Question

If \(x = \frac{{\sqrt 3 + 1}}{{\sqrt 3 - 1}}\) and \(y = \frac{{\sqrt 3 - 1}}{{\sqrt 3 + 1}}\), then the value of (x³ + y³) is
1. 64
2. 32
3. 76
4. None of these

Answer 1

Correct Answer - Option 4 : None of these

Given:

\(x = \frac{{√ 3 + 1}}{{√ 3 - 1}}\)

\(y = \frac{{√ 3 - 1}}{{√ 3 + 1}}\)

Formula used:

(a + b)² = a² + b² + 2ab

(a - b)2 = a2 + b2 - 2ab

(a + b)(a - b) = a² - b²

(a + b)² + (a - b)² = 4ab

Calculations:

Rationalize the value of x = \(x = \frac{{√ 3 + 1}}{{√ 3 - 1}} × \frac{{√ 3 + 1}}{{√ 3 + 1}}\)

⇒ (3 + 1 + 2√3)/(3 - 1)

⇒ (4 + 2√3)/2

⇒ 2 + √3   

Rationalize the value of  \(y = \frac{{√ 3 - 1}}{{√ 3 + 1}} × \frac{{√ 3 - 1}}{{√ 3 - 1}}\)

⇒ (3 + 1 - 2√3)/(3 - 1)

⇒ (4 - 2√3)/2

⇒ 2 - √3   

(x3 + y3) = (x + y)(x² + y² - xy)

⇒ [(2 + √3) + (2 - √3)][(2 + √3)² + ( 2 - √3)² - (2 - √3 )(2 + √3 )]

⇒ [2 + √3 + 2 - √3][4 + 3 + 4√3 + 4 + 3 - 4√3 - 4 + 3]

⇒ 4 × (13)

⇒ 52