Question

If α, β are the roots of x² - 2x + 2 = 0 then the equation whose roots are α⁴, β⁴
1. x² - 8x + 16 = 0
2. x² + 8x + 16 = 0
3. x² - 8x - 16 = 0
4. x² + 8x - 16 = 0

Answer 1

Correct Answer - Option 2 : x² + 8x + 16 = 0

Calculation:

α, β are the roots of x2 - 2x + 2 = 0 

Then, α + β = 2 , αβ = 2 

⇒ (α + β)² = 2²

⇒ α² + β² + 2αβ = 4

⇒ α2 + β2 = 4 - 4 = 0

⇒ (α2 + β2 )² = 0

⇒ α⁴ + β⁴ + 2α2β2 = 0

⇒ α4 + β4 = - 8 

⇒ α⁴β⁴ = 2⁴ = 16 

The equation whose roots are α4, β⁴

⇒  x² - (α4 + β4 )x + α4β⁴ = 0

⇒ x² + 8x + 16 = 0

∴ The equation whose roots are α4, β⁴ is x2 + 8x + 16 = 0.