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
191
A cube of side 4 cm is cut into small cubes of side 1 cm. What is the ratio of the surface area of the original cube to the sum of the surface areas of the smaller cubes?
Answer:
1:4
Step 1: Original surface area = 6(4)² = 96. Number of small cubes = 4³ / 1³ = 64. Step 2: Surface area of one small cube = 6(1)² = 6. Total SA of 64 cubes = 64 × 6 = 384. Step 3: Ratio = 96 : 384 = 1 : 4.
192
Two solid hemispheres of the same base radius r are joined together along their bases. The curved surface area of this new solid is:
Answer:
4πr²
Step 1: Joining two identical hemispheres along their bases forms a complete sphere. Step 2: The surface area of a sphere is 4πr². Step 3: Therefore, the curved surface area of the new solid is 4πr².
193
The diameter of a sphere is 6 cm. It is melted and drawn into a wire of diameter 2 mm. The length of the wire is:
Answer:
36 m
Step 1: Radius of sphere = 3 cm. Volume = (4/3)π(3)³ = 36π. Step 2: Wire radius = 1 mm = 0.1 cm. Volume = π(0.1)²h = 0.01πh. Step 3: 0.01πh = 36π -> h = 3600 cm = 36 m.
194
If the volume of two cubes is in the ratio 27:64, what is the ratio of their surface areas?
Answer:
9:16
Step 1: Volume ratio = a³ / b³ = 27 / 64. So, ratio of sides a/b = 3/4. Step 2: Surface area ratio = 6a² / 6b² = a² / b². Step 3: (3/4)² = 9/16.
195
A right triangle with sides 3 cm, 4 cm, and 5 cm is rotated about the side of 3 cm to form a cone. The volume of the cone so formed is:
Answer:
16π cm³
Step 1: Since it's rotated about the 3 cm side, the height of the cone h = 3 cm, and the radius r = 4 cm. Step 2: Volume = (1/3)πr²h. Step 3: V = (1/3)π(4)²(3) = 16π cm³.
196
A swimming pool is 24 m long and 15 m wide. If 3600 cubic meters of water is pumped into it, find the rise in water level.
Answer:
10 m
Step 1: Volume = length × breadth × height (rise in level). Step 2: 3600 = 24 × 15 × h. Step 3: 3600 = 360 × h -> h = 10 m.
197
The radii of two cylinders are in the ratio 2:3 and their curved surface areas are in the ratio 5:3. What is the ratio of their volumes?
Answer:
10:9
Step 1: CSA ratio = (r1 h1) / (r2 h2) = 5/3. Since r1/r2 = 2/3, (2/3)(h1/h2) = 5/3 -> h1/h2 = 5/2. Step 2: Volume ratio = (r1² h1) / (r2² h2) = (r1/r2)² × (h1/h2). Step 3: (2/3)² × (5/2) = (4/9) × (5/2) = 20/18 = 10:9.
198
The curved surface area of a right circular cylinder of height 14 cm is 88 cm². Find the diameter of the base.
Answer:
2 cm
Step 1: CSA = 2πrh = 88. Step 2: 2 × (22/7) × r × 14 = 88 -> 88r = 88 -> r = 1 cm. Step 3: Diameter = 2r = 2 cm.
199
A cone and a hemisphere have equal bases and equal volumes. Find the ratio of their heights.
Answer:
2:1
Step 1: Volume of cone = (1/3)πr²h. Volume of hemisphere = (2/3)πr³. Step 2: Equate volumes: (1/3)πr²h = (2/3)πr³. Step 3: h = 2r. The height of a hemisphere is its radius r. So ratio of their heights (h : r) = 2r : r = 2:1.
200
If the height of a cylinder is doubled, by what number must the radius be multiplied so that the volume remains the same?
Answer:
1/√2
Step 1: V1 = πr²h. V2 = π(r_new)²(2h). Step 2: For V1 = V2, πr²h = 2π(r_new)²h. Step 3: r² = 2(r_new)², so r_new = r / √2. The multiplier is 1/√2.