐

Condition for two perpendiculer line : 𝒎𝟏 𝒎𝟐 = −𝟏

Slope of tangent at point p :{𝒅𝒚

𝒅𝒙}

𝒑

Slope of normal at point p :𝟏

{𝒅𝒚

𝒅𝒙}

𝒑

Equation of a line : ( 𝒚𝟐 − 𝒚𝟏 ) = 𝒎 (𝒙𝟐 − 𝒙𝟏 )

1. Find the slope of tangent and the normal of the curve x2+ 3y + y2 = 5 at (1,1) . {-2/5 , 5/2}

2. Show that the tangent to the curve y=2x3 -3 at the point where x=2 and x= -2 are parallel.

3. Show that the tangent to the curve y=x2 -5x + 6 at the point where x=2 and x= 3 are at right angles.

4. The slope of cuve at 2 y2 = a x2+b at ( 1,-1 ) is -1 find a and b. {a=2 , b=0}

5. The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0. 6. Find the slope of tangent and normal of curve

m = tan φ = 𝑦2−𝑦1

𝑥2−𝑥1=

𝑑𝑦

𝑑𝑥

where φ is the angle made by line with positive x-axis


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X = 1- a sin Ɵ , y = b cos2 Ɵ at Ɵ= π/ 2 (2b/a, -a/2b)

7. Find the slope of tangent and normal of curve

X = a cos3Ɵ , y=a sin3Ɵ at Ɵ= π/ 4 (-1 , 1)

8. Find the point on the curve y = x3-2 x2 -x at which the tangent lines are parallel to the line y = 3x -2.

9. Find the point on the curve y = 2 x2 -6x-4 at which the tangent lines are parallel to x-axis. (3

2,

−17

2 )

10. Find the point on the curve y = x3 at which the tangent is equal to y-coordinate of the point.(2010)

( dy

dx= 𝑦1 𝑤ℎ𝑒𝑟𝑒 (𝑥1, 𝑦1)𝑖𝑠 required point )

11. Find the point on the curve y = x3 at which the tangent is equal to x-coordinate of the point.(2008)

( dy

dx= 𝑥1 𝑤ℎ𝑒𝑟𝑒 (𝑥1, 𝑦1)𝑖𝑠 required point ). (0 ,0) .( 1/3,1/27)

12. Find the point on the curve 𝑥2

9−

𝑦2

16= 1 where tangent is parallel to (i)x-axis ,(ii) y axis. {no point for x

– axis , and y – axis (3,0) and (-3 , 0)}


4+

𝑦2

25= 1 where tangent is parallel to (i)x-axis ,(ii) y axis. {(0, 5)and (

0 -5) for x – axis , and y – axis (2,0) and (-2 , 0)}


9+

𝑦2

16= 1 where tangent is parallel to (i)x-axis ,(ii) y axis. {(0, ±4) x

– axis , and y – axis (±3,0)}

15. Find the equation of tangent to the curve y=-5x + 6x + 7 at point ( ½ , 35/4) .{ y=x+ 33/4)

16. Find the equations of the tangent and normal to the given curves at the indicated points: 17. y = x4 - 6x3 + 13x2 - 10x + 5 at (0, 5) 18. y = x4 - 6x3 + 13x2 - 10x + 5 at (1, 3) 19. y = x3 at (1, 1) 20. y = x2 at (0, 0) 21. x = cos t, y = sint at t = π/4

22. Show that the tangents to the curve y = 7x3 + 11 at the points where x = 2 and x = - 2 are parallel.

23. Find the equation to tangent line to curve x= 1-cos Ɵ , y= y=Ɵ- sin Ɵ at Ɵ= π/ 4.

(√2 − 1) 𝑥 − 𝑦 = 2(√2 − 1) − π/4.

24. Find the equation tangent and normal to line to curve x=a 𝑎𝑠𝑖𝑛 3𝑡 , 𝑦 = 𝑏𝑐𝑜𝑠3𝑡. (2010)

( bx cost +ay sint )= ab sin t cos t).( ax sin t –b y cos t = a2 (𝑠𝑖𝑛 4𝑡 − 𝑏2𝑐𝑜𝑠4𝑡 ).

25. Show that the curve 𝑥

𝑎+

𝑦

𝑏= 1 touches the curve y=be−x/a at the point where it cross y – axis. (at

y-axis , x=0 by that we get the point of contact (o,b)).

26. Find the equation to tangent line to curve t=𝑥−7

(𝑥−2)(𝑥−3) at the point , where it cut x-axis .

27. (at x-axis , y=0 by that we get the point of contact (7,0)).

28. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the

point.

29. Find the points on the curve x2 + y2 - 2x - 3 = 0 at which the tangents are parallel to the x-axis.

30. Find the equation of the normal at the point (am2,am3) for the curve ay2 = x3

31. Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 =

0.

32. Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).


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33. Find the equation of the tangent to the curve y = √3x − 2 which is parallel to the line 4x-2y + 5 = 0.

34. The line y = x + 1 is a tangent to the curve y2 = 4x at the point.( point where slope of tangent is equal

to slope of line )

35. Find the angle of intersection of of curve xy=6 and x2y=12.

36. Find the angle of intersection of of curve y2=4x and x2=4y.

37. Prove that curve x=y2 and xy =k cut at right anglr if 8k2 =1.

38. Prove that curve 4 x=y2 and 4xy =k cut at right anglr if k2 =512.

39. Prove that curve 2x=y2 and 2xy =k cut at right anglr if k2 =8.

40. For the curve y = 4x3 - 2x5, find all the points at which the tangent passes through the origin.

(let point be (x1, y1) then y1=4x13 – 2x1

5 , then y’ = 12 x12 – 10x1

4 then equation of line from origen is 0-

y1=(12 x12 – 10x1

4 ) ?( 0-x1)⇒y1= 12x13-10x1

5(2) subtact 0 = - 8x13+8x1

5 then x= 0, ±1 put in (2) )

Let f be continuous on [a, b] and differentiable on the open interval (a,b).

Then f is increasing in [a,b] if f’(x) > 0 for each x ϵ (a, b)

f is decreasing in [a,b] if f’(x) < 0 for each x ϵ (a, b) f is a constant function in [a,b] if f’ (x) = 0 for each x ϵ (a, b).

Let f be continuous on [a, b] and differentiable on the open interval (a,b).

f is strictly increasing in (a, b) if f '(x) > 0 for each x ϵ (a, b) f is strictly decreasing in (a, b) if f '(x) < 0 for each x ϵ (a, b)

it to be noted that in case of SD or SI

we do not include point where f’(x) is zero and end points are not taken. Incase of only In or De we do take where f’(x) is zero points. F(x)=(x+1)3( x-3)3

F’(x)=6(x-1)(x-3)2(x+1)2 , then x= 1,3,-1

SD : (-∞ , -1)∪ (−1 ,1) SI : (1 ,3)∪ (3 , ∞)

D : ((-∞ , 1) I: (1, ∞)

f’(x) 0 0 0 ---------------- -1----------------------------------1+++++++++++++++++++ 3 +++++++++++


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Find the intervals in which the following functions are increasing or decreasing:

1) f (x) = x3 - 3x2 + 4x

2) f(x) = x2 - 4x + 6

3) f (x) = 4x3 - 6x2 - 72x + 30

4) f(x) = 2x2 - 3x

5) f(x) = 2x3 - 3x2 - 36x + 7

6) f(x) =x2 + 2x - 5

7) f(x) = 10 - 6x - 2x2

8) f(x) = -2x3 - 9x2 - 12x + 1

9) f(x) = 6 - 9x - x2

10) f(x) = (x + 1)3 (x - 3)3 {01C}

11) f(x) = (x - 1)3 (x - 2)2

12) f(x) = [x(x - 2)]2

13) f(x) = x2 - x + 1

14) f(x) = x100 + sin x -1

15) f(x) =x2 + ax + 1

16) f(x) = x3 - 3x2 + 3x – 100

17) f(x) =-x2 -2x + 15

18) f(x) = 4𝑥2+1

𝑥

19) f(x) = 𝑥3 + 1

𝑥2

20) f(x) =sin x +cos x

21) f(x) = 4 cos Ɵ

2+𝑐𝑜𝑠Ɵ [0, π/2]

22) prove that f(x) =x2 -x+ 1 is neither increasing nor decreasing on (-1,1).

23) Find the interval in which f(x) sin x – cos x , 0<x˂2π.

24) Which of the following functions are strictly decreasing on

(A) cos x (B) cos 2x (C) cos 3x (D) tan x

25) Show that the function given by f (x) = sin x is

a. strictly increasing in strictly decreasing in

b. neither increasing nor decreasing in (0, n)

26) Prove that the function given by f (x) = cos x is

a. strictly decreasing in (0, n)

b. strictly increasing in (n, 2n), and

c. neither increasing nor decreasing in (0, 2n).


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1. Find two positive numbers whose sum is 15 and the sum of whose squares is minimum.

2. Find two numbers whose sum is 24 and whose product is as large as possible.

3. Find two positive numbers x and y such that x + y = 60 and xy3 is maximum.

4. Find two positive numbers x and y such that their sum is 35 and the product x2 y5 is a maximum.

5. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

6. Show that all rectangle with a given perimeter , square has largest area.

7. Show that all rectangle of given area area , the square has smallest perimeter.

8. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum

area.

9. Show that the rectangele of maximum perimeterwhich can be inscribed in a circle of radius ’a’is

a suare of side √2 .

