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
161
A cone, a hemisphere and a cylinder have equal bases. If the heights of the cone and the cylinder are equal to their common radius, then the ratio of their volumes is:
Answer:
1:2:3
Step 1: Cone V = (1/3)πr²(r) = (1/3)πr³. Step 2: Hemisphere V = (2/3)πr³. Cylinder V = πr²(r) = πr³. Step 3: Ratio = 1/3 : 2/3 : 1 = 1:2:3.
162
If the radius of a cylinder is doubled and its volume remains the same, its new height will be:
Answer:
One-fourth of the original
Step 1: Original volume V = πr²h. Step 2: New radius = 2r. New volume = π(2r)²h' = 4πr²h'. Step 3: For volume to be the same, 4πr²h' = πr²h -> h' = h / 4. So the height becomes one-fourth.
163
The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 0.2 cm. Find the length of the wire.
Answer:
36 m
Step 1: Volume of sphere = (4/3)π(3)³ = 36π. Step 2: Volume of wire (cylinder) = π(0.1)²h = 0.01πh. Step 3: Equate them: 0.01πh = 36π -> h = 3600 cm = 36 m.
164
Find the total surface area of a half cylinder of radius 7 cm and height 10 cm. (Take π = 22/7)
Answer:
514 cm²
Step 1: A half cylinder has a curved surface, a rectangular flat surface, and two semicircular ends. Step 2: Curved area = (1/2)2πrh = πrh = 220. Rectangular base = 2r × h = 14 × 10 = 140. Two semicircles = πr² = 154. Step 3: TSA = 220 + 140 + 154 = 514 cm².
165
A solid piece of iron in the form of a cuboid of dimensions 49 cm × 33 cm × 24 cm is moulded to form a solid sphere. The radius of the sphere is:
Answer:
21 cm
Step 1: Volume of cuboid = 49 × 33 × 24 = 38808 cm³. Step 2: Volume of sphere = (4/3)πr³ = 38808. Step 3: (4/3) × (22/7) × r³ = 38808 -> r³ = 9261 -> r = 21 cm.
166
If the surface area of a sphere is reduced by 36%, its volume is reduced by:
Answer:
48.8%
Step 1: Original SA is 100%. New SA is 64%. Ratio of new SA to old = 64/100 = (8/10)². Step 2: This means the new radius is 8/10 of the original. Step 3: New volume = (8/10)³ = 512/1000 = 51.2% of original. So it is reduced by 100 - 51.2 = 48.8%.
167
A cylindrical rod of iron whose height is 8 times its radius is melted and cast into spherical balls of the same radius. The number of balls produced is:
Answer:
6
Step 1: Volume of cylinder = πr²h = πr²(8r) = 8πr³. Step 2: Volume of one spherical ball = (4/3)πr³. Step 3: Number of balls = (8πr³) / ((4/3)πr³) = 8 × (3/4) = 6.
168
If the areas of three adjacent faces of a cuboid are x, y, and z respectively, then the volume of the cuboid is:
Answer:
√(xyz)
Step 1: Let the dimensions be l, b, h. Given lb = x, bh = y, hl = z. Step 2: Multiply them: (lb)(bh)(hl) = xyz -> l²b²h² = xyz. Step 3: Since Volume V = lbh, V² = xyz. Thus, V = √(xyz).
169
A hemispherical tank of radius 1.75 m is full of water. It is connected with a pipe which empties it at the rate of 7 liters per second. How much time will it take to empty the tank completely?
Answer:
26.7 minutes
Step 1: Volume = (2/3)πr³ = (2/3) × (22/7) × (1.75)³ = 11.229 m³ = 11229 liters. Step 2: Time = Volume / Rate = 11229 / 7 seconds = 1604.14 seconds. Step 3: In minutes: 1604.14 / 60 ≈ 26.7 minutes.
170
The volume of a hemisphere is 19404 cm³. Find its total surface area. (Take π = 22/7)
Answer:
4158 cm²
Step 1: (2/3)πr³ = 19404 -> (2/3) × (22/7) × r³ = 19404 -> r³ = (19404 × 21) / 44 = 9261 -> r = 21 cm. Step 2: TSA of hemisphere = 3πr². Step 3: TSA = 3 × (22/7) × 441 = 66 × 63 = 4158 cm².