Question 1

Percent increase in cube surface area when side doubles: When each edge of a cube is doubled, by what percentage does its total surface area increase?

Accepted Answer

Introduction / Context:Surface area of a cube scales with the square of its edge. Doubling the edge multiplies surface area by 4, leading to a specific percentage increase. This is a scaling/ratio question. Given Data / Assumptions: Original edge a, SA1 = 6a^2 New edge 2a, SA2 = 6*(2a)^2 = 24a^2 Concept / Approach: Percent increase = ((SA2 − SA1) / SA1) * 100%. Step-by-Step Solution: Increase = 24a^2 − 6a^2 = 18a^2.Percent increase = (18a^2 / 6a^2) * 100% = 3 * 100% = 300%. Verification / Alternative check: Pick a = 1: SA1=6; SA2=24; increase 18 on base 6 ⇒ 300%. Why Other Options Are Wrong: 150%/50%/25%: These correspond to edge, area, or linear misinterpretations. Common Pitfalls: Assuming surface area doubles when edge doubles (it quadruples). Final Answer: 300%

Question 2

Sphere — If the radius of a sphere is doubled, by what percentage does its surface area increase?

Accepted Answer

Introduction / Context:This question checks scaling laws for 3D shapes. For a sphere of radius r, surface area S = 4 * π * r^2. If the radius changes by a factor k, the surface area scales by k^2. Given Data / Assumptions: Original radius = r New radius = 2r Surface area S = 4πr^2 Concept / Approach:Under r -> 2r, S -> 4π(2r)^2 = 4π * 4r^2 = 16πr^2. Compare new to old to get the factor and convert to a percent increase. Step-by-Step Solution:Original S = 4πr^2New S' = 4π(2r)^2 = 16πr^2Increase factor = 16πr^2 / (4πr^2) = 4 ⇒ new area is 4 times the oldPercentage increase = (4 − 1) * 100% = 300% Verification / Alternative check:Scaling rule: areas scale with the square of linear scale (2^2 = 4×). 4× corresponds to +300%. Why Other Options Are Wrong:100% corresponds to a doubling, not quadrupling; 200% means 3× total, not 4×; 50% and 25% are far too small. Common Pitfalls:Confusing area scaling (square of factor) with length scaling (factor itself). Final Answer:300%

Question 3

Cone — If only the height of a right circular cone is doubled while radius is unchanged, by what percentage does its volume increase?

Accepted Answer

Introduction / Context:Volume of a cone V = (1/3) * π * r^2 * h. If r is fixed and h changes, the volume scales linearly with h. Given Data / Assumptions: r is constant h → 2h V ∝ h (with r fixed) Concept / Approach:Double h ⇒ double V. A doubling corresponds to an increase of 100% (since new is 2× original). Step-by-Step Solution:Original V = (1/3)πr^2hNew V' = (1/3)πr^2(2h) = 2 * VPercentage increase = (2 − 1) * 100% = 100% Verification / Alternative check:Pick numbers: r = 1, h = 3 ⇒ V = π; double h to 6 ⇒ V = 2π (exactly twice). Why Other Options Are Wrong:200% or 300% imply 3× or 4× total; 400% is 5×; 50% is only 1.5×, not correct. Common Pitfalls:Forgetting that only height changes; mixing up with r scaling (which affects r^2). Final Answer:100%

Question 4

Sphere — If the radius of a sphere is doubled, by what percentage does its volume increase?

Accepted Answer

Introduction / Context:Volume of a sphere V = (4/3) * π * r^3. If radius scales by factor k, the volume scales by k^3. Given Data / Assumptions: r → 2r V ∝ r^3 Concept / Approach:Doubling r gives V' = (4/3)π(2r)^3 = 8 * (4/3)πr^3 = 8V. An 8× total equals a 700% increase. Step-by-Step Solution:Increase factor = 8Percentage increase = (8 − 1) * 100% = 700% Verification / Alternative check:Scaling: cubic dependence makes volume grow faster than surface area when radius increases. Why Other Options Are Wrong:100% and 200% are too small; 800% implies 9×; 300% implies 4× total, not correct. Common Pitfalls:Using square (area) scaling instead of cube (volume) scaling. Final Answer:700 %

Question 5

Cylinders — Radii in ratio 2:3 and heights in ratio 5:3. Find the ratio of their volumes.

