Question

If (λ² + λ – 2) x² + (λ + 2) x< 1 for all x∈ R then λ belongs to the interval –

• A. (–2, 1)
• B. $\left( - 2,\frac{2}{5} \right)$
• C. $\left( \frac{2}{5},1 \right)$
• D. None of these

Answer 1

Answer: (B)

$\left( - 2,\frac{2}{5} \right)$

Answer 2

(λ₂ + λ – 2) x² + (λ + 2) x – 1 < 0 for all x ∈ R if D < 0 and

a < 0, i.e., (λ + 2)² + 4 (λ² + λ – 2) < 0 and λ² + λ – 2 < 0.

5λ² + 8λ – 4 < 0 and λ² + λ – 2 < 0

⇒ (5λ – 2) (λ + 2) < 0 and (λ + 2) (λ – 1) < 0

–2 < λ < $\frac{2}{5}$ and –2 < λ < 1

i.e., –2 < λ < $\frac{2}{5}$ or λ ∈ $\left( - 2,\frac{2}{5} \right)$

Hence (2) is correct answer.d.