Given, x + 1⁄x = 5, then determine the value of 2x⁄(x² − 1)
- (a)(2√21)⁄21
- (b)(2√21)⁄25
- (c)(7√21)⁄25
- (d)(9√21)⁄29
Answer
Why
Correct — A. Divide top and bottom by x to bring in x − 1⁄x.
Divide by x: 2x⁄(x² − 1) = 2⁄(x − 1⁄x)
Identity: (x − 1⁄x)² = (x + 1⁄x)² − 4
Substitute: 5² − 4 = 21
Take the root x greater than 1: x − 1⁄x = √21
Value: 2⁄√21
Rationalise: (2 × √21)⁄(√21 × √21) = (2√21)⁄21 → option (a)
Why the others are wrong
- (b)(2√21)⁄25 — (2√21)⁄25 needs x − 1⁄x = 25⁄√21 ≈ 5.46, since 2 ÷ 0.367 ≈ 5.46. For positive x, x − 1⁄x is smaller than x + 1⁄x = 5, so this is impossible.
- (c)(7√21)⁄25 — (7√21)⁄25 ≈ 1.28 would need x − 1⁄x = 2 ÷ 1.28 ≈ 1.56, whose square is about 2.4. The identity demands (x − 1⁄x)² = 21.
- (d)(9√21)⁄29 — (9√21)⁄29 ≈ 1.42 would need x − 1⁄x ≈ 1.41, whose square is about 2.0, not the 21 the identity gives.
Concept
Given x + 1⁄x, you can reach x − 1⁄x without solving for x. Both squares expand to x² + 1⁄x² plus or minus 2, so they differ by exactly 4:
(x + 1⁄x)² − (x − 1⁄x)² = 4
The step to spot is that 2x⁄(x² − 1) becomes 2⁄(x − 1⁄x) once the numerator and the denominator are divided by x.
x + 1⁄x = 5 has two roots, (5 + √21)⁄2 and (5 − √21)⁄2, which are reciprocals. The smaller root gives x − 1⁄x = −√21 and the value −(2√21)⁄21.
The options print the positive value alone, so the key follows the larger root.
Key facts
- (x − 1⁄x)² = (x + 1⁄x)² − 4.
- x² + 1⁄x² = (x + 1⁄x)² − 2, so x + 1⁄x = 5 gives x² + 1⁄x² = 23.
- 2x⁄(x² − 1) = 2⁄(x − 1⁄x) for any x other than 0, 1 and −1.
Study next
Common traps
- Using (x + 1⁄x)² − 2, the x² + 1⁄x² identity, which gives 23 instead of 21.
- Leaving the answer as 2⁄√21 and hunting for it among the options without rationalising.
The same starting value, x + 1⁄x = 5, is on 13 Sep 2025, 12:30, Quant Q.25, which asks for x² + 1⁄x² (keyed 23).
On 19 Sep 2024, 12:30, Quant Q.20, x + 1⁄x = 7 and the stem asks for the positive value of x − 1⁄x, fixing the sign this item leaves open.
Related PYQs
No directly related past PYQ was found.