Accepted Answer

Introduction / Context:Volume of a cylinder V = π r^2 h. Ratios multiply across exponents: r contributes squared, h contributes linearly. Given Data / Assumptions: r1:r2 = 2:3 h1:h2 = 5:3 Concept / Approach:Compute V1:V2 = (r1^2 h1):(r2^2 h2) = (2^2 * 5) : (3^2 * 3). Step-by-Step Solution:V1:V2 = (4 * 5):(9 * 3) = 20:27 Verification / Alternative check:Plug sample values (e.g., r1=2, r2=3, h1=5, h2=3) to confirm the computed ratio. Why Other Options Are Wrong:27:20 inverts; 4:9 or 9:4 ignore heights; 5:3 ignores r^2. Common Pitfalls:Forgetting to square the radius in volume formula. Final Answer:20 : 27

Question 6

Cylinders (equal volumes) — Two cylinders have equal volume and heights in ratio 1:2. What is the ratio of their radii?

Accepted Answer

Introduction / Context:Equal volumes with different heights imply different radii. Since V = π r^2 h, the radius must compensate inversely with the square root. Given Data / Assumptions: V1 = V2 h1:h2 = 1:2 Concept / Approach:From π r1^2 h1 = π r2^2 h2 ⇒ r1^2 / r2^2 = h2 / h1 ⇒ r1 / r2 = √(h2/h1) = √2. Step-by-Step Solution:r1:r2 = √2:1 Verification / Alternative check:Test with h1=1, h2=2, r1=√2, r2=1 gives equal volumes. Why Other Options Are Wrong:1:√2 flips the correct ratio; 1:2 or 1:4 treat radius linear with height, not square root. Common Pitfalls:Missing the square on r in V = πr^2h. Final Answer:√2 : 1

Question 7

Spheres — Radii in ratio 1:2. What is the ratio of their surface areas?

Accepted Answer

Introduction / Context:Surface area of a sphere S = 4πr^2, so areas scale with r^2. Given Data / Assumptions: r1:r2 = 1:2 Concept / Approach:S1:S2 = r1^2:r2^2 = 1^2:2^2 = 1:4. Step-by-Step Solution:S1:S2 = 1:4 Verification / Alternative check:Pick radii 1 and 2; compute 4π and 16π. Why Other Options Are Wrong:1:2 uses linear scale; 1:√2 mixes up radius with area ratio; others are unrelated. Common Pitfalls:Forgetting the square in surface area. Final Answer:1 : 4

Question 8

Right circular cone — Base radius 6 cm and slant height 28 cm. Find the curved surface area (CSA).

Accepted Answer

Introduction / Context:Curved surface area (CSA) of a cone is π r l, where r is base radius and l is slant height. Given Data / Assumptions: r = 6 cm l = 28 cm Concept / Approach:Direct substitution into CSA = π r l. Step-by-Step Solution:CSA = π * 6 * 28 = 168π ≈ 528 sq. cm (using π ≈ 22/7) Verification / Alternative check:168 * (22/7) = 168 * 3.142857… = 528 exactly with 22/7. Why Other Options Are Wrong:462 corresponds to π * r^2 or other mix-ups; 268 and 396 are underestimates; 658 is an overestimate. Common Pitfalls:Using height instead of slant height in π r l. Final Answer:528 sq. cm

Question 9

Casting — How many spherical bullets (diameter 2 cm) can be made from a lead cube of edge 22 cm?

