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
31
If the radius of a cylinder is decreased by 50% and its height is increased by 50%, what is the net percentage change in its volume?
Answer:
62.5% decrease
Original Volume V = πr²h. New radius = 0.5r. New height = 1.5h. New Volume V' = π(0.5r)²(1.5h) = π(0.25r²)(1.5h) = 0.375πr²h = 0.375V. The new volume is 37.5% of the original. The decrease is 100% - 37.5% = 62.5%.
32
A tank is 5 m long, 4 m wide, and 3 m deep. How much will it cost to plaster its walls and bottom at the rate of Rs. 20 per m²?
Answer:
Rs. 1480
Area to be plastered = Area of 4 walls + Area of bottom = 2h(l + w) + lw. Area = 2 × 3 × (5 + 4) + (5 × 4) = 6 × 9 + 20 = 54 + 20 = 74 m². Cost = 74 × 20 = Rs. 1480.
33
What is the volume of the largest sphere that can be carved out of a cube of edge 6 cm? (Leave answer in terms of π)
Answer:
36π cm³
The diameter of the largest sphere equals the edge of the cube. Diameter = 6 cm, so radius r = 3 cm. Volume = (4/3)πr³ = (4/3)π(3³) = (4/3)π(27) = 36π cm³.
34
A right circular cylinder is circumscribing a hemisphere. What is the ratio of the volume of the cylinder to the volume of the hemisphere?
Answer:
3:2
If a cylinder circumscribes a hemisphere, its base radius is r and its height is equal to the radius of the hemisphere, h = r. Volume of cylinder = πr²h = πr³. Volume of hemisphere = (2/3)πr³. Ratio = πr³ / ((2/3)πr³) = 1 / (2/3) = 3/2, or 3:2.
35
If the areas of the circular bases of a cylinder and a cone are equal, and their volumes are equal, what is the ratio of their heights (cylinder to cone)?
Answer:
1:3
Volume of cylinder = πr²h₁. Volume of cone = (1/3)πr²h₂. Since they are equal: πr²h₁ = (1/3)πr²h₂. Dividing by πr² gives h₁ = (1/3)h₂, so h₁ / h₂ = 1/3. The ratio is 1:3.
36
If the surface area of a sphere is 4π cm², what is its volume?
Answer:
4/3 π cm³
Surface area = 4πr² = 4π, so r² = 1, meaning r = 1 cm. The volume is (4/3)πr³ = (4/3)π(1)³ = 4/3 π cm³.
37
A spherical ball of lead 3 cm in radius is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm respectively. Find the radius of the third ball.
Answer:
2.5 cm
Volume is conserved. (4/3)π(3)³ = (4/3)π(1.5)³ + (4/3)π(2)³ + (4/3)π(r)³. Canceling (4/3)π gives: 27 = 3.375 + 8 + r³. 27 = 11.375 + r³. r³ = 15.625. Since 2.5³ = 15.625, the radius of the third ball is 2.5 cm.
38
If the diameter of the base of a cone is 10 cm and its height is 12 cm, find its total surface area. (Use π = 3.14)
Answer:
282.6 cm²
Radius r = 5 cm, height h = 12 cm. Slant height l = √(5² + 12²) = 13 cm. Total surface area = πr(r + l) = 3.14 × 5 × (5 + 13) = 3.14 × 5 × 18 = 3.14 × 90 = 282.6 cm².
39
A cone and a cylinder have equal bases and equal heights. The volume of the cone is V. What is the volume of the cylinder?
Answer:
3V
Volume of a cone = (1/3) × base area × height. Volume of a cylinder = base area × height. Therefore, the volume of the cylinder is exactly 3 times the volume of the cone. If cone volume is V, cylinder volume is 3V.
40
A solid rectangular block of dimensions 4.4 m, 2.6 m, and 1 m is cast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
Answer:
112 m
Volume of block = 4.4 × 2.6 × 1 = 11.44 m³. Internal radius r = 30 cm = 0.3 m. External radius R = 30 + 5 = 35 cm = 0.35 m. Volume of hollow pipe = πh(R² - r²) = (22/7) × h × (0.35² - 0.30²) = (22/7) × h × (0.1225 - 0.09) = (22/7) × h × 0.0325. Equating volumes: (22/7) × h × 0.0325 = 11.44. h = (11.44 × 7) / (22 × 0.0325) = 80.08 / 0.715 = 112 m.