1. The diameter of one sphere is twice the diameter of another sphere. If the numerical value of the surface area of the larger sphere is equal to the numerical value of the volume of the smaller sphere, then what is the radius of the smaller sphere?

2. From a cube-shaped water tank, 64 buckets of equal size are removed, leaving the tank one-third full. If the side length of the tank is 1.2 meters, then how much water does each bucket hold in liters? (Assume: 1 cubic decimeter = 1 liter)

3. The length of one side is 6.2 cm, and the measures of the two angles adjacent to that side are 50° and 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

4. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

5. The bottom part of a solid is in the shape of a hemisphere, and the top part is in the shape of a right circular cone. If the surface areas of both parts are equal, then calculate and write the ratio of the radius to the height of the cone.

6. The largest cone that can be obtained from a cube has a diameter equal to the length of the cube's side.

7. If a cube has a side length of \(a\) units and a diagonal length of \(d\) units, then the relationship between \(a\) and \(d\) is—?

(a) \(\sqrt2a=d\) (b) \(\sqrt3a=d\) (c) \(a=\sqrt3d\) (d) \(a=\sqrt2d\)

8. If a triangle similar to one with sides 4 cm, 6 cm, and 8 cm has its largest side measuring 6 cm, what is the length of the smallest side of that triangle?

(a) 4 cm (b) 3 cm (c) 2 cm (d) 5 cm

9. If a right circular cone and a sphere have the same radius, and the numerical value of the cone's volume is equal to the numerical value of the sphere's curved surface area, then what is the height of the cone?

(a) 10 units (b) 11 units (c) 12 units (d) 13 units

10. The volume of a cube is equal to the area of one of its faces. What is the length of its edge?

(a) 4 units (b) 5 units (c) 6 units (d) 8 units

11. If a right circular cylinder has a radius of 3 cm and a height of 4 cm, then what is the length of the longest rod that can be placed inside it?

12. If the area of the square drawn on one side of any triangle is equal to the sum of the areas of the squares drawn on the other two sides, then prove that the angle opposite to the first side is a right angle.

13. Prove that in a triangle, if the area of the square constructed on one side is equal to the sum of the areas of the squares constructed on the other two sides, then the angle opposite the longest side is a right angle.

14. The volume of a sphere is directly proportional to the cube of its radius. If three solid spheres with radii of 3 cm, 4 cm, and 5 cm are melted to form a new solid sphere, and there is no loss in volume during melting, then find the radius of the new sphere.

15. A one-end open cylindrical pipe is 12 meters long and has a radius of 4.5 meters. What is the length of the longest rod that can be placed inside the cylinder?

16. A right-angled isosceles triangle in which each of the equal sides has a length of \(a\) units. The perimeter of the triangle is —

(a) \((1+\sqrt{2})a\) units (b) \((2+\sqrt{2})a\) units (c) \(3a\) units (d) \((3+2\sqrt{2})a\) units

17. If a right circular cone has a radius of \(\frac{r}{2}\) units and a slant height of \(2l\) units, then the total surface area is \(2πr(r + l)\).

18. 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 \(x\).

19. Two cubes with equal edge lengths are joined side by side to form a cuboid. If the volume of the cuboid is 1024 cubic centimeters, what is the length of each edge of the cubes?

20. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

21. The radius of a hemisphere and the height of a cone are equal. If their volumes are numerically equal, then what is the ratio of the radius of the hemisphere to the radius of the cone?

22. Draw an isosceles triangle with a base length of 5.2 cm and each of the equal sides measuring 7 cm. Then, construct the circumcircle of the triangle and measure the circumradius. (Only mark the construction steps).

23. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.

24. A solid object has its lower part in the shape of a hemisphere and its upper part in the shape of a right circular cone. If the surface areas of both parts are equal, determine the ratio of the radius to the height of the cone.

25. Prove that if the area of a square drawn on one side of any triangle is equal to the sum of the areas of squares drawn on the other two sides, then the angle opposite the first side is a right angle.

26. A circle has center ‘O’, and a point P lies 26 cm away from it. If the length of the tangent drawn from point P to the circle is 10 cm, then what is the radius of the circle?

27. In a circle centered at O, there are two parallel chords AB and CD with lengths 10 cm and 24 cm, positioned on opposite sides of the center. If the distance between the chords AB and CD is 17 cm, then calculate and write the radius of the circle.

28. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

29. A right-angled triangle in which the two sides adjacent to the right angle are 7 cm and 9 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

30. An isosceles triangle where the base is 7.8 cm and each of the equal sides is 6.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).