Accepted Answer

Introduction / Context:When recasting, volume is conserved. Number of bullets = (volume of cube) / (volume of one bullet). Use exact π = 22/7 for clean integers here. Given Data / Assumptions: Cube edge a = 22 cm ⇒ V_cube = a^3 = 10648 cm^3 Bullet diameter = 2 cm ⇒ radius = 1 cm Sphere volume V_s = (4/3)πr^3 Concept / Approach:Compute V_s with r = 1 using π = 22/7, then divide the cube volume by V_s and take the integer count. Step-by-Step Solution:V_s = (4/3) * (22/7) * 1^3 = 88/21 cm^3N = 10648 / (88/21) = (10648 * 21) / 88 = 121 * 21 = 2541 Verification / Alternative check:88 * 121 = 10648, so cancellation is exact. Why Other Options Are Wrong:2662 and 1347 come from rounding with π = 3.14; 5324 doubles the correct count. Common Pitfalls:Using diameter instead of radius in sphere volume; forgetting exact fraction 22/7. Final Answer:2541

Question 10

Recasting spheres — A lead sphere of radius 10 cm is melted into small spheres of radius 5 mm. How many small spheres are formed?

Accepted Answer

Introduction / Context:Number formed equals ratio of volumes. For spheres, V ∝ r^3; π cancels in the ratio. Given Data / Assumptions: Big radius R = 10 cm Small radius r = 0.5 cm (since 5 mm = 0.5 cm) Concept / Approach:N = (R^3) / (r^3) = (10^3) / (0.5^3). Step-by-Step Solution:N = 1000 / 0.125 = 8000 Verification / Alternative check:Scaling by factor 20 in radius would give 20^3 = 8000 if r had been 0.5 vs 10 (factor 20). Why Other Options Are Wrong:800 and 400 are off by factors of 10 or 20; 125 is 5^3, unrelated. Common Pitfalls:Using diameter instead of radius; forgetting units (mm vs cm). Final Answer:8000

Question 11

Carving a cone from a block — From a 10 × 5 × 2 cm block, a cone of radius 3 cm and height 7 cm is carved. What percentage of wood is wasted?

Accepted Answer

Introduction / Context:Waste percentage = (block volume − cone volume) / block volume * 100%. Use V_block = l * b * h and V_cone = (1/3)πr^2h. Given Data / Assumptions: Block: 10 × 5 × 2 ⇒ V_block = 100 cm^3 Cone: r = 3 cm, h = 7 cm Concept / Approach:Compute cone volume exactly with π ≈ 22/7, then percentage waste. Step-by-Step Solution:V_cone = (1/3) * π * 9 * 7 = 21π ≈ 65.973 cm^3Waste = 100 − 21π ≈ 34.027 cm^3Percentage waste ≈ 34.027% Verification / Alternative check:Using π = 22/7 gives V_cone = 21 * 22/7 = 66 exactly; waste = 34/100 = 34%. Why Other Options Are Wrong:46%, 54%, 66% are too large; 29% underestimates waste. Common Pitfalls:Using diameter instead of radius or forgetting the 1/3 factor in cone volume. Final Answer:34%

Question 12

Cone → cylinder (same radius) — Material of a cone is reshaped into a cylinder of equal radius. If cylinder height is 5 cm, find the original cone’s height.

Accepted Answer

Introduction / Context:Equal volume under reshaping and same radius implies a simple relation between heights using volume formulas of cone and cylinder. Given Data / Assumptions: Cylinder height H_c = 5 cm Same radius r for both shapes V_cone = (1/3)πr^2h, V_cyl = πr^2H_c Concept / Approach:Equate volumes and solve for cone height h. Step-by-Step Solution:(1/3)πr^2h = πr^2H_c ⇒ h = 3H_c = 15 cm Verification / Alternative check:Plug back: cone volume equals cylinder volume if height triples. Why Other Options Are Wrong:10 or 12 cm would give smaller cone volume; 18 or 24 cm would overshoot. Common Pitfalls:Forgetting the 1/3 in the cone volume. Final Answer:15 cm

Question 13

Cylinder → cone (same radius) — A cylinder of radius 2 cm and height 6 cm is melted into a cone of the same radius. Find the cone’s height.

