All Categories MCQs
Topic Notes: All Categories
General Description
Plato
- Biography: Ancient Greek philosopher (427–347 BCE), student of Socrates and teacher of Aristotle, founder of the Academy in Athens.
- Important Ideas:
- Theory of Forms
- Philosopher-King
- Ideal State
2771
What is the volume of a tetrahedron with an edge length of 6 cm?
Answer:
18√2 cm³
The volume of a regular tetrahedron is V = (a³√2) / 12. V = (6³√2) / 12 = (216√2) / 12 = 18√2 cm³.
2772
If a right square pyramid has a base side of 10 cm and a volume of 400 cm³, what is its height?
Answer:
12 cm
Volume = (1/3) × Base Area × Height. 400 = (1/3) × (10 × 10) × h. 400 = (100/3) × h. h = (400 × 3) / 100 = 12 cm.
2773
A hemispherical bowl has a diameter of 12 cm. It is full of liquid. The liquid is emptied into cylindrical bottles of radius 3 cm and height 4 cm. How many bottles are needed?
Answer:
4
Let's say the cylindrical vessel has height h. Its radius is 1.5h. Volume of bowl = (2/3)π(6)³ = 144π. Let's make an assumption that the vessels have radius 3 cm. Then height = 2 cm. Volume = π(3²)(2) = 18π. Number of vessels = 144π / 18π = 8. (Since the question doesn't define the vessel size strictly, I'll adjust the options or the question. Let's ask: If the cylinder has radius 6 cm, h = 4 cm. V = π(36)(4) = 144π. Then 1 vessel. If the question implies the cylinder's dimensions match the hemisphere somehow, let's substitute standard values. Let's change the question: the vessels have radius 3cm and height 4cm. V = 36π. 144π/36π = 4 vessels.)
2774
What is the ratio of the volume of a cylinder to the volume of a sphere inscribed perfectly inside it?
Answer:
3:2
If a sphere is inscribed in a cylinder, the cylinder's radius is r and its height is 2r. Volume of cylinder = πr²(2r) = 2πr³. Volume of sphere = (4/3)πr³. Ratio = 2πr³ : (4/3)πr³ = 2 : 4/3 = 6:4 = 3:2.
2775
The radii of two cylinders are in the ratio 2:3 and their heights are in the ratio 5:3. The ratio of their volumes is:
Answer:
20:27
Volume ratio = (r₁/r₂)² × (h₁/h₂) = (2/3)² × (5/3) = (4/9) × (5/3) = 20/27, which is 20:27.
2776
If a sphere and a cube have equal surface areas, what is the ratio of the volume of the sphere to the volume of the cube?
Answer:
√6 : √π
Given 4πr² = 6a², so r/a = √(6/(4π)) = √(3/(2π)). Volume ratio = (4/3)πr³ / a³ = (4/3)π(r/a)³ = (4/3)π(√(3/(2π)))³. Expanding: (4/3)π × (3/(2π)) × √(3/(2π)) = 2 × √(3/(2π)) = √(4 × 3 / 2π) = √(6/π) = √6 : √π.
2777
A right circular cylinder and a right circular cone have the same base radius and height. If the ratio of their curved surface areas is 8:5, find the ratio of the radius to the height.
Answer:
3:4
CSA of cylinder = 2πrh. CSA of cone = πr√(r² + h²). Ratio = 2πrh / (πr√(r² + h²)) = 8/5. So, 2h / √(r² + h²) = 8/5. Squaring both sides: 4h² / (r² + h²) = 64/25. 100h² = 64r² + 64h². 36h² = 64r². r²/h² = 36/64 = 9/16. So, r/h = 3/4, or 3:4.
2778
The diameter of a hollow cone is 14 cm and its height is 24 cm. Find its curved surface area.
Answer:
550 cm²
Radius r = 7 cm. Height h = 24 cm. Slant height l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm. Curved Surface Area = πrl = (22/7) × 7 × 25 = 550 cm².
2779
If the areas of the adjacent faces of a rectangular block are in the ratio 2:3:4 and its volume is 9000 cm³, find the length of the shortest edge.
Answer:
15 cm
Let the areas be 2x, 3x, 4x. Their product is (lw)(wh)(lh) = l²w²h² = V². So, (2x)(3x)(4x) = 9000². 24x³ = 81,000,000. x³ = 3,375,000. x = 150. The areas are 300, 450, 600. The edges are V/Area: 9000/600=15, 9000/450=20, 9000/300=30. The shortest edge is 15 cm.
2780
Find the length of the longest pole that can be kept in a room 10 m long, 10 m wide, and 5 m high.
Answer:
15 m
The longest pole matches the body diagonal of the room. Diagonal = √(L² + W² + H²) = √(10² + 10² + 5²) = √(100 + 100 + 25) = √225 = 15 m.