Class 12 Math Ch-6 Application of Derivatives MCQs Exam 2027

Topic 1: Rate of Change of Quantities

1. The rate of change of the area of a circle with respect to its radius $r$ at $r = 6$ cm is: 2021, 2023

(A) $10\pi$

(B) $12\pi$

(C) $8\pi$

(D) $11\pi$

2. A spherical balloon is being inflated at the rate of $35 \text{ cc/min}$. The rate of increase of its surface area when its radius is $14$ cm is: 2016, 2024

(A) $5 \text{ sq. cm/min}$

(B) $10 \text{ sq. cm/min}$

(C) $2.5 \text{ sq. cm/min}$

(D) $1 \text{ sq. cm/min}$

3. Ek circle ki radius $0.7 \text{ cm/s}$ ki dar se badh rahi hai. Iske circumference ke badhne ki dar kya hogi? 2021

(A) $1.4\pi \text{ cm/s}$

(B) $0.7\pi \text{ cm/s}$

(C) $0.4\pi \text{ cm/s}$

(D) None

4. Cube ka volume $9 \text{ cm}^3\text{/s}$ ki dar se badh raha hai. Jab edge $10 \text{ cm}$ ho, toh surface area kis dar se badhega? 2022

(A) $3.6 \text{ cm}^2\text{/s}$

(B) $4 \text{ cm}^2\text{/s}$

(C) $1.8 \text{ cm}^2\text{/s}$

(D) $2 \text{ cm}^2\text{/s}$

5. Sphere ka volume $V = \frac{4}{3}\pi r^3$ hai, toh $r$ ke respect mein change ki rate hogi: 2019

(A) $4\pi r^2$

(B) $2\pi r$

(C) $4\pi r$

(D) $3\pi r^2$

6. Ek square ki side $3$ cm/s ki rate se badh rahi hai. Area kis rate se badhega jab side $10$ cm ho? 2022

(A) 30

(B) 60

(C) 90

(D) 100

7. Kisi circle ki radius $3$ cm hai. Iska area $r$ ke respect mein change hoga: 2021

(A) $6\pi$

(B) $3\pi$

(C) $9\pi$

(D) $2\pi$

8. Cone ka volume $V = \frac{1}{3}\pi r^2 h$. Agar $h$ constant ho toh $dV/dr$ hoga: 2023

(A) $\frac{2}{3}\pi rh$

(B) $\pi r^2$

(C) $\frac{1}{3}\pi r^2$

(D) None

9. Cylinder ka Total Surface Area $S = 2\pi rh + 2\pi r^2$. $dS/dr$ kya hoga? 2021

(A) $2\pi h + 4\pi r$

(B) $2\pi h$

(C) $4\pi r$

(D) $2\pi r$

10. Rate of change of volume of sphere jab $r=2$ units ho: 2024

(A) $16\pi$

(B) $8\pi$

(C) $4\pi$

(D) $32\pi$

11. Circle ka area $A = \pi r^2$. $dA/dt$ nikalen jab $r=5$ aur $dr/dt=2$: 2022

(A) $20\pi$

(B) $10\pi$

(C) $5\pi$

(D) $25\pi$

12. Radius of sphere $r=7$ cm. Error in volume agar $r$ mein error $0.1$ cm ho: 2018

(A) $19.6\pi$

(B) $4.9\pi$

(C) $3.6\pi$

(D) None

13. Rate of change of perimeter of circle w.r.t radius: 2023

(A) $2\pi$

(B) $\pi$

(C) $2$

(D) $4\pi$

14. Area of circle increases at $2 \text{ cm}^2\text{/s}$. Find rate of change of $r$ at $r=4$: 2020

(A) $1/4\pi$

(B) $1/2\pi$

(C) $\pi/4$

(D) $4/\pi$

15. Rate of change of $y = \sqrt{x^2+16}$ w.r.t $x$ at $x=3$: 2021

(A) 3/5

(B) 4/5

(C) 1

(D) 0

16. Radius of circle $r$ is increasing at $3$ cm/s. Area rate at $r=10$: 2022

(A) $60\pi$

(B) $30\pi$

(C) $100\pi$

(D) $3\pi$

17. Displacement $s = t^2 – 4t + 3$. Acceleration is: 2024

(A) 2

(B) $2t-4$

(C) 0

(D) 4

18. Particle ki position $s = t^3 – 6t^2 + 9t$. Velocity zero kab hogi? 2017

(A) $t=1, 3$

(B) $t=2, 4$

(C) $t=0$

(D) $t=1, 2$

Topic 2: Tangents and Normals (Sparsh Rekha aur Abhilamb)

