.
CHAPTER 6 STD 12 Date : 13/12/25
APPLICATION OF DERIVATIVES Maths
//X Section A
• Write the answer of the following questions. [Each carries 3 Marks] [42]
1. Find the intervals in which the function f given by f(x) = 2x3 – 3x2 – 36x + 7 is
(a) increasing (b) decreasing
2. 3) Find the intervals in which the following functions are strictly increasing or decreasing :
–2x3 – 9x2 – 12x + 1
3. Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the
same function in [–3, –1].
4sin q é pù
4. Prove that y = - q is an increasing function of q in ê0, ú .
(2 + cos q) ë 2û
5. 5) Find the intervals in which the following functions are strictly increasing or decreasing :
(x + 1)3 (x – 3)3
6. Find all the points of local maxima and local minima of the function f given by
f(x) = 2x3 – 6x2 + 6x + 5.
2x
7. Show that y = log(1 + x) – , x > –1, is an increasing function of x throughout its domain.
2 + x
8. Find absolute maximum and minimum values of a function f given by
4 1
f ( x ) = 12x 3 - 6 x 3 , x Î [–1, 1]
9. Find the values of x for which y = [x(x – 2)]2 is an increasing function.
10. Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
1
11. Let I be any interval disjoint from [–1, 1]. Prove that the function f given by f(x) = x + is
x
increasing on I.
1
Find the intervals in which the function f given by f ( x ) = x + , x ¹ 0 is
3
12.
x3
(i) increasing (ii) decreasing.
13. Find the maxim um and minim um values of x + sin(2x) on [0, 2p].
14.4) Find the local maxima and local minima, if any, of the following functions. Find also the local
maximum and the local minimum values, as the case may be : f(x) = sin x – cos x ; 0 < x < 2p
//X Section B
• Write the answer of the following questions. [Each carries 4 Marks] [64]
15. 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.
16. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius
2R
R is . Also find the maximum volume.
3
17. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height
is tan–1 2.
18. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
4sin x - 2 x - x cos x
19. Find the intervals in which the function f given by f ( x ) = is (i) increasing
2 + cos x
(ii) decreasing.
20. A water tank has the shape of an inverted right circular 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
meter per hour. Find the rate at which the level of the water is rising at the instant when the depth
of water in the tank is 4 m.
, A water tank has the shape of an inverted right circular 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
meter per hour. Find the rate at which the level of the water is rising at the instant when the depth
of water in the tank is 4 m.
21. 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.
22. Find local maximum and local minimum values of the function f given by
f(x) = 3x4 + 4x3 – 12x2 + 12.
23. Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 – 15x2 + 36x
+ 1 on the interval [1, 5].
24. 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.
8
25. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of
27
the volume of the sphere.
26. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its
depth is 2m and volume is 8m3. If building of tank costs ` 70 per sq. metres for the base and `
45 per square metre for sides. What is the cost of least expensive tank ?
27. 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.
28. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a
4r
sphere of radius r is .
3
29. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show
2 2 3
that the minimum length of the hypotenuse is (a 3 + b 3 ) 2 .
30. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is
æ 1ö
sin -1 ç ÷ .
è 3ø
, .
