Question

If the roots of quadratic equation x² – 2kx + k² + k – 5 = 0 are less than 5, then k lies in the interval –

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

Answer 1

Answer: (C)

(–∞, 4)

Answer 2

Both roots are less than 5,

(i) D ≥ 0 (ii) af(k) > 0 (iii) $\frac{–b}{2a}$ < k

4k²–4(k²+k–8)≥0 1f(5) > 0 $\frac{2k}{2}$ < 2

k ≤ 5 25–10k+k²+k–5>0 k < 5

k² – 9k + 20>0

k ∈ (–∞, 4) ∪ (5, ∞)

∴ k ∈ (–∞, 4)