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
61
If the lengths of the diagonals of a rhombus are 16 cm and 12 cm, what is its area?
Answer:
96 cm²
The area of a rhombus is given by (1/2) × d1 × d2, where d1 and d2 are the lengths of the diagonals. Area = (1/2) × 16 × 12 = 96 cm².
62
The height of a conical tent is 14 m and its floor area is 346.5 m². How much canvas, 1.1 m wide, will be required to make it? (Use π = 22/7)
Answer:
525 m
Base area = πr² = 346.5 => (22/7)r² = 346.5 => r² = 110.25 => r = 10.5 m. Slant height l = √(r² + h²) = √(110.25 + 196) = √306.25 = 17.5 m. Canvas area = Curved Surface Area = πrl = (22/7) × 10.5 × 17.5 = 577.5 m². Length of canvas = Area / width = 577.5 / 1.1 = 525 m.
63
The total surface area of a solid hemisphere is 108π cm². What is its volume?
Answer:
144π cm³
TSA of hemisphere = 3πr² = 108π. So, r² = 36, meaning r = 6 cm. Volume = (2/3)πr³ = (2/3)π(6³) = (2/3)π(216) = 144π cm³.
64
A rectangular block 6 cm by 12 cm by 15 cm is cut into an exact number of equal cubes. Find the least possible number of cubes.
Answer:
40
To find the least number of cubes, we need the largest possible cube. The side length of the largest cube is the HCF of the block's dimensions (6, 12, 15), which is 3 cm. Volume of block = 6 × 12 × 15 = 1080 cm³. Volume of cube = 3³ = 27 cm³. Number of cubes = 1080 / 27 = 40.
65
A right circular cylinder and a sphere have equal volumes and equal radii. What is the ratio of the height of the cylinder to the diameter of the sphere?
Answer:
2:3
Volume of cylinder = Volume of sphere. πr²h = (4/3)πr³, which gives h = 4r/3. The diameter of the sphere is D = 2r. The ratio is h / D = (4r/3) / 2r = 4/6 = 2:3.
66
If the diagonal of a cube is √12 cm, find its volume in cm³.
Answer:
8
The main diagonal of a cube is a√3. Given a√3 = √12. So a = √(12/3) = √4 = 2 cm. The volume is a³ = 2³ = 8 cm³.
67
A cube of edge 5 cm is cut into cubes of edge 1 cm. What is the ratio of the total surface area of the original cube to the sum of the total surface areas of the smaller cubes?
Answer:
1:5
Number of smaller cubes = 5³ / 1³ = 125. Surface area of large cube = 6 × 5² = 150 cm². Surface area of one small cube = 6 × 1² = 6 cm². Total surface area of all small cubes = 125 × 6 = 750 cm². Ratio = 150:750 = 1:5.
68
A hemispherical bowl of internal radius 9 cm contains a liquid. This liquid is to be filled into small cylindrical bottles of diameter 3 cm and height 4 cm. How many bottles are needed?
Answer:
72
Volume of hemisphere = (2/3)π(9)³ = 486π. Radius of bottle = 3/2 = 1.5 cm. Volume of one bottle = π(1.5)² × 4 = 9π. Number of bottles = 486π / 9π = 54.
69
A sector of a circle of radius 12 cm and central angle 120° is rolled up so that its two bounding radii are joined together to form a cone. Find the radius of the base of the cone.
Answer:
4 cm
The arc length of the sector becomes the circumference of the cone's base. Arc length = (θ/360) × 2πr_sector = (120/360) × 2π × 12 = 8π cm. Let r_cone be the radius of the cone. 2πr_cone = 8π, so r_cone = 4 cm.
70
The curved surface area of a cylindrical pillar is 264 m² and its volume is 924 m³. Find the diameter of the pillar.
Answer:
14 m
Volume / CSA = (πr²h) / (2πrh) = r/2. So, r/2 = 924 / 264 = 3.5. Therefore, radius r = 7 m. The diameter is 2r = 14 m.