CHAPTER 6 STD 12 Date : 13/12/25
APPLICATION OF DERIVATIVES Maths
Section [ A ] : 3 Marks Questions
No Ans Chap Sec Que Universal_QueId
1. - Chap 6 S2 5 QP25P11B1213_P1C6S2Q5
2. - Chap 6 S2 6.3 QP25P11B1213_P1C6S2Q6.3
3. - Chap 6 S3 10 QP25P11B1213_P1C6S3Q10
4. - Chap 6 S2 9 QP25P11B1213_P1C6S2Q9
5. - Chap 6 S2 6.5 QP25P11B1213_P1C6S2Q6.5
6. - Chap 6 S5 18 QP25P11B1213_P1C6S5Q18
7. - Chap 6 S2 7 QP25P11B1213_P1C6S2Q7
8. - Chap 6 S5 28 QP25P11B1213_P1C6S5Q28
9. - Chap 6 S2 8 QP25P11B1213_P1C6S2Q8
10. - Chap 6 S3 15 QP25P11B1213_P1C6S3Q15
11. - Chap 6 S2 15 QP25P11B1213_P1C6S2Q15
12. - Chap 6 S4 4 QP25P11B1213_P1C6S4Q4
13. - Chap 6 S3 12 QP25P11B1213_P1C6S3Q12
14. - Chap 6 S3 3.4 QP25P11B1213_P1C6S3Q3.4
Section [ B ] : 4 Marks Questions
No Ans Chap Sec Que Universal_QueId
15. - Chap 6 S3 24 QP25P11B1213_P1C6S3Q24
16. - Chap 6 S4 14 QP25P11B1213_P1C6S4Q14
17. - Chap 6 S3 25 QP25P11B1213_P1C6S3Q25
18. - Chap 6 S3 19 QP25P11B1213_P1C6S3Q19
19. - Chap 6 S4 3 QP25P11B1213_P1C6S4Q3
20. - Chap 6 S5 31 QP25P11B1213_P1C6S5Q31
21. - Chap 6 S5 25 QP25P11B1213_P1C6S5Q25
22. - Chap 6 S5 20 QP25P11B1213_P1C6S5Q20
23. - Chap 6 S5 27 QP25P11B1213_P1C6S5Q27
24. - Chap 6 S5 24 QP25P11B1213_P1C6S5Q24
25. - Chap 6 S3 23 QP25P11B1213_P1C6S3Q23
26. - Chap 6 S4 6 QP25P11B1213_P1C6S4Q6
27. - Chap 6 S5 29 QP25P11B1213_P1C6S5Q29
28. - Chap 6 S4 12 QP25P11B1213_P1C6S4Q12
29. - Chap 6 S4 9 QP25P11B1213_P1C6S4Q9
Welcome To Future - Quantum Paper
CHAPTER 6 STD 12 Date : 13/12/25
APPLICATION OF DERIVATIVES Maths
//X Section A
• Write the answer of the following questions. [Each carries 3 Marks] [42]
1. Find the intervals in which the function f given by f(x) = 2x3 – 3x2 – 36x + 7 is
(a) increasing (b) decreasing
2. 3) Find the intervals in which the following functions are strictly increasing or decreasing :
–2x3 – 9x2 – 12x + 1
3. Find the maximum value of 2x3 – 24x + 107 in the interval [1, 3]. Find the maximum value of the
same function in [–3, –1].
4sin q é pù
4. Prove that y = - q is an increasing function of q in ê0, ú .
(2 + cos q) ë 2û
5. 5) Find the intervals in which the following functions are strictly increasing or decreasing :
(x + 1)3 (x – 3)3
6. Find all the points of local maxima and local minima of the function f given by
f(x) = 2x3 – 6x2 + 6x + 5.
2x
7. Show that y = log(1 + x) – , x > –1, is an increasing function of x throughout its domain.
2 + x
8. Find absolute maximum and minimum values of a function f given by
4 1
f ( x ) = 12x 3 - 6 x 3 , x Î [–1, 1]
9. Find the values of x for which y = [x(x – 2)]2 is an increasing function.
10. Find two positive numbers x and y such that their sum is 35 and the product x2y5 is a maximum.
1
11. Let I be any interval disjoint from [–1, 1]. Prove that the function f given by f(x) = x + is
x
increasing on I.
1
Find the intervals in which the function f given by f ( x ) = x + , x ¹ 0 is
3
12.
x3
(i) increasing (ii) decreasing.
13. Find the maxim um and minim um values of x + sin(2x) on [0, 2p].
14.4) Find the local maxima and local minima, if any, of the following functions. Find also the local
maximum and the local minimum values, as the case may be : f(x) = sin x – cos x ; 0 < x < 2p
//X Section B
• Write the answer of the following questions. [Each carries 4 Marks] [64]
15. 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.
16. Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius
2R
R is . Also find the maximum volume.
3
17. Show that the semi-vertical angle of the cone of the maximum volume and of given slant height
is tan–1 2.
18. Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.
