SSC CGL 2025 Tier-I · 12 September 2025, Shift 2 · 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.

Let x = √3 + √5 and y = √8 + √2. Which is greater?

(a)x=y
(b)x>y
(c)x<y
(d)Cannot be determined

Answer

Why

Correct — C. Both numbers are positive, so compare their squares.

x² = 3 + 5 + 2√(3 × 5) = 8 + 2√15
y² = 8 + 2 + 2√(8 × 2) = 10 + 2 × 4 = 18

Is 8 + 2√15 below 18? Subtract 8: is 2√15 < 10, i.e. √15 < 5?
Square both: 15 < 25, which is true.

So x² < y², hence x < y → option (c)

Why the others are wrong

(a)
x=y — Equality needs x² = 18, but x² = 8 + 2√15 ≈ 15.75. In decimals x ≈ 3.97 and y ≈ 4.24, so the two sums are not equal.
(b)
x>y — This reverses the inequality. x² ≈ 15.75 is below y² = 18, and for positive numbers the larger square belongs to the larger number, so y is greater.
(d)
Cannot be determined — x and y are fixed positive numbers, not variables, so they can always be compared. Squaring settles it: 8 + 2√15 against 18.

Concept

To compare sums of square roots without a calculator, square both sides. For positive numbers squaring keeps the order, so the larger square marks the larger number.

(√a + √b)² = a + b + 2√(ab), so each square becomes a whole number plus one root, and those are easy to compare.

Here y also simplifies directly: √8 = 2√2, so y = 2√2 + √2 = 3√2 = √18.

Decimals work too if you know √2 ≈ 1.414, √3 ≈ 1.732 and √5 ≈ 2.236: x ≈ 3.968 and y ≈ 4.243. Squaring is the safer route when two values are closer than your decimals are accurate.

Key facts

(√a + √b)² = a + b + 2√(ab).
For positive p and q, p < q exactly when p² < q².
√8 + √2 = 2√2 + √2 = 3√2 = √18.
√15 ≈ 3.873, so 8 + 2√15 ≈ 15.75.

Study next

Comparing √a + √b with √c + √d when a + b = c + d (compare the products ab and cd)Simplifying surds such as √8, √18 and √50 to multiples of √2Comparing differences √a − √b by rationalising to (a − b)⁄(√a + √b)

Common traps

Comparing term by term: √8 beats √5 but √3 beats √2, so that look settles nothing.
Choosing 'Cannot be determined' because the terms point different ways, when any two fixed numbers can be compared.

Here the stem names the two sums x and y, and the options are the possible orderings plus 'Cannot be determined'.

Squaring also settles the surd-sum comparisons at 18 Sep 2025, 12:30, Quant Q.3 (√6 + √2 against √5 + √3) and 19 Sep 2025, 09:00, Quant Q.3 (√7 + √2 against √6 + √3).

Related PYQs

No directly related past PYQ was found.

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