SSC CGL 2025 Tier-I · 12 September 2025, Shift 3 · Quantitative Aptitude · Question 3 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 11 September 2026.

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Simplify: (1⁄(7 + √2)) + (1⁄(7 − √2)) − (14⁄(49 − 2))

(a)0
(b)1
(c)2
(d)√2

Answer

Why

Correct — A. Add the first two fractions over one denominator.

Numerator: (7 − √2) + (7 + √2) = 14, the √2 terms cancel
Denominator: (7 + √2)(7 − √2) = 7² − (√2)² = 49 − 2 = 47
So the first two terms = 14⁄47

The third term is 14⁄(49 − 2) = 14⁄47
14⁄47 − 14⁄47 = 0 → option (a)

Why the others are wrong

(b)
1 — For 1, the first two fractions would have to total 1 + 14⁄47 = 61⁄47. They total 14⁄47, exactly the term being subtracted.
(c)
2 — For 2, the first pair would need to exceed 2. In decimals it is about 0.119 + 0.179 ≈ 0.298, which is 14⁄47 and nowhere near 2.
(d)
√2 — The √2 cancels in the numerator, (7 − √2) + (7 + √2) = 14, and the denominator 49 − 2 = 47 is rational, so no √2 survives.

Concept

7 + √2 and 7 − √2 are conjugates: the same two terms with the sign of the surd flipped. Their product is a difference of squares, (a + √b)(a − √b) = a² − b, so the surd disappears.

Adding their reciprocals gives 1⁄(a + √b) + 1⁄(a − √b) = 2a⁄(a² − b). With a = 7 and b = 2 that is 14⁄47.

The third term, 14⁄(49 − 2), is the sum of the first two written out, so the expression is built to cancel. Seeing that first saves the working.

Key facts

(a + √b)(a − √b) = a² − b.
1⁄(a + √b) + 1⁄(a − √b) = 2a⁄(a² − b).
(√2)² = 2, so (7 + √2)(7 − √2) = 49 − 2 = 47.

Study next

Rationalising a single denominator such as 1⁄(√5 − √3)Squaring a surd sum: (√a + √b)² = a + b + 2√(ab)Simplifying √(x + 2√y) as √m + √n

Common traps

Writing (7 + √2)(7 − √2) as 49 − √2: the product subtracts the square of √2, which is 2.
Adding the denominators: 1⁄(7 + √2) + 1⁄(7 − √2) is not 2⁄14, because fractions add over a common denominator.

Here the stem hands you a conjugate pair and prints their sum as the third term. A conjugate product also drives 21 Sep 2025, 16:00, Quant Q.24, where (5 + 2√6)(5 − 2√6) = 25 − 24 = 1 settles √(5 + 2√6) ÷ √(5 − 2√6).

Related PYQs

No directly related past PYQ was found.

Topic: Square roots — all 21 SSC CGL questions on it, every solved shift.