1. A hollow metallic sphere has an inner radius of 3 cm and an outer radius of 5 cm. It is melted to form a solid right circular cylinder of height \(\frac{8}{3}\) cm. Find the diameter of the base of the cylinder.

2. The volume of a sphere is directly proportional to the cube of its radius. Three solid spheres with diameters of \(1\frac{1}{2}\) m, 2 m, and \(2\frac{1}{2}\) m are melted and recast into a single solid sphere. Using the concept of proportionality, determine the diameter of the new sphere.

3. The length of a rectangular field is 36 meters more than its breadth. The area of the field is 460 square meters. Form a quadratic equation in one variable from this statement and determine the coefficients of \(x^2\), \(x\), and \(x^0\).

4. Let the smaller pillar be AB \(= x\) meters and the larger pillar be CD \(= 2x\) meters.
The midpoint of the base connection BD is O; the angles of elevation from O to the tops of the pillars are \(\angle\)AOB = \(\theta\) and \(\angle\)COD = 90\(^o-\theta\).
Since O is the midpoint of BD, we have BO = OD = \(\frac{120}{2}\) meters = 60 meters.

From \( \triangle \)ABO, we get:
\(\cfrac{AB}{BO} = \tan \theta\)
Or, \(\cfrac{x}{60} = \tan\theta\) --------(i)

From \( \triangle \)COD, we get:
\(\cfrac{CD}{OD} = \tan(90^o - \theta)\)
Or, \(\cfrac{2x}{60} = \cot\theta\) --------(ii)

Multiplying equations (i) and (ii), we get:
\(\cfrac{x}{60} \times \cfrac{2x}{60} = \tan\theta \times \cot\theta\)
Or, \(\cfrac{2x^2}{60 \times 60} = 1\)
Or, \(x^2 = \cfrac{60 \times \cancel{60}30}{\cancel{2}}\)
Or, \(x = 30\sqrt2\)

\(\therefore\) The height of the smaller pillar is \(30\sqrt2\) meters.
And the height of the larger pillar is \(30\sqrt2 \times 2\) meters \(= 60\sqrt2\) meters.
(Proved).

5. Three jars with equal diameter and height were partially filled with dilute sulfuric acid: - the first jar was filled to \(\frac{2}{3}\) of its height, - the second to \(\frac{5}{6}\), and - the third to \(\frac{7}{9}\). If the acid from these three jars is poured into a single jar with a diameter of 2.1 decimetres, and the height of the acid in this jar becomes 4.1 decimetres, then calculate and write the height of each of the original three jars, given that their diameter was 1.4 decimetres.

6. Relative to a point on a horizontal plane, a kite is observed at an elevation angle of \(45^\circ\) when it is at a height of 45 meters, and at an elevation angle of \(60^\circ\) when it is at a height of 60 meters. > Show that the distance between these two positions of the kite is > \[ 5\sqrt{6(23 - 12\sqrt{3})} \text{ meters}, \] > assuming the horizontal distance from the point to the kite remains constant.

7. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

8. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \[ s = 6d^2 \]

9. 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 \(x\).

10. A solid cuboid has a length, width, and height ratio of \(4:3:2\), and its total surface area is \(468\) square cm. Determine the volume of the cuboid.

11. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \(s^3 = d^2\).

12. If a circle has a radius of 5 cm and a tangent is drawn from an external point \(P\) to the circle with a length of 12 cm, what is the distance from the center to point \(P\)?

13. Three circles with centers \(A\), \(B\), and \(C\) externally touch each other. If \(AB = 5\) cm, \(BC = 7\) cm, and \(CA = 6\) cm, then find the radius of the circle centered at \(C\).

14. In a joint business, the capital ratio of A, B, and C is \(\frac{1}{2}:\frac{1}{3}:\frac{1}{4}\). After 4 months, A withdraws half of his capital, and after another 4 months, the total profit amounts to 61,050 rupees. Determine each person's share of the profit.

