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
2781
A cone of radius 5 cm and height 12 cm is melted and recast into a sphere. Find the radius of the sphere.
Answer:
∛(75) cm
Volume of cone = (1/3)πr²h = (1/3)π(5²)(12) = 100π. Volume of sphere = (4/3)πR³ = 100π. R³ = (100 × 3) / 4 = 75. Therefore, R = ∛75 cm.
2782
The total surface area of a cube is 96 cm². What is the length of its main diagonal?
Answer:
4√3 cm
Total Surface Area = 6a² = 96. a² = 16, so a = 4 cm. The main diagonal of a cube is given by the formula a√3. Therefore, the diagonal is 4√3 cm.
2783
If the height of a cylinder is doubled and the radius remains the same, the volume will:
Answer:
Double
Volume of a cylinder = πr²h. Because the volume is directly proportional to the height, if the height is doubled (h becomes 2h) while the radius remains constant, the new volume will be πr²(2h) = 2(πr²h), which is exactly double the original volume.
2784
The slant height of a conical tomb is 25 m and its base diameter is 14 m. Find the cost of whitewashing its curved surface at the rate of Rs. 210 per 100 m².
Answer:
Rs. 1155
Radius r = 7 m. Slant height l = 25 m. CSA = πrl = (22/7) × 7 × 25 = 550 m². Cost = Area × (Rate/100) = 550 × (210 / 100) = 5.5 × 210 = Rs. 1155.
2785
A cylindrical tank is 14 m in diameter and 10 m high. How many liters of water can it hold? (Use π = 22/7, 1 m³ = 1000 L)
Answer:
1,540,000 L
Radius r = 7 m. Volume = πr²h = (22/7) × 7² × 10 = 1540 m³. Since 1 m³ = 1000 liters, the capacity is 1540 × 1000 = 1,540,000 liters.
2786
What is the total surface area of a solid hemisphere of radius 'r'?
Answer:
3πr²
A solid hemisphere consists of a curved surface area (half of a sphere's area, so 2πr²) and a flat circular base (πr²). The total surface area is 2πr² + πr² = 3πr².
2787
A rectangular sheet of metal 22 cm long and 12 cm broad is rolled along its length to form a cylinder. Find the volume of the cylinder.
Answer:
462 cm³
When rolled along its length, the length becomes the circumference of the base, so 2πr = 22 => 2 × (22/7) × r = 22 => r = 7/2 = 3.5 cm. The breadth becomes the height, so h = 12 cm. Volume = πr²h = (22/7) × (7/2) × (7/2) × 12 = 22 × 0.5 × 3.5 × 12 = 462 cm³.
2788
Three solid spheres of radii 3 cm, 4 cm, and 5 cm are melted to form a single solid sphere. What is the radius of the new sphere?
Answer:
6 cm
The sum of the volumes of the three spheres equals the volume of the new sphere. (4/3)π(3³ + 4³ + 5³) = (4/3)πR³. This simplifies to 27 + 64 + 125 = R³, which means 216 = R³. Taking the cube root, R = 6 cm.
2789
What is the curved surface area of a frustum of a cone with radii 3 cm and 6 cm, and height 4 cm?
Answer:
45π cm²
First, find the slant height l of the frustum: l = √(h² + (R - r)²) = √(4² + (6 - 3)²) = √(16 + 9) = √25 = 5 cm. CSA of frustum = πl(R + r) = π × 5 × (6 + 3) = 45π cm².
2790
A hemispherical dome has an inner radius of 14 m. Find the cost of painting its inner surface at the rate of Rs. 10 per m². (Use π = 22/7)
Answer:
Rs. 12320
Inner surface area = Curved Surface Area = 2πr² = 2 × (22/7) × 14 × 14 = 2 × 22 × 2 × 14 = 1232 m². Cost = Area × Rate = 1232 × 10 = Rs. 12320.