SSC CGL 2024 Tier-I · 9 September 2024, Shift 1 · Quantitative Aptitude · Question 10 of 100

Text as printed on a candidate's response sheet for this shift; answer per SSC's tentative key as marked on that sheet. Solution last reviewed 6 September 2026.

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SSC does not publish CGL question papers or a public answer key. The questions and options here are reproduced as printed on a candidate's own response sheet for this shift — the Commission's text, unchanged. Each answer is SSC's tentative key as marked on that sheet; SSC allowed challenges to it and never publishes a final key. The solutions are written by UnlockIAS.

The radii of the two cones are in the ratio of 2 : 5 and their volumes are in the ratio of 3 : 5. What is the ratio of their heights?

(a)13 : 11
(b)4 : 11
(c)11 : 15
(d)15 : 4

Answer

Why

Correct — D. A cone's volume is V = (1⁄3)πr²h, so in a ratio the (1⁄3)π cancels and the radius is squared while the height is not.

Set up the ratio: (r₁²h₁) ⁄ (r₂²h₂) = 3⁄5
Put in the radii: (2² × h₁) ⁄ (5² × h₂) = 3⁄5
Square them: (4h₁) ⁄ (25h₂) = 3⁄5

Isolate the heights: h₁⁄h₂ = (3⁄5) × (25⁄4) = 75⁄20
Reduce: h₁ : h₂ = 15 : 4 → option (d)

Why the others are wrong

(a)
13 : 11 — 13 : 11 is barely above 1, but squaring the radii shrinks that term to 4 : 25, which alone multiplies the height ratio by 6.25 — even after the 3⁄5 volume factor the answer must be near 3.75, not 1.18.
(b)
4 : 11 — 4 : 11 keeps the 4 from squaring the radius 2 but pairs it with a denominator no step produces, and it points the wrong way: the narrower cone has to be taller, not shorter, to hold three-fifths the volume.
(c)
11 : 15 — 11 : 15 is less than 1, so it makes the narrow cone the shorter one. The squaring forces the opposite: cross-sections of 4 : 25 against volumes of 3 : 5 leave a height ratio well above 1.

Concept

Two cones are compared, not measured. Because V = (1⁄3)πr²h, the constant (1⁄3)π disappears from any ratio and only r²h survives.

So a radius ratio of 2 : 5 enters the volume ratio as 4 : 25, not as 2 : 5. That single squaring is the whole question.

Once the volume ratio 3 : 5 is written as (4h₁) : (25h₂), the heights come out by simple cross-multiplication.

Nothing here says the cones are similar. If they were, the heights would be forced into the same 2 : 5 ratio as the radii and the given volume ratio would be impossible.

Key facts

Cone volume is V = (1⁄3)πr²h — the radius is squared, the height is not.
For two cones V₁ : V₂ = r₁²h₁ : r₂²h₂, because (1⁄3)π cancels out of the ratio.
Here h₁ : h₂ = (3⁄5) ÷ (4⁄25) = 75 : 20 = 15 : 4.

Study next

The same r²h ratio for cylinders, where the (1⁄3) is absentVolume and surface-area ratios of similar solids (k³ against k²)Volume of a frustum, and the cone cut off by a plane parallel to the base

Common traps

Using the radii as 2 : 5 in the volume ratio instead of squaring them to 4 : 25
Inverting the final division, which turns 15 : 4 into 4 : 15
Assuming the two cones are similar, which contradicts the given volume ratio

SSC gives two of the three ratios — radii, heights, volumes — and asks for the third, changing which one is hidden from shift to shift. The arithmetic is always the same squaring step.

Related PYQs

No directly related past PYQ was found.