Question

If both the roots of the quadratic equation $x^2$-2kx+$k^2$+k-5=0 are less than 5, then k lies in the interval.

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

Answer 1

The correct option is A

(-∞, 4)

x${}^2$ - 2kx + k${}^2$ + k - 5 = 0

Roots are less than 5, D ≥ 0

4k${}^2$ - 4 (k${}^2$ + k - 5) ≥ 0 ............(i)

⇒ k ≤ 5

f(5) > 0 ..........(ii)

25-10k+k${}^2$+k-5>0

k${}^2$-9k+20>0 $\Rightarrow$ (k-5)(k-4)>0

⇒ k ∈ (-∞, 4) ∪ (5, ∞): -$($ $\frac{2k}{2}$ $)$ < 5 ⇒ k<5 .................(iii)

from (i), (ii) and (iii), k ∈ (-∞, 4)