Accepted Answer

Introduction / Context:Volume is conserved. With same radius, the height must adjust to preserve volume when the shape changes from cylinder to cone. Given Data / Assumptions: Cylinder: r = 2 cm, h = 6 cm ⇒ V = πr^2h = 24π Cone: r = 2 cm, height H unknown Concept / Approach:Set (1/3)πr^2H = 24π and solve for H. Step-by-Step Solution:(1/3) * π * 4 * H = 24π ⇒ (4/3)H = 24 ⇒ H = 18 cm Verification / Alternative check:Check: cone volume (1/3)*π*4*18 = 24π equals cylinder volume. Why Other Options Are Wrong:Other values do not equate the volumes. Common Pitfalls:Missing the 1/3 factor for cones. Final Answer:18 cm

Question 14

Cones — Volumes in ratio 1:4; diameters in ratio 4:5. Find the ratio of their heights.

Accepted Answer

Introduction / Context:For cones, V ∝ r^2 h. With diameter ratio 4:5, radius ratio is 2:2.5 = 4:5. Use the volume ratio to solve for height ratio. Given Data / Assumptions: V1:V2 = 1:4 r1:r2 = 4:5 ⇒ r1^2:r2^2 = 16:25 Concept / Approach:(r1^2 h1):(r2^2 h2) = 1:4 ⇒ (16 h1):(25 h2) = 1:4 ⇒ h1/h2 = (1/4) * (25/16) = 25/64. Step-by-Step Solution:h1:h2 = 25:64 Verification / Alternative check:Check by plugging into V ∝ r^2h. Why Other Options Are Wrong:They ignore r^2 or invert ratios. Common Pitfalls:Using diameter directly without squaring radius in volume. Final Answer:25 : 64

Question 15

Cone — Vertical height 24 cm and volume 1232 cm^3. Find the curved surface area (CSA).

Accepted Answer

Introduction / Context:Given cone volume and height, find radius, then slant height and CSA. Use V = (1/3)πr^2h, l = √(r^2 + h^2), CSA = π r l. Given Data / Assumptions: h = 24 cm V = 1232 cm^3 Use π = 22/7 for neat integers Concept / Approach:Compute r from 8πr^2 = 1232 (since (1/3)*24 = 8), then l, then CSA. Step-by-Step Solution:8πr^2 = 1232 ⇒ r^2 = 154/π = 49 (with π = 22/7) ⇒ r = 7 cml = √(r^2 + h^2) = √(49 + 576) = √625 = 25 cmCSA = π r l = (22/7) * 7 * 25 = 550 cm2 Verification / Alternative check:Volume back-check: (1/3)*π*7^2*24 = 1232 cm^3 ✔ Why Other Options Are Wrong:704, 616, 1254, and 154 cm2 result from arithmetic or formula mix-ups. Common Pitfalls:Using height instead of slant height in CSA. Final Answer:550 cm2

Question 16

Rise in water level — A cylinder (diameter 60 cm) is partially filled. A sphere of diameter 60 cm is submerged. By how much does the water rise?

Accepted Answer

Introduction / Context:Water rise height h is found by equating displaced volume (sphere volume) to cylinder's additional water volume: πR^2 h = (4/3)πr^3. Given Data / Assumptions: Cylinder radius R = 30 cm Sphere radius r = 30 cm Same π cancels Concept / Approach:Compute h directly from the equality of volumes. Step-by-Step Solution:πR^2 h = (4/3)πr^3 ⇒ h = (4/3) * r^3 / R^2h = (4/3) * 30^3 / 30^2 = (4/3) * 30 = 40 cm Verification / Alternative check:Units are consistent (cm). Why Other Options Are Wrong:15, 20, 25, 30 cm arise from halving or linear thinking rather than volume conservation. Common Pitfalls:Using diameter in place of radius in formulas. Final Answer:40 cm

Question 17

Cone — Base area = 154 cm2 and height = 14 cm. Find the curved surface area (CSA).

Accepted Answer

Introduction / Context:We are given base area and height. From base area A_b = πr^2, find r, then slant height l = √(r^2 + h^2), and finally CSA = π r l. Given Data / Assumptions: A_b = 154 cm2 = π r^2 h = 14 cm Use π = 22/7 Concept / Approach:Compute r from r^2 = 154 / π, then l, then CSA symbolically for an exact surd answer. Step-by-Step Solution:r^2 = 154 / (22/7) = 49 ⇒ r = 7 cml = √(r^2 + h^2) = √(49 + 196) = √245 = 7√5CSA = π r l = (22/7) * 7 * 7√5 = 154√5 cm2 Verification / Alternative check:Dimensional check: area units cm2. Why Other Options Are Wrong:154√2 cm2 uses wrong sum under root; 11 and 5324 cm2 are invalid magnitudes; 308 cm2 would be π r h or misapplied formula. Common Pitfalls:Using height in place of slant height in CSA; not extracting r correctly from base area. Final Answer:154√5 cm2

Question 18

Hemisphere — Volume is 19404 cm^3. Find its total surface area (TSA).

