Question

Let α and β be the roots of the equation x² + (2i – 1) = 0. Then, the value of |α⁸ + β⁸| is equal to ______.

विकल्प

• 50

• 250

• 1250

• 1500

Answer 1

Let α and β be the roots of the equation x² + (2i – 1) = 0. Then, the value of |α⁸ + β⁸| is equal to 50.

Explanation:

Given: α and β be the roots of x² + (2i – 1) = 0

⇒ x² + (2i – 1) = 0

⇒ x² = 1 – 2i

⇒ α² = 1 – 2i and β² = 1 – 2i

⇒ α² = β²

⇒ (α²)⁴ = (β²)⁴

⇒ α⁸ = β⁸

∴ α⁸ + β⁸ = 2α⁸

∴ α⁸ + β⁸ = 2(α²)⁴

⇒ α⁸ + β⁸ = 2(1 – 2i)⁴

⇒ |α⁸ + β⁸| = 2|1 – 2i|⁴

⇒ |α⁸ + β⁸| = `2(sqrt((1)^2 + (-2)^2))^4`

⇒ |α⁸ + β⁸| = `2(sqrt(5))^4`

⇒ |α⁸ + β⁸| = 2(25) = 50