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
41
What is the volume of the largest right circular cone that can be cut out of a cube of edge 14 cm? (Use π = 22/7)
Answer:
718.66 cm³
The base of the largest cone will touch the sides of the cube, so its diameter is 14 cm, giving a radius r = 7 cm. Its height will be equal to the edge of the cube, h = 14 cm. Volume = (1/3)πr²h = (1/3) × (22/7) × 7² × 14 = (1/3) × 22 × 7 × 14 = 2156 / 3 ≈ 718.66 cm³.
42
A metallic sphere of radius 6 cm is melted and drawn into a wire of uniform circular cross-section. If the length of the wire is 72 m, find its radius.
Answer:
2 mm
Volume of sphere = (4/3)π(6)³ = 288π cm³. The wire is a cylinder. Let its radius be r cm. Length h = 36 m = 3600 cm. Volume of wire = πr²h = πr²(3600). Equating volumes: 3600πr² = 288π. r² = 288 / 3600 = 8 / 100 = 0.08. Wait, 288/3600 = 8/100 is wrong. 288/36 = 8, so 288/3600 = 8/100. Wait, √0.08 is not an integer. Let me re-calculate: V_sphere = 4/3 * π * 216 = 288π. V_wire = π * r² * 3600. r² = 288/3600 = 144/1800 = 72/900 = 36/450 = 4/50 = 2/25 = 0.08. That gives an irrational radius. Let me adjust the problem to radius 2 mm (0.2 cm). r² = 0.04. Then 3600 * 0.04 = 144. So sphere volume should be 144π, meaning R_sphere = ³√(108) which is not integer. Let me select an option close or change the length. If h = 36m, R=6cm. r² = 288/3600 = 0.08 cm² = 8 mm² -> r = √8 mm. Let me fix the height to 36 m? No, if V=288π. Let h = 72 m = 7200 cm. r² = 288/7200 = 4/100 = 0.04 cm². So r = 0.2 cm = 2 mm. I will assume the question intended length to be 72 m for option B. But since I can't edit the question easily, let me re-evaluate 288 / 3600 = 0.08. r = √0.08 cm = √8 mm ≈ 2.82 mm. I will rewrite the question to length = 72m.
43
A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to the height of the cone is:
Answer:
1:3
Volume of cylinder = πr²h1. Volume of cone = (1/3)πr²h2. Since volumes are equal: πr²h1 = (1/3)πr²h2. This implies h1 = h2 / 3, so h1/h2 = 1/3, making the ratio 1:3.
44
Find the curved surface area of a frustum of a cone with radii 7 cm and 14 cm, and a slant height of 10 cm. (Use π = 22/7)
Answer:
660 cm²
The curved surface area of a frustum of a cone is given by the formula πl(r1 + r2). Substituting the given values: Area = (22/7) × 10 × (14 + 7) = (22/7) × 10 × 21 = 22 × 10 × 3 = 660 cm².
45
If a solid cylinder of radius 5 cm and height 12 cm is melted and cast into small spherical balls of radius 1 cm, how many balls can be formed?
Answer:
225
Volume of cylinder = πr²h = π(5)²(12) = 300π cm³. Volume of one ball = (4/3)πr³ = (4/3)π(1)³ = 4π/3 cm³. Number of balls = (300π) / (4π/3) = (300 × 3) / 4 = 900 / 4 = 225.
46
Water flows through a cylindrical pipe of internal diameter 7 cm at 2 m/s. Calculate the volume of water (in liters) delivered in 1 minute. (Use π = 22/7)
Answer:
462 liters
Radius r = 3.5 cm = 0.035 m. Speed of water = height of the water column formed per second = 2 m. Volume per second = πr²h = (22/7) × (0.035)² × 2 = 22 × 0.005 × 0.035 × 2 = 0.0077 m³. Volume per minute = 0.0077 × 60 = 0.462 m³. In liters: 0.462 × 1000 = 462 liters.
47
The volume of a sphere is 4851 cm³. Find its curved surface area. (Use π = 22/7)
Answer:
1386 cm²
Volume = (4/3)πr³ = 4851. So, (4/3) × (22/7) × r³ = 4851. r³ = (4851 × 21) / 88. Simplifying: r³ = (441 × 21) / 8 = 9261 / 8. r = 21/2 = 10.5 cm. Surface Area = 4πr² = 4 × (22/7) × (21/2) × (21/2) = 22 × 3 × 21 = 1386 cm².
48
What is the ratio of the volume of a sphere to the volume of a cylinder that perfectly encloses it?
Answer:
2:3
If a cylinder perfectly encloses a sphere of radius r, the cylinder's radius is r and its height is 2r. Volume of sphere = (4/3)πr³. Volume of cylinder = πr²(2r) = 2πr³. Ratio = ((4/3)πr³) / (2πr³) = (4/3) / 2 = 2/3, or 2:3.
49
If the base radius of a cone is halved and its height is doubled, what happens to its volume?
Answer:
Halved
Original volume V = (1/3)πr²h. New volume V' = (1/3)π(r/2)²(2h) = (1/3)π(r²/4)(2h) = (1/2) × (1/3)πr²h = V / 2. The volume is halved.
50
The diameter of a garden roller is 1.4 m and its length is 2 m. How much area will it cover in 5 revolutions? (Use π = 22/7)
Answer:
44 m²
Area covered in 1 revolution = CSA of cylinder = πdh = (22/7) × 1.4 × 2 = 22 × 0.2 × 2 = 8.8 m². Area covered in 5 revolutions = 5 × 8.8 = 44 m².