SSC CGL 2024 Tier-I · 19 September 2024, Shift 2 · Quantitative Aptitude · Question 20 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.

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 (x + 1⁄x) = 7, x > 0. The positive value of (x − 1⁄x) is:

Answer

Why

Correct — B. The stem is printed as an image and reads: If (x + 1/x) = 7, x > 0, the positive value of (x − 1/x) is. The four options are images too, and option (b) shows 3√5.

Square what you are given:
(x + 1/x)² = x² + 2 + 1/x² = 49
so x² + 1/x² = 49 − 2 = 47

Now build the difference from the same two terms:
(x − 1/x)² = x² − 2 + 1/x² = 47 − 2 = 45
x − 1/x = √45 = √(9 × 5) = 3√5 → option (b)

Why the others are wrong

(a)
Option (a) shows 5√4, which is not a surd at all — it is 5 × 2 = 10, and 10² = 100. The difference squared has to be 45.
(c)
Option (c) shows 5√5, whose square is 125 — larger than (x + 1/x)² = 49. The difference squared is always the smaller of the two, by exactly 4.
(d)
Option (d) shows 4√5, whose square is 80. The two squares differ by 4 and no more, so 49 − 4 = 45 is the only value (x − 1/x)² can take.

Concept

The two symmetric expressions in x and 1/x sit exactly four apart once squared, because the cross term is 2 × x × (1/x) = 2, positive in one and negative in the other:

(x + 1/x)² − (x − 1/x)² = 4.

So the whole question is one subtraction. You never need x itself, and hunting for it is the slow road: solving x + 1/x = 7 gives the awkward x = (7 ± 3√5)/2.

The rider x > 0 matters. Both roots of x + 1/x = 7 are positive, and they give x − 1/x = +3√5 and −3√5, which is why the stem asks for the positive value rather than the value.

Key facts

(x − 1/x)² = (x + 1/x)² − 4, because the cross term 2 × x × (1/x) = 2 is added in one square and subtracted in the other.
√45 = 3√5, because 45 = 9 × 5.
x + 1/x = 7 has roots (7 + 3√5)/2 and (7 − 3√5)/2, both positive.
x² + 1/x² = (x + 1/x)² − 2 = 47 here, the same identity with 2 instead of 4.

Study next

x³ + 1/x³ built from x + 1/x through the cube identityx⁴ + 1/x⁴ built from x² + 1/x² by squaring once moreSimplifying surds such as √45 into the a√b form

Common traps

Subtracting 2 rather than 4 when moving from the sum squared to the difference squared.
Stopping at 45 and forgetting the square root.
Reading 5√4 as a surd when it is the whole number 10.

SSC hands you one symmetric expression and asks for another, so the intended work is an identity and never a solve for x.

Watch which of the pair is wanted. The sum squared, the difference squared and x² + 1/x² are three values two steps apart, and the option list here carries plausible surds for more than one of them.

Related PYQs

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