If \(\rm\left(x−\frac{1}{x}\right)\) = 2√2, and x > 1, what is the value of \(\rm\left(x^{6}−\frac{1}{x^{6}}\right)\)?

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 20 Jul 2023 Shift 3)

Answer 1

Given:

x - 1/x = 2√2

Concept Used:

a² - b² = (a + b)(a - b)

a⁶ - b⁶ = (a² - b²)³ + 3(a² - b²)

x - 1/x = k then, x + 1/x = √(k² + 4)

Calculation:

As per the question,

x - 1/x = 2√2

x + 1/x = √{(2√2)² + 4

(x + 1/x) = √12 = 2√3

Again,

(x² - 1/x² ) = (x - 1/x)(x + 1/x)

⇒ 2√2 × 2√3

⇒ 4√6

Then,

x⁶ - 1/x6 = (x2 - 1/x2)³ +3 (x2 - 1/x2 )

⇒ (4√6)³ + 12√6

⇒ 384√ 6 + 12√6 = 396√6

∴ The correct answer is Option 3.