DELHI PUBLIC SCHOOL RUBY PARK, KOLKATA

WORKSHEET ON APPLICATION OF DERIVATIVE

CLASS- XII
MATHEMATICS

Multiple Choice Questions ( 1mark each)

Choose the correct option :

1. The minimum value of x x ( x  0) is

e
1
1
−
(a) 1 (b) e e
(c)   (d) none
e

2. The rate measure of the function y = 2 x − x 2 at x = 4 is :

(a) -6 (b) -8 (c) 6 (d) 8

3. The function f ( x) = 4 x − x 2 − 3 has a maximum value at

(a) x = 3 (b) x = 2 (c) x = −2 (d) none

4. The minimum value of the function y = x 2 − 6 x + 11 is

(a) 2 (b) -2 (c) 3 (d) -3

2 3 2 2
5. The critical points of the function f ( x) =
x − x − 2 x + 5 are-
3 3
1 1 1 1
(a) , −2 (b) − , 2 (c) ,2 (d) − , −2
2 2 2 2
6. If x + y = 10 , then the maximum value of xy is

(a) 25 (b) 21 (c) 16 (d) None of these

7. The minimum value of the 3sin x − 4cos x + 5 is

(a) 10 (b) 5 (c) 0 (d) None of these

8. The rate of change of area of a circle with respect to its radius r at r = 6 cm is

(a) 10π (b) 12π (c) 8π (d) 11π

9. The total revenue in Rupees received from the sale of x units of a product is given by

R ( x) = 3x 2 + 36 x + 5 . The marginal revenue, when x = 15 is

CLASS-XII/PRACTICE PAPER/MATHEMATICS/AOD/2024-25/ PAGE 1 of 7

 (a) 116 (b) 96 (c) 90 (d) 126

10. The point on the curve x 2 = 2 y which is nearest to the point (0, 5) is

(a) (2 2,4) (b) (2 2,0) (c) (0, 0) (d) (2, 2)

11. If the function f ( x ) = 2 x 2 − kx + 5 is increasing on [1, 2] , then k lies in the interval

(a) ( −, 4) (b) (4,  ) (c) ( −, 8) (d) (8,  )

12. The volume of a sphere is increasing at the rate of 4 cm 3 / sec . The rate of increase of the radius
when the volume is 288 cm 3 , is

1 1 1 1
(a) cm/s (b) cm/s (c) cm/s (d) cm/s
4 12 36 9

13. Side of an equilateral triangle expands at the rate of 2 cm / s . The rate of change of its area when

each side is 10 cm is

(a) 10 2 cm 2 / s (b) 10 3 cm 2 / s (c) 10 cm 2 / s (d) 5 cm 2 / s

14. A cylindrical tank of radius 10 m is being filled with wheat at the rate of 100 m 3 / hr .

The depth of the wheat is increasing at the rate of

(a) 1 m / hr (b) 0.1 m / hr (c) 1.1 m / hr (d) 0.5 m / hr

15. In the interval x  (1,2) , function f ( x ) = 2 | x − 1 | + 3 | x − 2 | is

(a) increasing (b) decreasing (c) constant (d) None of these

16. For x  R , the function f ( x ) = tan −1 x − x is

(a) increasing (b) decreasing (c) constant (d) None of these

17. The interval on which the function f ( x ) = 2 x 3 + 9 x 2 + 12 x − 1 is decreasing is :

(a) [−1,  ) (b) [−2, − 1] (c) [−, − 2] (d) [−1, 1]

18. The maximum value of sin2 x cos 2 x is

1 1
(a) (b) (c) 2 (d) 2 2
4 2

CLASS-XII/PRACTICE PAPER/MATHEMATICS/AOD/2024-25/ PAGE 2 of 7

19. The lengths of the three sides of a triangle are 10 + x 2 ,10 + x 2 and 20 − 2 x 2 units. If for x = k , the are
of the triangle is maximum , then the value of 3k 2 is
(a) 10 (b) 11 (c) 12 (d) 8

20. A rectangle is inscribed in an equilateral triangle of side length 2 2 units. Then the largest area of the
rectangle is
(a) 3 (b) 5 (c) 3 (d) 6

ASSERTION-REASON BASED QUESTIONS (1 mark each )

In the following questions, a statement of assertion (A) is followed by a statement of Reason (R).
Choose the correct answer out of the following choices.

(a) Both A and R are true and R is the correct explanation of A.

(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
21. Assertion (A):The rate of change of area of a circle with respect to is radius r when r = 6 cm is
12 cm 2 / cm .
dA
Reason (R): Rate of change of area of a circle with respect to its radius r is , where A is the area of the
dr
circle.
22. Assertion (A): f ( x) = tan x − x always increasing.
dy
Reason (R): Any function y = f ( x ) is increasing if  0.
dx
23. Assertion (A) : For the function y = x 3 , the point x = 0 is a point of inflection.
Reason (R) : If the second derivative is always zero at a point x = a , then it is a point of inflection.
24. Assertion (A):The absolute maximum of the function f ( x ) = 2 x 3 − 15 x 2 + 36 x + 1
on the interval 1, 5 is 56.
Reason (R) : In a closed interval , a function always attains the maximum value at the critical points.
25. Assertion (A): Of all the rectangles inscribed in a given area , square has the largest area.
Reason (R) : For maximum area, the length and the width of the rectangle will be equal to 2 times the radius
of the circle.