19. The line $y = x + 1$ is a tangent to the curve $y^2 = 4x$ at the point: 2019, 2024

(A) $(1, 2)$

(B) $(2, 1)$

(C) $(1, -2)$

(D) $(-1, 2)$

20. Curve $y = x^3 – x$ ki tangent ka slope $x = 2$ par kya hoga? 2023

(A) 10

(B) 11

(C) 12

(D) 13

21. Curve $x = a \cos^3 \theta, y = a \sin^3 \theta$ ke liye $\theta = \pi/4$ par normal ka slope hai: 2019, 2022

(A) 1

(B) -1

(C) 0

(D) Not defined

22. Curve $y = 2x^2 + 3\sin x$ ke point $x = 0$ par tangent ka slope hai: 2021

(A) 3

(B) 0

(C) 2

(D) 5

23. Yadi tangent $x$-axis ke parallel ho, toh $\frac{dy}{dx}$ ka maan hota hai: 2018, 2024

(A) 1

(B) -1

(C) 0

(D) $\infty$

24. Curve $y^2 = x$ ke point $(1, 1)$ par normal ki equation hai: 2017

(A) $x + 2y – 3 = 0$

(B) $2x + y – 3 = 0$

(C) $x – 2y + 1 = 0$

(D) $2x – y – 1 = 0$

25. Curve $y = x^2 – x$ par kis point par tangent $x$-axis ke parallel hai? 2023

(A) $(1/2, -1/4)$

(B) $(1/2, 1/4)$

(C) $(0, 0)$

(D) $(1, 1)$

26. Agar $y = x^4 – 10$ ho, toh $x=2$ par dy/dx (slope) kya hoga? 2024

(A) 32

(B) 20

(C) 16

(D) 40

27. Curve $x^2 + y^2 = 25$ ke point $(3, 4)$ par tangent ka slope: 2023

(A) 3/4

(B) -3/4

(C) 4/3

(D) -4/3

28. Curve $y = x^3$ ke point $(1, 1)$ par normal ki equation: 2018

(A) $x+3y-4=0$

(B) $3x+y-4=0$

(C) $x+3y+4=0$

(D) None

29. Point jahan $y = x^2 – 4x + 5$ ki tangent $x$-axis ke parallel hai: 2021

(A) $(2, 1)$

(B) $(1, 2)$

(C) $(0, 5)$

(D) $(2, 0)$

30. Tangent ka slope curve $y = \sqrt{x}$ par point $(4, 2)$ par: 2022

(A) 1/2

(B) 1/4

(C) 1

(D) 4

31. $y = x^3 – 3x^2 – 9x + 5$ ka point jahan tangent horizontal hai: 2019

(A) $x=1, -3$

(B) $x=-1, 3$

(C) $x=0, 3$

(D) $x=1, 3$

32. Curve $x = a \cos \theta, y = b \sin \theta$ ke liye $dy/dx$: 2021

(A) $-(b/a) \cot \theta$

(B) $(b/a) \tan \theta$

(C) $b/a$

(D) None

33. Normal ka equation $y = x$ par origin par: 2020

