Question

If α, β, γ, σ are the roots of the equation

x⁴ + 4x³ – 6x² + 7x – 9 = 0, then the value of

(1 + α²) (1 + β²) (1 + γ²) (1 + σ²) is –

• A. 5
• B. 9
• C. 11
• D. 13

Answer 1

Answer: (D)

13

Answer 2

Since α, β, γ and σ are the roots of the given equation.

Therefore

x⁴ + 4x³ – 6x² + 7x – 9 = (x – α) (x – β) (x – γ) (x – σ)

Putting x = i and then x = –i, we get

1 – 4i + 6 + 7i – 9 = (i – α) (i – β) (i – γ) (i – σ)

and 1 + 4i + 6 – 7i – 9 = (–i – α) (–i – β) (–i – γ) (–i – σ)

Multiplying these two, we get

(–2 + 3i) (–2 – 3i) = (1 + α²) (1 + β²) (1 + γ²) (1 + σ²)

⇒ 13 = (1 + α²) (1 + β²) (1 + γ²) (1 + σ²).