10. A wire of length 36 m is to be cut into two pieces. One of the pieces is to be made into a square

and the other into a circle. What should be the length of the two pieces so that the combined area

of the square and the circle is minimum? (36π/π+4)

11. A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square

and the other into a circle. What should be the length of the two pieces so that the combined area

of the square and the circle is minimum?

12. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the

sum of their areas is least when the side of square is double the radius of the circle.

13. A square piece of tin of side 24 cm is to be made into a box without top, by cutting a square from

each corner and folding up the flaps to form the box. What should be the side of the square to be

cut off so that the volume of the box is the maximum possible. ( side =4, 1024)

14. A square piece of tin of side 18 cm is to be made into a box without top, by cutting a square from

each corner and folding up the flaps to form the box. What should be the side of the square to be

cut off so that the volume of the box is the maximum possible.

15. Rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off

square from each corner and folding up the flaps. What should be the side of the square to be cut

off so that the volume of the box is maximum ?

16. An open topped box is to be constructed by removing equal squares from each corner of a 3

metre by 8 metre rectangular sheet of aluminium and folding up the sides. Find the volume of the

largest such box.

17. A window is in the form of a rectangle surmounted by a semicircular opening. Find the

dimensions of the window to admit maximum light through the whole opening.

18. A window is in the form of a rectangle surmounted by a semicircular opening. The total

perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light

To find maximum and minimum value of a given function when interval [a,b] is given

Step 1 : Find f’(x)

Step 2 : Put f’(x) = 0 and get value of x1,x2 x3,x4….. etc

Step 3: Then find f’(a) , f’(x1), f’(x2), f’(x3), …..f’(b)

Lagest value so obtained is called absolute maximum , smallest is called absolute minimum.

We can also use line method in which if value change from + to – then point is maximum and

when value change from – to + then minimum point or do not change then inflect ion point.


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through the whole opening.

19. Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.

20. Show that a cylinder of a given volume which is open at the top , has minimum total surface area

, provided its height is eual to the radius of the base.

21. Show that the right circular cylinder of given surface and maximum volume is such that its

height is equal to the diameter of the base.(ncert)

22. Show that the height of a cylinder , which is open from top , having a given surface area greatest

volume , is equal to the radius of base.

23. Show that the right circular cone of least curved surface and given volume has an altitude equal

to √2 time the radius of the base. .(ncert)

24. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of

radius R is 2𝑅

√3. Also find the maximum volume ( ncert)

25. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height

is tan−1 √2 .(ncert)

26. Show that semi-vertical angle of right circular cone of given surface area and maximum volume

is sin-1 (1/3) .(ncert)

27. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8

27of the

volume of the sphere. .(ncert)

28.Show that the altitude of the right circular cone of maximum volume that can be inscribed in a

sphere of radius r is4𝑟

3

29. Prove that the radius of the right circular cylinder of greatest curved surface area which can be

inscribed in a given cone is half of that of the cone.(ncert)

30. Show that volume of the greatest cylinder which can be inscribed un a cone of hight h and semi

vertical angle α is 4

27 πℎ3𝑡𝑎𝑛2𝛼 . also show that height of cylinder is

ℎ

3 .( 01c,07,08,10)

31. An open box with square base is made out of is to be made out of a given quantity of caed

board of area 𝑐2 square units . show that the maximum volume of the box is 𝑐3

6√3 cubic units.

32. Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimeters,

find the dimensions of the can which has the minimum surface area?(ncert)

33. Let AP and BQ be two vertical poles at points A and B , respectively. If AP = 16 m, BQ = 22 m

and AB = 20 m, then find the distance of a point R on AB from the point A such that RP2 +RQ2

is minimum.

34. An Apache helicopter of enemy is flying along the curve given by y = x2 + 7. A soldier, placed at

(3, 7), wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

35. Find the point on the curve x2 = 2y which is nearest to the point (0, 5).

36. Find the shortest distance of the point (0, c) from the parabola y = x2, where 0 < c < 5.

37. If length of three sides of a trapezium other than base are equal to 10cm, then find the area of the

trapezium when it is maximum.

38. Manufacturer can sell x items at a price of rupees {5 −𝑥

100} each. The cost price of x items is

Rs (5 + 500) . Find the number of items he should sell to earn maximum profit.

39. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its

depth is 2 m and volume is 8 m3. If building of tank costs Rs 70 per sq metres for the base and Rs

45 per square metre for sides. What is the cost of least expensive tank?


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MATHEMATICS for CLASS 12 - deepak sir13) √ 3− 2−7 +8 , 1 √ 2− ë,√𝑙 𝑔{sin[ ë 2 3...

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Transcript of MATHEMATICS for CLASS 12 - deepak sir13) √ 3− 2−7 +8 , 1 √ 2− ë,√𝑙 𝑔{sin[ ë 2 3...