(A) $y = x$

(B) $y = -x$

(C) $x = 0$

(D) $y = 0$

34. Curve $y = e^{2x}$ ke liye $(0, 1)$ par tangent ka slope: 2024

(A) 1

(B) 2

(C) 0

(D) $e$

35. Chord $y = x – 2$ curve $y = (x-2)^2$ par tangent ke parallel kab hogi? 2021

(A) $(2, 0)$

(B) $(3, 1)$

(C) $(2.5, 0.25)$

(D) None

36. Curve $y^2 = 4ax$ ke $(at^2, 2at)$ par tangent ka slope: 2019

(A) $t$

(B) $1/t$

(C) $-t$

(D) $-1/t$

37. Tangent ki equation $y = x^2$ par $(1, 1)$ par: 2023

(A) $2x – y – 1 = 0$

(B) $x – 2y + 1 = 0$

(C) $2x + y – 3 = 0$

(D) $y = x$

38. Slope of normal to $x = a \cos \theta, y = a \sin \theta$: 2021

(A) $\tan \theta$

(B) $\cot \theta$

(C) $-\tan \theta$

(D) $-\cot \theta$

39. $y = x^2 + 3x – 4$ ka $x = 1$ par tangent slope: 2024

(A) 5

(B) 4

(C) 3

(D) 2

40. Angle between $y^2 = x$ and $x^2 = y$ at origin: 2016

(A) $0^\circ$

(B) $90^\circ$

(C) $45^\circ$

(D) $30^\circ$

41. Equation of tangent to $y = e^x$ at $(0, 1)$: 2022

(A) $y = x+1$

(B) $y = x$

(C) $y = 1$

(D) $x+y=1$

42. Normal to $x^2 = 4y$ passing through $(1, 2)$ slope: 2021

(A) -1

(B) 1

(C) 2

(D) -2

43. Tangent to $y = \cos x$ at $x = \pi/2$ slope: 2023

(A) 0

(B) -1

(C) 1

(D) $\infty$

44. At what point on $y = x^2$ is the slope of tangent equal to $x$-coordinate? 2020

(A) $(0, 0)$

(B) $(1, 1)$

(C) $(1/2, 1/4)$

(D) $(2, 4)$

45. Tangent to $xy = c^2$ at $(ct, c/t)$ slope: 2018

(A) $t^2$

(B) $-1/t^2$

(C) $1/t^2$

(D) $-t^2$

46. Slope of normal to $y^2 = 4x$ at $(1, 2)$: 2019

(A) 1

(B) -1

(C) 2

(D) -2

47. The slope of the normal to the curve $y = 2x^2 + 3 \sin x$ at $x = 0$ is: 2026

(A) 3

(B) 1/3

(C) -1/3

(D) -3

48. The slope of the tangent to the curve $y = x^3 – x$ at $x = 2$ is: 2026

(A) 11

(B) 12

(C) 10

(D) 6

Topic 3: Increasing and Decreasing Functions (Vardhman aur Hrasman Phalan)