4sin x - 2 x - x cos x
19. Find the intervals in which the function f given by f ( x ) = is (i) increasing
2 + cos x
(ii) decreasing.
20. A water tank has the shape of an inverted right circular 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
meter per hour. Find the rate at which the level of the water is rising at the instant when the depth
of water in the tank is 4 m.
, A water tank has the shape of an inverted right circular 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
meter per hour. Find the rate at which the level of the water is rising at the instant when the depth
of water in the tank is 4 m.
21. 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.
22. Find local maximum and local minimum values of the function f given by
f(x) = 3x4 + 4x3 – 12x2 + 12.
23. Find the absolute maximum and minimum values of a function f given by f(x) = 2x3 – 15x2 + 36x
+ 1 on the interval [1, 5].
24. 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.
8
25. Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is of
27
the volume of the sphere.
26. A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its
depth is 2m and volume is 8m3. If building of tank costs ` 70 per sq. metres for the base and `
45 per square metre for sides. What is the cost of least expensive tank ?
27. 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.
28. Show that the altitude of the right circular cone of maximum volume that can be inscribed in a
4r
sphere of radius r is .
3
29. A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Show
2 2 3
that the minimum length of the hypotenuse is (a 3 + b 3 ) 2 .
30. Show that semi-vertical angle of right circular cone of given surface area and maximum volume is
æ 1ö
sin -1 ç ÷ .
è 3ø
, .
CHAPTER 6 STD 12 Date : 13/12/25
APPLICATION OF DERIVATIVES Maths
Section [ A ] : 3 Marks Questions
No Ans Chap Sec Que Universal_QueId
1. - Chap 6 S2 5 QP25P11B1213_P1C6S2Q5
2. - Chap 6 S2 6.3 QP25P11B1213_P1C6S2Q6.3
3. - Chap 6 S3 10 QP25P11B1213_P1C6S3Q10
4. - Chap 6 S2 9 QP25P11B1213_P1C6S2Q9
5. - Chap 6 S2 6.5 QP25P11B1213_P1C6S2Q6.5
6. - Chap 6 S5 18 QP25P11B1213_P1C6S5Q18
7. - Chap 6 S2 7 QP25P11B1213_P1C6S2Q7
8. - Chap 6 S5 28 QP25P11B1213_P1C6S5Q28
9. - Chap 6 S2 8 QP25P11B1213_P1C6S2Q8
10. - Chap 6 S3 15 QP25P11B1213_P1C6S3Q15
11. - Chap 6 S2 15 QP25P11B1213_P1C6S2Q15
12. - Chap 6 S4 4 QP25P11B1213_P1C6S4Q4
13. - Chap 6 S3 12 QP25P11B1213_P1C6S3Q12
14. - Chap 6 S3 3.4 QP25P11B1213_P1C6S3Q3.4
Section [ B ] : 4 Marks Questions
No Ans Chap Sec Que Universal_QueId
15. - Chap 6 S3 24 QP25P11B1213_P1C6S3Q24
16. - Chap 6 S4 14 QP25P11B1213_P1C6S4Q14
17. - Chap 6 S3 25 QP25P11B1213_P1C6S3Q25
18. - Chap 6 S3 19 QP25P11B1213_P1C6S3Q19
19. - Chap 6 S4 3 QP25P11B1213_P1C6S4Q3
20. - Chap 6 S5 31 QP25P11B1213_P1C6S5Q31
21. - Chap 6 S5 25 QP25P11B1213_P1C6S5Q25
22. - Chap 6 S5 20 QP25P11B1213_P1C6S5Q20
23. - Chap 6 S5 27 QP25P11B1213_P1C6S5Q27
24. - Chap 6 S5 24 QP25P11B1213_P1C6S5Q24
25. - Chap 6 S3 23 QP25P11B1213_P1C6S3Q23
26. - Chap 6 S4 6 QP25P11B1213_P1C6S4Q6
27. - Chap 6 S5 29 QP25P11B1213_P1C6S5Q29
28. - Chap 6 S4 12 QP25P11B1213_P1C6S4Q12
29. - Chap 6 S4 9 QP25P11B1213_P1C6S4Q9
Welcome To Future - Quantum Paper
