Vardaan Learning Institute

1. Two cubes each of volume 64 cm$^3$ are joined end to end. The surface area of the resulting cuboid is:

1. 128 cm$^2$
2. 160 cm$^2$
3. 140 cm$^2$
4. 180 cm$^2$

2. A solid is in the form of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.

1. $\pi$ cm$^3$
2. $2\pi$ cm$^3$
3. $\frac{4}{3}\pi$ cm$^3$
4. $\frac{2}{3}\pi$ cm$^3$

3. A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm. Find the height of the cylinder.

1. 2.74 cm
2. 2.84 cm
3. 2.64 cm
4. 2.54 cm

4. The volumes of two spheres are in the ratio 64:27. The ratio of their surface areas is:

1. 1:2
2. 2:3
3. 9:16
4. 16:9

5. If the diagonal of a cube is $6\sqrt{3}$ cm, then its total surface area is:

1. 144 cm$^2$
2. 216 cm$^2$
3. 180 cm$^2$
4. 108 cm$^2$

6. The surface area of a sphere is 616 cm$^2$. Its radius is:

1. 7 cm
2. 14 cm
3. 21 cm
4. 28 cm

7. A cylinder with base radius 8 cm and height 2 cm is melted to form a cone of height 6 cm. The radius of the cone is:

1. 4 cm
2. 6 cm
3. 8 cm
4. 5 cm

8. If two solid hemispheres of same base radius r are joined together along their bases, then curved surface area of this new solid is:

1. $4\pi r^2$
2. $6\pi r^2$
3. $3\pi r^2$
4. $8\pi r^2$

9. The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm is:

1. 3
2. 5
3. 4
4. 6

10. Assertion (A): The total surface area of a cylinder of radius $r$ and height $h$ is $2\pi rh$.
Reason (R): $2\pi rh$ is the curved surface area of the cylinder.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is not the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

11. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.

12. A tent is in the shape of a cylinder surmounted by a conical top. If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent.

13. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm$^2$.

14. Three metallic solid cubes whose edges are 3 cm, 4 cm and 5 cm are melted and formed into a single cube. Find the edge of the cube so formed.

15. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 10 cm, and its base is of radius 3.5 cm, find the total surface area of the article.

16. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

17. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm$^3$ of iron has approximately 8g mass. (Use $\pi = 3.14$)

18. A well of diameter 3m is dug 14m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4m to form an embankment. Find the height of the embankment.

19. Water in a canal, 6 m wide and 1.5 m deep, is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed?

20. A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 20 cm and the diameter of the cylinder is 7 cm. Find the total volume of the solid.

21. A gulab jamun, contains sugar syrup up to about 30% of its volume. Find approximately how much syrup would be found in 45 gulab jamuns, each shaped like a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm.

22. A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled?

23. Case Study: Construction Block
Rasheed got a playing top (lattu) as his birthday present, which surprisingly had no colour on it. He wanted to colour it with his crayons. The top is shaped like a cone surmounted by a hemisphere. The entire top is 5 cm in height and the diameter of the top is 3.5 cm.

(i) Find the height of the conical part. (1 Mark)
(ii) Calculate the slant height of the cone. (2 Marks)
(iii) Find the Total Surface Area of the toy to be colored. (1 Mark)