Question

If both roots of the quadratic equation

x² – 2kx + (k² + k – 5) = 0 are less than 5 then k lies in interval –

• A. (5, 6)
• B. (6, ∞)
• C. (– ∞, 4)
• D. [4, 5]

Answer 1

Answer: (C)

(– ∞, 4)

Answer 2

D ≥ 0, $\frac{–b}{2a}$ < 5, f(5) > 0 ; D ≥ 0 ⇒ 4[k² – (k² + k – 5)] ≥ 0

k – 5 ≤ 0 ⇒ k ≤ 5 .......(i)

$–\frac{b}{2a}$ < 5 ⇒ $\frac{(–2k)}{2}$ < 5 ⇒ k < 5 .....(ii)

f(5) > 0 ⇒ 25 – 10k + k² + k – 5 > 0

k² – 9k + 20 > 0 ; (k – 5) (k – 4) > 0

k ∈ (– ∞, 4) ∪ (5, ∞) ...(iii)

⇒ k ∈ (– ∞ , 4)