Accepted Answer

Introduction / Context:For a hemisphere, volume V = (2/3)πr^3 and total surface area (TSA) = 3πr^2 (curved area + base). Given Data / Assumptions: V = 19404 cm^3 Use π = 22/7 for exact integers Concept / Approach:Find r from volume, then compute TSA. Step-by-Step Solution:(2/3)πr^3 = 19404 ⇒ (44/21)r^3 = 19404 ⇒ r^3 = 19404 * 21 / 44 = 441 * 21 = 9261 = 21^3r = 21 cmTSA = 3πr^2 = 3 * (22/7) * 441 = 66 * 63 = 4158 sq. cm Verification / Alternative check:Back-substitute r into volume: (2/3)*(22/7)*9261 = 19404 ✔ Why Other Options Are Wrong:27772 is far too large; 5544 and 1386 mismatch 3πr^2. Common Pitfalls:Using curved area 2πr^2 instead of TSA 3πr^2. Final Answer:4158 sq. cm

Question 19

Recasting spheres — A 3 cm diameter sphere is recast into three spheres of diameters 1.5 cm, 2 cm, and a third unknown. Find the third diameter.

Accepted Answer

Introduction / Context:Volume is conserved when melting and recasting. For spheres, volume depends on the cube of radius: V = (4/3)πr^3. Work exactly to avoid rounding errors. Given Data / Assumptions: Initial diameter = 3 cm ⇒ r0 = 1.5 cm Two new diameters: 1.5 cm (r1 = 0.75 cm) and 2 cm (r2 = 1 cm) Third radius r3 unknown Concept / Approach:Set total initial volume equal to sum of volumes of the three smaller spheres and solve for r3. Step-by-Step Solution:V0 = (4/3)π(1.5)^3 = (4/3)π * 3.375 = 4.5πV1 = (4/3)π(0.75)^3 = (4/3)π * 0.421875 = 0.5625πV2 = (4/3)π(1)^3 = 1.333...πV3 = V0 − (V1 + V2) = 4.5π − 1.895833...π = 2.604166...π = (125/48)π(4/3)π r3^3 = (125/48)π ⇒ r3^3 = 125/64 ⇒ r3 = 5/4 = 1.25 cm ⇒ diameter = 2.50 cm Verification / Alternative check:Exact fraction arithmetic confirms r3^3 = 125/64. Why Other Options Are Wrong:2.66 cm is a rounded overestimate; 2.25 cm is too small; 3 cm and 3.5 cm exceed available volume. Common Pitfalls:Using diameters directly (cube of diameter/2) or rounding π prematurely. Final Answer:2.50 cm

Question 20

Painting walls — Painting the four walls of a room costs Rs. 350. If a new room has each dimension (L, B, H) tripled, what will be the new cost?

Accepted Answer

Introduction / Context:Cost to paint four walls is proportional to their total area: A = perimeter * height = 2(L + B) * H. If all dimensions triple, area scales by 3 (perimeter) * 3 (height) = 9. Given Data / Assumptions: Original cost C = Rs. 350 L → 3L, B → 3B, H → 3H Concept / Approach:Area and cost scale by the same factor for the same paint rate. Step-by-Step Solution:Scale factor = 3 * 3 = 9New cost = 9 * 350 = Rs. 3150 Verification / Alternative check:Perimeter scales ∝ (L + B) by 3; height by 3; product is 9. Why Other Options Are Wrong:1050 is only 3×; 1400 and 2800 are partial scalings; 4200 is 12×, not 9×. Common Pitfalls:Assuming surface area of all six faces; the problem specifies four walls (no ceiling/floor). Final Answer:Rs. 3150

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