Question

The set of possible value of λ for which x² – (λ² –5λ + 5) x + (2λ² – 3λ – 4) = 0 has roots whose sum and product are both less than 1, is –

• A. $\left( - 1,\frac{5}{2} \right)$
• B. (1, 4)
• C. $\left\lbrack 1,\frac{5}{2} \right\rbrack$
• D. $\left( 1,\frac{5}{2} \right)$

Answer 1

Answer: (D)

$\left( 1,\frac{5}{2} \right)$

Answer 2

Sum of Roots < 1

λ² – 5λ + 5 < 1

⇒ λ² – 5λ + 4 < 0

⇒ (λ – 5) (λ – 1) < 0 1 < λ < 5 …(1)

also product < 1

⇒ 2λ² – 3λ – 4 < 1

⇒ 2λ² – 3λ – 5 < 0

⇒ 2λ² – 5λ + 2λ – 5 < 0

⇒ (2λ – 5) (λ + 1) < 0

⇒ –1 < λ < $\frac{5}{2}$ …(2)

From (1) & (2)

1 < l < $\frac{5}{2}$.