SSC CGL 2024 Tier-I · 18 September 2024, Shift 3 · Quantitative Aptitude · Question 10 of 98

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 10 September 2026.

Where this paper and its answers come from

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.

If 3a + 2b = 27 and 27a³ + 8b³ = 1458, then find 2ab.

(a)72
(b)75
(c)70
(d)77

Answer

Why

Correct — B. Read 27a³ + 8b³ as (3a)³ + (2b)³, because 27 = 3³ and 8 = 2³. The cubes are the cubes of the very expressions the linear equation already gives you.

Put x = 3a and y = 2b. Then x + y = 27 and x³ + y³ = 1458.

Identity: x³ + y³ = (x + y)³ − 3xy(x + y)
1458 = 19683 − 81xy
81xy = 18225
xy = 225

xy is 3a × 2b = 6ab, so ab = 225 ⁄ 6 = 37.5.
2ab = 75 → option (b).

Why the others are wrong

(a)
72 — 72 requires ab = 36, that is xy = 216. The identity pins xy at 225, and 225 ⁄ 6 = 37.5, so 36 is only reached by forcing ab to be a whole number.
(c)
70 — 70 requires xy = 210. The subtraction gives 19683 − 1458 = 18225, and 18225 ⁄ 81 = 225 exactly, leaving no room for 210.
(d)
77 — 77 requires xy = 231. Only xy = 225 satisfies x³ + y³ = 1458 together with x + y = 27, so 231 never arises.

Concept

Two-thirds of the work here is reading the coefficients. 27a³ = (3a)³ and 8b³ = (2b)³, so the cube condition and the linear condition are about the same pair of quantities.

Once the substitution is made, one identity finishes it: x³ + y³ = (x + y)³ − 3xy(x + y). It converts a statement about cubes into a linear equation in the product xy.

The matching subtraction form, x³ − y³ = (x − y)³ + 3xy(x − y), handles the version where the paper prints a difference instead of a sum.

The question asks for 2ab, not ab. The substitution gives you 6ab straight away, so divide by 6 and then double — or note that 2ab = xy ⁄ 3 = 225 ⁄ 3 = 75 in one step.

Key facts

x³ + y³ = (x + y)³ − 3xy(x + y).
27a³ + 8b³ is (3a)³ + (2b)³, since 27 = 3³ and 8 = 2³.
With 3a + 2b = 27 and 27a³ + 8b³ = 1458, the product (3a)(2b) = 225, so ab = 37.5.
The subtraction counterpart is x³ − y³ = (x − y)³ + 3xy(x − y).

Study next

The cube identities a³ + b³ and a³ − b³ in both factored and expanded formSpotting perfect-cube coefficients — 8, 27, 64 and 125 as 2³, 3³, 4³ and 5³Questions that give a + b and ab and ask for a³ + b³ or a² + b²

Common traps

Substituting as though the cubes were a³ and b³, which breaks the pairing with 3a + 2b.
Expanding (3a + 2b)³ by hand and dropping the 3xy(x + y) middle term.
Reaching ab = 37.5 and marking the nearest whole-number option instead of doubling it.

SSC pairs a linear equation with a cube or square condition and asks for the product term, so the identity that contains xy is always the one to reach for.

Identity work is also set at Quant Q.11 of this shift, where a trinomial has to be recognised as a perfect square.

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