1. The outer radius and inner radius of a hollow sphere made of iron are 21 cm and 7 cm respectively. If the weight of iron is 30 grams per cubic centimeter, what is the weight of the sphere?

(a) 2112.12 kg (b) 2212.21 kg (c) 1121.12 kg (d) 1122.21 kg

2. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?

3. A hollow cylindrical iron pipe open at both ends has an inner radius and outer radius of 4 cm and 5 cm respectively. If the total surface area of the pipe is 1188 square centimeters, what is the length of the pipe?

4. A vertical hollow cylindrical pipe made of iron has both ends open. Its height is 35 cm, the outer radius is 12 cm, and the inner radius is 10 cm. What is the volume of iron in the pipe in cubic centimeters?

5. The outer and inner radii of a hollow cylinder are 11 cm and 10 cm respectively, and its height is 14 cm. What is the volume of the metallic part of the cylinder?

(a) 880 cubic cm (b) 968 cubic cm (c) 88 cubic cm (d) 924 cubic cm

6. A hollow cylindrical iron pipe is 6 meters long. Its internal and external diameters are 3.5 cm and 4.2 cm respectively. Calculate the volume of iron in the pipe. If the weight of iron is 5 kg per cubic decimeter, then find the total weight of the pipe.

7. A wooden box with a 5 cm thick lid is made from wooden planks. Its weight is 115.5 kg, and when filled with rice, it weighs 880.5 kg. The internal length and breadth of the box are 12 decimeters and 8.5 decimeters respectively, and the weight of rice per cubic decimeter is 1.5 kg. **(1) Calculate and write the internal height of the box.** **(2) At a rate of ₹1.50 per square decimeter, calculate the total cost to paint the outer surface of the box (excluding the base).**

8. "A hollow cylindrical iron pipe is open at both ends. The external diameter of its mouth is 30 cm, the internal diameter is 26 cm, and the length of the pipe is 14.7 metres. If tar is applied to its entire surface at the rate of ₹2.25 per square decimetre, calculate and write the total cost."

9. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, calculate the length of the pipe.

10. "A hollow cylindrical iron pipe is 6 meters long. The internal and external diameters of the pipe are 3.5 cm and 4.2 cm respectively. Calculate the amount of iron used in the pipe. If the weight of iron is 5 kg per cubic decimeter, calculate the weight of the pipe."

11. The difference between the outer and inner curved surface areas of a hollow cylindrical pipe is 44 square centimeters, and the length of the pipe is 14 cm. The volume of the material of the pipe is 99 cubic centimeters. Find the outer and inner radii of the pipe.

12. In two concentric circles, a chord of the larger circle is tangent to the smaller circle. If the radii of the two circles are 10 cm and 4 cm respectively, then what is the length of the chord?

(a) 8 cm (b) 9 cm (c) 11 cm (d) 12 cm

13. The outer dimensions of a box with a lid are 12 cm in length, 10 cm in width, and 8 cm in height. The total inner surface area of the box is 376 square cm. If the walls of the box have equal thickness, what is their thickness?

14. In front of Anowar's house, there is a solid iron pillar with a cylindrical lower part and a conical upper part. The diameter of the base is 20 cm, the height of the cylindrical part is 2.8 meters, and the height of the conical part is 42 cm. If the weight of 1 cubic centimeter of iron is 7.5 grams, then calculate and write the total weight of the iron pillar.

15. The length of a right rectangular prism is three times its width and five times its height. If its volume is 14,400 cubic centimeters, then what is its total surface area?

(a) 4300 square cm (b) 4320 square cm (c) 4500 square cm (d) 4520 square cm

16. If the length of a cuboid is increased by 10%, the width by 20%, and the height is decreased by 30%, what is the percentage change in its volume?

(a) 6.5 increased (b) 7.6 increased (c) 6.5decreased (d) 7.6 decreased

17. A right circular cylinder made of iron sheet is open at both ends. Its height is 42 cm, thickness is 1 cm, and its outer radius is 10 cm. Find the amount of iron used to make the cylinder.

18. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent between the two circles.

19. To make a hollow cylindrical pipe of uniform thickness, a solid iron sphere of radius 6 cm is melted. The pipe has a length of 18 cm and an outer radius of 10 cm. Find the thickness of the pipe.

20. Given that the combined (weighted) mean of the following frequency distribution is 50 and the total frequency is 120, find the values of \( f_1 \) and \( f_2 \). | Class Interval | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | |----------------|------|--------|--------|--------|---------| | Frequency | 17 | \( f_1 \) | 32 | \( f_2 \) | 19 |

21. A classroom needs to be built for 40 students in such a way that each student gets 5 square meters of floor area and 25 cubic meters of space. If the length of the room is 20 meters, what should be its height?

(a) 5 meters (b) 7 meters (c) 9 meters (d) 11 meters

22. The radii of two circles are 4 cm and 3 cm respectively, and the distance between their centers is 13 cm. Find the length of a direct common tangent to the two circles.

23. A rectangular box has its length, breadth, and height in the ratio 3:2:1, and its volume is 384 cubic centimeters. Find the surface area of the box.

24. A vertical hollow cylindrical pipe has an outer radius of 16 cm and an inner radius of 12 cm, with a height of 36 cm. If the pipe is melted and solid cylindrical rods are made from it, each with a diameter of 2 cm and a length of 6 cm, determine how many such solid rods can be produced.

25. A hollow cylindrical iron pipe, open at both ends, has a height of 2.8 meters. The inner circumference of the pipe is 4.6 decimeters, and it is made of 84.48 cubic decimeters of iron. Find the outer circumference of the pipe.

26. A brass plate has a square base with side length \(x\) cm, thickness 1 mm, and a total weight of 4725 grams. If the weight of 1 cubic centimeter of brass is 8.4 grams, then calculate and write the value of \(x\).

27. A cylindrical wooden log of uniform density has a curved surface area of 440 square decimeters. The weight of 1 cubic decimeter of wood is 1.5 kilograms, and the total weight of the log is 9.24 quintals. Calculate the diameter and height of the log.

28. In a business, three friends have their capital investment in the ratio 2:3:4. If in the following year they increase their capital by 10%, 15%, and 20%, respectively, in what ratio will the profit be distributed among them at the end of the second year?

(a) 44 : 69 : 96 (b) 69 : 44 : 96 (c) 96 : 69 : 44 (d) 45 : 96 : 63

29. A rectangular box has a length, width, and height ratio of 3:2:1. If its volume is 384 cubic centimeters, find the total surface area of the box.

30. Translate the following: "The internal length, width, and height of a tea box are 7.5 decimeters, 6 decimeters, and 5.4 decimeters respectively. The weight of the box when filled with tea is 52 kilograms 350 grams. But when empty, the box weighs 3.75 kilograms. Determine the weight of tea per cubic decimeter."