SSC CGL 2024 Tier-I · 19 September 2024, Shift 3 · Quantitative Aptitude · Question 25 of 99

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.

In a rectangular field, the sum of the length and the breadth is 9 cm and area is 18 cm². If the length is 'a' and breadth is 'b', then find the value of (a² − ab + b²).

(a)18
(b)36
(c)24
(d)27

Answer

Why

Correct — D. The stem gives the sum of the sides and their product, which is all an identity needs.

a + b = 9 and ab = 18 (the area)

(a + b)² = a² + 2ab + b²
81 = a² + b² + 2(18)
a² + b² = 81 − 36 = 45

a² − ab + b² = 45 − 18 = 27 → option (d)

Check with the actual sides, 6 and 3: 36 − 18 + 9 = 27.

Why the others are wrong

(a)
18 — 18 is the area, ab, lifted straight from the stem. It is the quantity you subtract from 45, not the value asked for.
(b)
36 — 36 is 2ab, the middle term of (a + b)². It is what you take off 81 to reach a² + b² = 45, which is still one step short.
(c)
24 — 24 fails the direct check: the sides are 6 and 3, and 6² − 6(3) + 3² = 36 − 18 + 9 = 27.

Concept

For a rectangle, the sum of the length and the breadth and the area are precisely the sum and the product of the two sides. Every symmetric expression in a and b can then be built without ever solving for a and b.

The bridge is one identity: a² + b² = (a + b)² − 2ab = 81 − 36 = 45.

The target a² − ab + b² is just a² + b² with one more ab removed, so 45 − 18 = 27. The sides here do come out whole, 6 and 3, but the identity route works when they do not.

Read 9 cm carefully — it is the sum of the length and the breadth, which is half the perimeter. Treating it as the perimeter would give a + b = 4.5 with ab = 18, and no real rectangle has those measurements.

Key facts

a² + b² = (a + b)² − 2ab converts a sum and a product into a sum of squares.
Here (a + b)² = 81 and 2ab = 36, so a² + b² = 45.
a² − ab + b² = (a² + b²) − ab = 45 − 18 = 27.
The sides are the roots of t² − 9t + 18 = 0, namely 6 and 3.

Study next

Building a² + b², a³ + b³ and a² − ab + b² out of a + b and abThe factorisation a³ + b³ = (a + b)(a² − ab + b²), where this exact expression reappearsForming t² − (a + b)t + ab = 0 to recover two sides from their sum and product

Common traps

Treating the 9 cm as the perimeter rather than as the sum of the two sides.
Getting the sign wrong and computing a² + ab + b² = 63.
Solving the quadratic for the sides first, which costs time the identity saves.

Identity-manipulation items of this family are set at 9 Sep 2024, 16:00, Quant Q.8, which gives a + b = 2c and asks which expression follows.

The plain mensuration version, a rectangle's area from its length and breadth, is at 23 Sep 2024, 16:00, Quant Q.8.

Related PYQs

No directly related past PYQ was found.