CLASS-XII/PRACTICE PAPER/MATHEMATICS/AOD/2024-25/ PAGE 3 of 7

 2marks/3 marks questions

26. The total revenue in Rupees received from the sale of x units of a product is given by
R ( x) = 13 x 2 + 26 x + 15 . Find the marginal revenue when x = 7

27. An edge of a variable cube is increasing at the rate of 3cm/s. How fast is the volume of the
cube increasing when the edge is 10 cm long?

 3 
28. Prove that the function f ( x ) = log e sin x is increasing on  , 2  .
 2 
29. Find the maximum profit that a company can make, if the profit function is given by
p( x ) = 41 − 72 x − 18 x 2 .

30. Find the interval in which the function f ( x ) = 10 − 6 x − 2 x 2 is increasing .

31. Prove that the function f ( x ) = x 3 − 3 x 2 + 3 x + 107 is increasing in R .
32. Find the maximum and minimum values of the function : f ( x ) = − | x + 2 | + 5, x  R .
 
33. Find the intervals in which the function given by f ( x ) = sin 3 x , x   0,  is strictly decreasing.
 2
34. Find the absolute maximum value of 2 x − 24 x + 107 in the interval [1, 3].
3

35. Find the absolute minimum value for the function: f ( x ) = sin x + cos x , x  [0,  ]
36. A man of height 2 metres walks at a uniform speed of 5 km/h away from a lamp post which is 6 metres
high. Find the rate at which the length of his shadow increases.
37. A particle moves along the curve 6 y = x 3 + 2 . Find the points on the curve at which the y-coordinate is

changing 8 times as fast as the x-coordinate.

38. . Find the two numbers whose sum is 24 and whose product is as large as possible.
39. Find the local maximum and local minimum values of the function

f ( x ) = x 3 − 6 x 2 + 9 x + 15

4sin x  
40. Prove that f ( x ) = − x is an increasing function of x in  0,  .
2 + cos x  2

5 marks questions
41. Find the interval in which the function f ( x ) = ( x + 1)3 ( x − 3)3 is

(a) strictly increasing (b) strictly decreasing.

42. Find the intervals in which the function f ( x ) = sin x + cos x, 0  x  2 is

(a) strictly increasing (b) strictly decreasing

CLASS-XII/PRACTICE PAPER/MATHEMATICS/AOD/2024-25/ PAGE 4 of 7

 43. 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 a box. What should be the side of the square to be cut off so that the volume of the box is
maximum? Also, find the maximum volume.
44. A water tank has a shape of an inverted cone with its axis vertical and vertex lowermost. Its semi-vertical angle
is tan −1 (0.5) . Water is poured into it at a constant rate of 5 cubic metre per minute. Find the rate at which the
level of the water is rising at the instant when the depth of water in the tank is 10 m.

45. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is
8
of the volume of the sphere.
27
46. 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 through the whole opening.
47. A garden in the form of a circular sector is of perimeter 10 cm. Find the maximum area of the garden.

CASE STUDY BASED QUESTIONS

48. Arvind wants to construct a rectangular plastic tank for his house that can hold 80 cubic ft of water.
The top of the tank is open. The width of tank will be 5 ft but the length and heights are variables.
Building the tank costs Rs. 20 per sq. foot for the base and Rs.10 per sq. foot for the side.
Let the length is x ft and height is y ft .

Based on the above information, answer the following questions:

(i) Find the relation between x and y . [1]
(ii) Find the total cost in terms of x . [1]

(iii) (a) Arvind wants to know the value of x for which cost is minimum. What is the value of x ? [2]

CLASS-XII/PRACTICE PAPER/MATHEMATICS/AOD/2024-25/ PAGE 5 of 7

 OR
(iii) (b) Arvind wants to know the minimum cost. What is that cost? [2]

49. Western music concert is organized every year in the stadium that can hold 36000 spectators. With
ticket price of Rs. 10, the average attendance has been 24000. Some financial expert estimated that
price of a ticket should be determined by the function
x
p ( x ) = 15 − where x is the number of tickets sold.
3000

Based on the above information, answer the following questions:

(i) Calculate the revenue R in terms of x . [1]
(ii) Find the value of x for which R is maximum . [1]
(iii) Find the maximum revenue. [2]
OR
(iii) When the revenue is maximum then calculate the price of each ticket (in rupees). [2]
50. A ladder 5m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the
wall, at the rate of 2cm/s.
Based on the above information, answer the following questions:

(i) The height of the wall y m and distance of the foot of ladder is x m away from the wall then what is the
relation between x, y & length of ladder? [2]

(ii) How fast is its height on the wall decreases when the foot of the ladder x m away from the ground? [2]

*************************************************************************

CLASS-XII/PRACTICE PAPER/MATHEMATICS/AOD/2024-25/ PAGE 6 of 7