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Question:

Let α and β be the roots of the equation x² + x + 1 = 0. Then for y ≠ 0 in R, |y + αβ αy + β 1 β1y + α| is equal to?

y(y² - 3)

y³ - 1;

y³

y(y² - 1;)

Solution:

Correct option is A. y³
Roots of the equation x² + x + 1 = 0 are α = ω and β = ω², where ω, ω² are complex cube roots of unity
∴ Δ = |y + ωω² ωy + ω² 1 ω² 1y + ω|
Expanding along R1, we get
→ Δ = y[1(y² + ωy + ω²y + ω³ - 1) - 1(ωy + ω² - ω²) + 1(ω - ω²y)]
Δ = y(y² + ωy + ω²y + 1 - 1 - ωy + ω² + ω - ω²y)
Δ = y(y²)
→ D = y³.