49. The interval in which $y = x^2 e^{-x}$ is increasing is: 2018, 2023

(A) $(-\infty, \infty)$

(B) $(-2, 0)$

(C) $(2, \infty)$

(D) $(0, 2)$

50. Function $f(x) = 2x^3 – 15x^2 + 36x + 1$ kis interval mein decreasing hai? 2020

(A) $(2, 3)$

(B) $(-\infty, 2)$

(C) $(3, \infty)$

(D) None of these

51. Phalan $f(x) = \log(\sin x)$ kis interval mein strictly increasing hai? 2021

(A) $(0, \pi/2)$

(B) $(\pi/2, \pi)$

(C) $(0, \pi)$

(D) None

52. $f(x) = x^x$ ka decreasing interval hai: 2015, 2023

(A) $(0, e)$

(B) $(0, 1/e)$

(C) $(1/e, \infty)$

(D) None

53. Sabhi real numbers $x$ ke liye, function $f(x) = e^x$ hota hai: 2022

(A) Strictly Increasing

(B) Strictly Decreasing

(C) Constant

(D) None

54. $y = x(x-3)^2$ kis interval mein badh (increasing) raha hai? 2020

(A) $(1, 3)$

(B) $(0, 1)$

(C) $(-\infty, 1) \cup (3, \infty)$

(D) None

55. Function $f(x) = 2x – \tan^{-1} x$ hamesha hota hai: 2021

(A) Increasing

(B) Decreasing

(C) Constant

(D) None

56. $f(x) = e^{2x}$ ka nature kya hai? 2020

(A) strictly increasing

(B) strictly decreasing

(C) Neither

(D) Constant

57. $f(x) = x^2 – x + 1$ interval $(0, 1)$ mein kaisa hai? 2022

(A) Increasing

(B) Decreasing

(C) Neither

(D) Constant

58. $y = \cos x$ interval $(0, \pi)$ mein hota hai: 2019

(A) Increasing

(B) Decreasing

(C) Constant

(D) None

59. Function $f(x) = \tan x – x$ hamesha: 2023

(A) Badhta hai

(B) Ghatta hai

(C) Sthir rehta hai

(D) None

60. $f(x) = x^3$ kaisa function hai? 2018

(A) Strictly Increasing

(B) Strictly Decreasing

(C) Constant

(D) None

61. $y = \log(1+x) – \frac{2x}{2+x}$ kaisa phalan hai $x > -1$ ke liye? 2021

(A) Vardhman

(B) Hrasman

(C) Dono

(D) None

62. $f(x) = \cos x$ in $[0, \pi/2]$: 2024

(A) Increasing

(B) Decreasing

(C) Constant

(D) None

63. Function $f(x) = \sin x$ is strictly decreasing in: 2022

(A) $(0, \pi/2)$

(B) $(\pi/2, \pi)$

(C) $(0, \pi)$

(D) $(-\pi/2, \pi/2)$

64. $f(x) = x^{100} + \sin x – 1$ is decreasing in: 2021

(A) $(0, 1)$

(B) $(\pi/2, \pi)$

(C) $(0, \pi/2)$

(D) None

65. Interval of $f(x) = 2x^2 – 3x$ strictly increasing: 2023

(A) $(3/4, \infty)$

(B) $(-\infty, 3/4)$

(C) $(0, \infty)$

(D) None

66. If $f(x) = kx – \sin x$ is increasing, then $k$ is: 2017

(A) $k > 1$

(B) $k < 1$

(C) $k = 0$

(D) $k < -1$

Topic 4: Maxima and Minima (Uchchisht aur Nimnisht)

