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
81
Find the ratio of the volume of a cube to the volume of the largest sphere that can fit perfectly inside it.
Answer:
6 : π
Let the side of the cube be a. Volume of cube = a³. The largest sphere has a diameter equal to a, so radius r = a/2. Volume of sphere = (4/3)π(a/2)³ = (4/3)π(a³/8) = πa³/6. Ratio = a³ : (πa³/6) = 1 : (π/6) = 6 : π.
82
If the diameter of a sphere is decreased by 25%, by what percent does its curved surface area decrease?
Answer:
43.75%
Surface area is proportional to the square of the diameter. If the new diameter is 75% (0.75) of the original, the new area is (0.75)² = 0.5625 of the original area. The decrease is 1 - 0.5625 = 0.4375, which is 43.75%.
83
Water flows into a tank measuring 200 m by 150 m through a rectangular pipe 1.5 m by 1.2 m at the rate of 20 km/hr. In what time will the water level rise by 2 meters?
Answer:
100 mins
Volume required = 200 × 150 × 2 = 60,000 m³. Flow rate = Area of pipe × speed = (1.5 × 1.2) m² × 20,000 m/hr = 1.8 × 20,000 = 36,000 m³/hr. Time = Volume / Flow rate = 60,000 / 36,000 = 5/3 hours = 100 minutes.
84
A hemispherical bowl has an internal radius of 9 cm. It is completely filled with liquid. The liquid is to be poured into cylindrical bottles, each of radius 1.5 cm and height 4 cm. How many bottles are required?
Answer:
54
Volume of bowl = (2/3)π(9)³ = (2/3)π(729) = 486π cm³. Volume of one bottle = π(1.5)²(4) = π(2.25)(4) = 9π cm³. Number of bottles = 486π / 9π = 54.
85
A metallic sphere of radius 10.5 cm is melted and recast into small right circular cones, each of base radius 3.5 cm and height 3 cm. The number of cones formed is:
Answer:
126
Volume of sphere = (4/3)π(10.5)³. Volume of one cone = (1/3)π(3.5)²(3) = π(3.5)². Number of cones = ((4/3)π(10.5)³) / (π(3.5)²) = (4/3) × (10.5/3.5)³ × 3.5 = (4/3) × 3³ × 3.5 = 4 × 9 × 3.5 = 126.
86
The total surface area of a hemisphere is 27π cm². What is the volume of this hemisphere?
Answer:
18π cm³
Total surface area = 3πr² = 27π, which implies r² = 9, so r = 3 cm. Volume = (2/3)πr³ = (2/3)π(27) = 18π cm³.
87
If the curved surface area of a cylinder is equal to its volume (numerically), then its radius must be:
Answer:
2 units
We are given CSA = Volume. So, 2πrh = πr²h. Dividing both sides by πrh (since radius and height are non-zero), we get 2 = r. Thus, the radius is 2 units.
88
A hollow iron pipe is 21 cm long and its exterior diameter is 8 cm. If the thickness of the pipe is 1 cm and iron weighs 8 g/cm³, then the weight of the pipe is:
Answer:
3.696 kg
External radius R = 4 cm. Internal radius r = 4 - 1 = 3 cm. Volume of iron = πh(R² - r²) = (22/7) × 21 × (4² - 3²) = 66 × (16 - 9) = 66 × 7 = 462 cm³. Weight = 462 × 8 = 3696 g = 3.696 kg.
89
A road roller is a cylinder 2 m long and has a diameter of 1.4 m. It takes 500 complete revolutions to level a playground. Find the area of the playground.
Answer:
4400 m²
Area covered in one revolution = Curved Surface Area = 2πrh = πdh. CSA = (22/7) × 1.4 × 2 = 22 × 0.2 × 2 = 8.8 m². Total area leveled in 500 revolutions = 500 × 8.8 = 4400 m².
90
A sphere is inscribed in a cube of side 6 cm. What is the volume of the sphere? (Use π = 3.14)
Answer:
113.04 cm³
The diameter of the inscribed sphere is equal to the side length of the cube, so diameter = 6 cm, making radius r = 3 cm. Volume = (4/3)πr³ = (4/3) × 3.14 × 27 = 4 × 3.14 × 9 = 113.04 cm³.