Question

If α and β are the distinct roots of the equation x² + (3)^1/4x + 3^1/2 = 0, then the value of α⁹⁶(α¹² – 1) + β⁹⁶(β¹² –1) is equal to :

(1) 56 × 3²⁵

(2) 56 × 3²⁴

(3) 52 × 3²⁴

(4) 28 × 3²⁵

Answer 1

Correct option is (3) 52 x 3²⁴

Given x² + (3)^1/4x + 3^1/2 = 0

⇒ x² + √3 = -3^1/4x

Squaring both sides,

⇒ x⁴ + 2√3x² + 3 = √3x²

⇒ x⁴ + √3x² + 3 = 0

⇒ x⁴ + 3 = -√3x²

Now squaring both the sides again,

⇒ x⁸ + 6x⁴ + 9 = 3x⁴

⇒ x⁸ + 3x⁴ + 9 = 0

Put x = α, α⁸ = –9α – 3α⁴

∴ α¹² = –9α⁴ – 3α⁸ = –9α⁴ –3(–9 – 3α⁴) = 27

Similarly β¹² = 27

⇒ α⁹⁶(α¹² – 1) + β⁹⁶(β¹² – 1)

= (27)⁸ × 26 + (27)⁸ × 26

= 52 × (27)⁸

= 52 × 3²⁴

Answer 2

Answer is: (3) 52 × 3²⁴