Mathematics/General Ability MCQs
Topic Notes: Mathematics/General Ability
MCQs and preparation resources for competitive exams, covering important concepts, past papers, and detailed explanations.
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
151
A right circular cylinder and a right circular cone have the same radius and same height. The ratio of their curved surface areas is 8:5. What is the ratio of their radius to their height?
Answer:
3:4
Step 1: Ratio = 2πrh / πr√(r²+h²) = 8/5. So 2h / √(r²+h²) = 8/5 -> h / √(r²+h²) = 4/5. Step 2: Square both sides: h² / (r² + h²) = 16/25. Step 3: 25h² = 16r² + 16h² -> 9h² = 16r² -> r²/h² = 9/16 -> r:h = 3:4.
152
The dimensions of a room are 10 m, 8 m, and 3.3 m. How many persons can be accommodated if each person requires 3 cubic meters of space?
Answer:
88
Step 1: Find the volume of the room: V = 10 × 8 × 3.3 = 264 m³. Step 2: Each person needs 3 m³. Step 3: Number of persons = 264 / 3 = 88.
153
How many spherical bullets can be made out of a lead cylinder 28 cm high and with base radius 6 cm, each bullet being 1.5 cm in diameter?
Answer:
1792
Step 1: Volume of cylinder = πr²h = π(6)²(28) = 1008π. Step 2: Volume of one bullet = (4/3)π(0.75)³ = (4/3)π(27/64) = 9π/16. Step 3: Number = 1008π / (9π/16) = 1008 × 16 / 9 = 112 × 16 = 1792.
154
If the base radius of a cone is doubled and its height is halved, the new volume will be what fraction of the original?
Answer:
Double
Step 1: Original volume V1 = (1/3)πr²h. Step 2: New volume V2 = (1/3)π(2r)²(h/2) = (1/3)π(4r²)(h/2) = 2 × (1/3)πr²h. Step 3: V2 = 2 × V1. The volume becomes double.
155
A solid cylinder has a total surface area of 231 cm². If its curved surface area is two-thirds of the total surface area, find the volume of the cylinder.
Answer:
269.5 cm³
Step 1: CSA = (2/3) × 231 = 154 cm². Base areas = 231 - 154 = 77 cm². Step 2: 2πr² = 77 -> 2 × (22/7) × r² = 77 -> r² = 49/4 -> r = 3.5 cm. Step 3: CSA = 2πrh = 154 -> 2 × (22/7) × 3.5 × h = 154 -> 22h = 154 -> h = 7 cm. Vol = πr²h = (22/7) × (12.25) × 7 = 269.5 cm³.
156
What is the volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm?
Answer:
19.404 cm³
Step 1: The diameter of the cone's base is equal to the cube's edge (4.2 cm), so r = 2.1 cm. The height h = 4.2 cm. Step 2: V = (1/3)πr²h = (1/3) × (22/7) × (2.1)² × 4.2. Step 3: V = 22 × 0.3 × 2.1 × 1.4 = 19.404 cm³.
157
The diameter of a metallic sphere is 6 cm. It is melted and drawn into a wire having diameter of the cross-section as 0.2 cm. Find the length of the wire.
Answer:
36 m
Step 1: Radius of sphere = 3 cm. Volume = (4/3)π(3)³ = 36π cm³. Step 2: Radius of wire = 0.1 cm. Volume = π(0.1)²L = 0.01πL. Step 3: 0.01πL = 36π -> L = 3600 cm = 36 m.
158
A cylindrical tank has a base radius of 14 m and depth of 10 m. Find the cost of painting its inner curved surface at Rs. 5 per m².
Answer:
Rs. 4400
Step 1: Inner CSA = 2πrh = 2 × (22/7) × 14 × 10 = 880 m². Step 2: Cost = Area × Rate. Step 3: Cost = 880 × 5 = Rs. 4400.
159
The total surface area of a solid hemisphere is 462 cm². Find its volume. (Take π = 22/7)
Answer:
718.67 cm³
Step 1: TSA = 3πr² = 462. So r² = (462 × 7) / (3 × 22) = 49. r = 7 cm. Step 2: Volume = (2/3)πr³ = (2/3) × (22/7) × 343. Step 3: Volume = 2156 / 3 = 718.66... cm³.
160
A metallic sphere of radius 3 cm is melted and recast into a cylinder of radius 3 cm. Find the height of the cylinder.
Answer:
4 cm
Step 1: Volume of sphere = (4/3)π(3)³ = 36π. Step 2: Volume of cylinder = π(3)²h = 9πh. Step 3: Equating volumes: 9πh = 36π -> h = 4 cm.