SSC CGL 2025 Tier-I · 17 September 2025, Shift 1 · Quantitative Aptitude · Question 1 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 13 September 2026.

Where this paper and its answers come from

SSC does not publish CGL question papers or a public answer key. From 2025 the candidate response sheet prints every question and option as a picture, so the text here is transcribed from those pictures for this shift and checked against them independently; where the layout carries meaning, the picture is shown as well. Each answer is the key SSC colour-marked on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. The solutions are written by UnlockIAS.

Simplify: √(12 + 6√3)

(a)2 + √3
(b)3 + √3
(c)√3 + 2√3
(d)3 + √9

Answer

Why

Correct — B. Look for x and y with (x + y)² = x² + y² + 2xy = 12 + 6√3.

Surd part: 2xy = 6√3, so xy = 3√3
Rational part: x² + y² = 12
x = 3 and y = √3 fit both: 3 × √3 = 3√3 and 9 + 3 = 12

Check by squaring: (3 + √3)² = 9 + 6√3 + 3 = 12 + 6√3
Square root: 3 + √3 → option (b)

Why the others are wrong

(a)
2 + √3 — (2 + √3)² = 7 + 4√3, not 12 + 6√3. Both the rational part (7) and the surd part (4√3) fall short.
(c)
√3 + 2√3 — √3 + 2√3 = 3√3, and (3√3)² = 27, a whole number with no √3 term. It cannot equal 12 + 6√3.
(d)
3 + √9 — √9 = 3, so this option is just 3 + 3 = 6, and 6² = 36. The radicand 12 + 6√3 is about 22.39, whose root is about 4.73.

Concept

A nested surd √(a + b√c) simplifies when a + b√c is a perfect square (x + y)², with x or y a surd.

Expand (x + y)² = x² + y² + 2xy. The 2xy term carries the surd, so halve the surd part first: half of 6√3 is 3√3, the product xy.

Then pick the pair with that product whose squares add to the rational part. For 3√3 the pair is 3 and √3, since 9 + 3 = 12.

A decimal check agrees: 6√3 ≈ 10.39, so the radicand ≈ 22.39 and its root ≈ 4.73.

The options come to 2 + √3 ≈ 3.73, 3 + √3 ≈ 4.73, 3√3 ≈ 5.20 and 3 + √9 = 6.

Key facts

(x + y)² = x² + y² + 2xy, so the surd part of the radicand is 2xy.
√(12 + 6√3) = 3 + √3, since (3 + √3)² = 12 + 6√3.
√(12 − 6√3) = 3 − √3, with the larger term first so the root stays positive.
√3 ≈ 1.732, so 3 + √3 ≈ 4.732.

Study next

Square roots of surds of the form √(a − b√c)Rationalising denominators such as 1⁄(√7 − √3)Comparing sums of surds such as √7 + √2 and √6 + √3

Common traps

Halving the wrong thing: xy is half the surd part (6√3 ÷ 2 = 3√3), not 6√3 itself.
Treating 3 + √9 as a surd answer. √9 = 3, so that option is the whole number 6.

The same (x + y)² matching settles 19 Sep 2025, 9:00, Quant Q.1, where √(9 + 4√5) = √5 + 2 because (√5 + 2)² = 9 + 4√5. It also opens 21 Sep 2025, 16:00, Quant Q.24, where √(5 + 2√6) = √3 + √2.

Related PYQs

No directly related past PYQ was found.

Topic: Square roots — all 48 SSC CGL questions on it, every solved shift.