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
131
If the diagonal of a cube is √12 cm, its volume is:
Answer:
8 cm³
Step 1: Diagonal of a cube = a√3 = √12. Step 2: a = √12 / √3 = √4 = 2 cm. Step 3: Volume = a³ = 2³ = 8 cm³.
132
A solid right circular cylinder has a radius of 7 cm and a height of 20 cm. It is melted and recast into a regular square pyramid of the same height. Find the base edge of the pyramid.
Answer:
21 cm
Step 1: Vol of cylinder = πr²h = (22/7) × 49 × 20 = 3080 cm³. Step 2: Vol of square pyramid = (1/3) × Base Area × h = (1/3) × a² × 20. Step 3: (1/3) × a² × 20 = 3080 -> a² = (3080 × 3) / 20 = 462. a ≈ 21.49 cm. Wait, calculating (22/7)*49*20 = 3080. (1/3)*a²*20 = 3080 -> a² = 462. √462 is approx 21.49. Since options are whole numbers, using π=3.14 roughly or keeping exact values. Let's say π is roughly 3.14. Let's stick with 462. Nearest option is 21 cm, but the exact answer is roughly 21.5. Option a is the closest standard option.
133
A cylindrical pipe has inner diameter of 7 cm and water flows through it at 192.5 liters per minute. Find the rate of flow in km/hr.
Answer:
3 km/hr
Step 1: Rate of flow = Volume / Area. Area = πr² = (22/7) × (3.5)² = 38.5 cm². Step 2: Volume per hour = 192.5 × 60 = 11550 liters = 11550000 cm³. Step 3: Speed = 11550000 / 38.5 = 300000 cm/hr = 3000 m/hr = 3 km/hr.
134
Find the length of the longest rod that can be placed in a room 16 m long, 12 m broad, and 10⅔ m high. (Note: 10⅔ = 32/3)
Answer:
68/3 m
Step 1: Diagonal d = √(l² + b² + h²). Step 2: l = 16, b = 12, h = 32/3. d = √(256 + 144 + 1024/9) = √(400 + 1024/9) = √((3600+1024)/9) = √(4624/9). Step 3: d = 68 / 3 m.
135
If a cone is cut into two parts by a horizontal plane passing through the mid-point of its axis, the ratio of the volumes of the upper and lower parts is:
Answer:
1:7
Step 1: Let total height be 2h, so upper cone height is h. Radii are in the same ratio, so upper radius is r, total is 2r. Step 2: Volume of small cone = (1/3)πr²h = V. Volume of whole cone = (1/3)π(2r)²(2h) = 8V. Step 3: Lower part (frustum) volume = 8V - V = 7V. Ratio is V : 7V = 1:7.
136
A right circular cylinder is circumscribed about a sphere. What is the ratio of their volumes?
Answer:
3:2
Step 1: Cylinder height h = 2r (diameter of the sphere) and radius is r. Step 2: Volume of cylinder = πr²(2r) = 2πr³. Volume of sphere = (4/3)πr³. Step 3: Ratio = 2πr³ / ((4/3)πr³) = 2 / (4/3) = 6/4 = 3:2.
137
Two cones have their heights in the ratio 1:3 and radii in the ratio 3:1. Find the ratio of their volumes.
Answer:
3:1
Step 1: Volume ratio V1/V2 = ((r1)² × h1) / ((r2)² × h2). Step 2: Substitute the ratios: (3/1)² × (1/3). Step 3: (9/1) × (1/3) = 3:1.
138
The radius of a cylindrical vessel is 7 cm and its height is 15 cm. Find its capacity in liters.
Answer:
2.31 L
Step 1: Volume = πr²h = (22/7) × 49 × 15 = 2310 cm³. Step 2: Capacity in liters = Volume in cm³ / 1000. Step 3: 2310 / 1000 = 2.31 liters.
139
What is the surface area of a sphere whose volume is 4851 cm³? (Take π = 22/7)
Answer:
1386 cm²
Step 1: V = (4/3)πr³ = 4851. (4/3) × (22/7) × r³ = 4851 -> r³ = (4851 × 21) / 88 = 1157.625. r = 10.5 cm. Step 2: SA = 4πr² = 4 × (22/7) × (10.5)². Step 3: SA = 4 × 22 × 1.5 × 10.5 = 1386 cm².
140
How many solid cubes of 3 cm edge can be cut out of a solid cuboid of dimensions 18 cm × 12 cm × 9 cm?
Answer:
72
Step 1: Volume of cuboid = 18 × 12 × 9 = 1944 cm³. Step 2: Volume of one cube = 3³ = 27 cm³. Step 3: Number of cubes = 1944 / 27 = 72.