67. The maximum value of the function $f(x) = \sin x + \cos x$ is: 2017, 2021

(A) $\sqrt{2}$

(B) 2

(C) 1

(D) 0

68. For all real values of $x$, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is: 2021

(A) $0$

(B) $1$

(C) $3$

(D) $1/3$

69. $f(x) = (2x-1)^2 + 3$ ka minimum value kya hai? 2021

(A) 0

(B) 2

(C) 3

(D) 1

70. $x + y = 10$ ho toh $xy$ ka maximum value hoga: 2019

(A) 20

(B) 25

(C) 30

(D) 50

71. Function $f(x) = x^{1/x}$ ka maximum value kis point par hota hai? 2016, 2022

(A) $x = e$

(B) $x = 1/e$

(C) $x = 1$

(D) $x = 0$

72. $f(x) = 3x^2 + 6x + 8$ ka minimum value hai: 2023

(A) 5

(B) 8

(C) 2

(D) 10

73. Sine function $f(x) = \sin x$ ka interval $[0, \pi]$ mein maximum value hai: 2020

(A) 0

(B) 1

(C) -1

(D) $1/2$

74. $f(x) = x^3 – 27x + 5$ ka maximum value kis point par hoga? 2018

(A) 3

(B) -3

(C) 0

(D) 27

75. $f(x) = \log x / x$ ka maximum value $x = ?$ par hota hai: 2015

(A) 1

(B) $e$

(C) $1/e$

(D) 2

76. Function $f(x) = \sin 2x$ ka maximum value interval $[0, \pi]$ mein: 2024

(A) 1

(B) 2

(C) 1/2

(D) 0

77. Function $f(x) = x + 1/x$ ka local minimum value: 2017

(A) 1

(B) -1

(C) 2

(D) -2

78. $f(x) = |x|$ ka $x=0$ par minimum value: 2022

(A) 1

(B) -1

(C) 0

(D) Does not exist

79. $f(x) = 7 – 24x – 5x^2$ ka maximum value: 2021

(A) 7

(B) 35.8

(C) 24

(D) 40

80. $y = x^2$ par wo point jo $(0, 5)$ ke sabse kareeb hai: 2022

(A) $(2, 4)$

(B) $(\sqrt{2}, 2)$

(C) $(0, 0)$

(D) $(2, 2)$

81. $f(x) = x^4 – 62x^2 + ax + 9$ ka $x=1$ par maximum hai, toh $a = ?$: 2018

(A) 120

(B) 110

(C) 100

(D) 150

82. $f(x) = \sin^4 x + \cos^4 x$ ka maximum value: 2020

(A) 1

(B) 1/2

(C) 3/4

(D) 2

83. $y = 2x^3 – 3x^2 – 12x + 5$ ka local maximum point: 2023

(A) $x = -1$

(B) $x = 2$

(C) $x = 0$

(D) $x = 1$

84. Maximum area of rectangle with perimeter 20 cm: 2024

(A) 20

(B) 25

(C) 100

(D) 50

85. $f(x) = x^2 – 4x + 3$ ka minimum value: 2021

(A) -1

(B) 0

(C) 1

(D) 3

86. Shortest distance between line $y – x = 1$ and $x = y^2$: 2020

(A) $3\sqrt{2}/8$

(B) $2\sqrt{3}/8$

(C) $1/2$

(D) $\sqrt{2}/4$

87. $y = x^3 – 3x + 2$ ka maximum value in $[0, 2]$: 2019

(A) 4

(B) 2

(C) 0

(D) 6

88. Maximum value of $x + 1/x$ for $x > 0$: 2023

(A) 2

(B) 1

(C) 0

(D) None (only minimum exists)

89. $f(x) = x^2$ in $[-1, 1]$ has minimum at $x = ?$: 2022

(A) 0

(B) 1

(C) -1

(D) $1/2$

90. Local maximum of $f(x) = x^3 – 3x$: 2024

(A) $x = -1$

(B) $x = 1$

(C) $x = 0$

(D) $x = 2$

Topic 5: Theorems & Critical Points (Rolle’s & Mean Value)

91. If $f(x) = x^2 – 1$, then in the interval $[1, 2]$, the value of $c$ for Rolle’s Theorem is: 2020

(A) $3/2$

(B) $2/3$

(C) $1/2$

(D) Rolle’s Theorem is not applicable

92. Rolle’s Theorem function $f(x) = x^2 – 4$ ke liye $[-2, 2]$ mein ‘c’ ka value: 2016

(A) 0

(B) 1

(C) 2

(D) -1

93. Mean Value Theorem $f(x) = x^2$ ke liye $[2, 4]$ mein ‘c’ ka value: 2019

(A) 2

(B) 3

(C) 4

(D) 3.5

94. $f(x) = 2x^3 – 6x^2 + 6x + 5$ ke liye $f'(x)$: 2021

(A) $6(x-1)^2$

(B) $6x^2$

(C) $x-1$

(D) 0

95. Rolle’s theorem value $c$ for $f(x) = \sin x$ in $[0, \pi]$: 2023

(A) $\pi/2$

(B) $\pi/4$

(C) $0$

(D) $\pi$