15. When the angle of elevation of the sun increases from 45° to 60°, the length of a pole’s shadow shortens by 3 meters. Determine the height of the pole. [Take \(\sqrt{3} = 1.732\) and find the approximate value correct to three decimal places.]

16. A solid sphere with a radius of \(r\) units is melted and recast into a solid right circular cone with a height of \(r\) units. Find the radius of the base of the cone.

(a) 2r units (b) 3r units (c) r unit (d) 4r units

17. Two identical solid hemispheres, each with a base radius of \(r\) units, are joined together along their flat surfaces. The total surface area of the resulting solid body will be \(6πr^3\) square units.

18. Assume \(PQ\) is the tower and \(AB\) is the stick. \(∴ BC = 4\) cm, \(QC = 28\) meters Triangles \(∆PQC\) and \(∆ABC\) are similar right-angled triangles. Therefore, they are similar. \(∴ \cfrac{PQ}{AB} = \cfrac{QC}{BC} \) Or, \(\cfrac{PQ}{6} = \cfrac{28}{4}\) Or, \(PQ = \cfrac{28 \times 6}{4} = 42\) meters ∴ The height of the tower is 42 meters.

19. A cylindrical wooden log with a diameter of \(12\sqrt{2}\) cm and a length of 21 meters is shaped into a cuboid with a square cross-section, such that the least amount of wood is wasted. Find the volume of wood remaining in the cuboid log.

20. If a right circular cylinder with radius \(2x\) cm and height \(x\) cm is melted to form a sphere, then the radius of the resulting sphere will be –

(a) \(\sqrt[3]3{x}\) cm (b) \(\sqrt[3]{3x}\) (c) \(x\) (d) \(2x\)

21. The elevation angle of a kite is 60°, and the length of the string is \(20\sqrt{3}\) meters. So, the height of the kite above the ground is—

(a) 20 meters (b) 30 meters (c) 40 meters (d) 50 meters

22. The radius of a circle with center \(O\) is 5 cm. Point \(P\) is located at a distance of 13 cm from \(O\). From point \(P\), two tangents \(PQ\) and \(PR\) are drawn to the circle. Find the area of the quadrilateral \(PQOR\).

(a) \(60\) square cm (b) \(30\) square cm (c) \(120\) square cm (d) \(150\) square cm

23. 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) \(a=\sqrt2d\)

24. If two circles with radii \(r_1\) and \(r_2\) touch each other externally, and the distance between their centers is \(d\), then which of the following is correct?

(a) \(r_1+d=r_2\) (b) \(r_2+d=r_1\) (c) \(r_1+r_2=d\) (d) \(r_1-r_2=d\)

25. If water flows through a pipe with an internal diameter of 5.6 cm at a rate of 200 meters per minute, how long will it take to fill \(\frac{5}{8}\) of a tank that is 2.8 meters long, 2.2 meters wide, and 1.6 meters deep?

26. As the sun's angle of elevation increases from 30° to 60°, the length of the monument's shadow decreases by \(50\sqrt{3}\) meters. What is the height of the monument?

27. Three friends purchase a bus by investing ₹1,20,000, ₹1,50,000, and ₹1,10,000 respectively. The first friend works as the driver, while the other two work as conductors. They decide that \(\frac{2}{5}\) of the income will be distributed based on work in the ratio 3 : 2 : 2, and the remaining amount will be divided according to their capital investment. If the total income in a certain month is ₹29,260, determine how much each person will receive.

28. The heights of two pillars are \(h_1\) meters and \(h_2\) meters respectively. If the angle of elevation from the base of the second pillar to the top of the first pillar is 60°, and the angle of elevation from the base of the first pillar to the top of the second pillar is 45°, show that \(h_1^2 = 3h_2^2\).

29. In a semicircle with a radius of 4 cm, AB is the diameter and ∠ACB is an angle inscribed in the semicircle. If BC = \(2\sqrt{7}\) cm, find the length of AC.

30. In a circle, two chords PQ and PR are perpendicular to each other. If the radius of the circle is \(r\) cm, find the length of the chord QR.