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
2831
The total surface area of a solid hemisphere is 108π cm². What is its volume?
Answer:
144π cm³
TSA of hemisphere = 3πr² = 108π. So, r² = 36, meaning r = 6 cm. Volume = (2/3)πr³ = (2/3)π(6³) = (2/3)π(216) = 144π cm³.
2832
A rectangular block 6 cm by 12 cm by 15 cm is cut into an exact number of equal cubes. Find the least possible number of cubes.
Answer:
40
To find the least number of cubes, we need the largest possible cube. The side length of the largest cube is the HCF of the block's dimensions (6, 12, 15), which is 3 cm. Volume of block = 6 × 12 × 15 = 1080 cm³. Volume of cube = 3³ = 27 cm³. Number of cubes = 1080 / 27 = 40.
2833
A right circular cylinder and a sphere have equal volumes and equal radii. What is the ratio of the height of the cylinder to the diameter of the sphere?
Answer:
2:3
Volume of cylinder = Volume of sphere. πr²h = (4/3)πr³, which gives h = 4r/3. The diameter of the sphere is D = 2r. The ratio is h / D = (4r/3) / 2r = 4/6 = 2:3.
2834
If the diagonal of a cube is √12 cm, find its volume in cm³.
Answer:
8
The main diagonal of a cube is a√3. Given a√3 = √12. So a = √(12/3) = √4 = 2 cm. The volume is a³ = 2³ = 8 cm³.
2835
A cube of edge 5 cm is cut into cubes of edge 1 cm. What is the ratio of the total surface area of the original cube to the sum of the total surface areas of the smaller cubes?
Answer:
1:5
Number of smaller cubes = 5³ / 1³ = 125. Surface area of large cube = 6 × 5² = 150 cm². Surface area of one small cube = 6 × 1² = 6 cm². Total surface area of all small cubes = 125 × 6 = 750 cm². Ratio = 150:750 = 1:5.
2836
A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into small cylindrical bottles of diameter 3 cm and height 4 cm. How many bottles are needed?
Answer:
72
Volume of hemisphere = (2/3)π(9)³ = 486π. Radius of bottle = 3/2 = 1.5 cm. Volume of one bottle = π(1.5)² × 4 = 9π. Number of bottles = 486π / 9π = 54.
2837
A sector of a circle of radius 12 cm and central angle 120° is rolled up so that its two bounding radii are joined together to form a cone. Find the radius of the base of the cone.
Answer:
4 cm
The arc length of the sector becomes the circumference of the cone's base. Arc length = (θ/360) × 2πr_sector = (120/360) × 2π × 12 = 8π cm. Let r_cone be the radius of the cone. 2πr_cone = 8π, so r_cone = 4 cm.
2838
The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the diameter of the pillar.
Answer:
14 m
Volume / CSA = (πr²h) / (2πrh) = r/2. So, r/2 = 924 / 264 = 3.5. Therefore, radius r = 7 m. The diameter is 2r = 14 m.
2839
What is the volume of a right prism whose base is an equilateral triangle of side 4 cm and whose height is 10 cm?
Answer:
40√3 cm³
Volume of prism = Base Area × Height. Base is an equilateral triangle, so Area = (√3/4)a² = (√3/4) × 4² = 4√3 cm². Volume = 4√3 × 10 = 40√3 cm³.
2840
The diameter of a roller is 84 cm and its length is 120 cm. It takes 500 complete revolutions to roll once over a playground. What is the area of the playground in m²? (Use π = 22/7)
Answer:
1584 m²
Diameter = 84 cm = 0.84 m, so r = 0.42 m. Length (h) = 120 cm = 1.2 m. Area of 1 revolution = CSA = 2πrh = 2 × (22/7) × 0.42 × 1.2 = 3.168 m². Area of playground = 500 × 3.168 = 1584 m².