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Question

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


A

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B

[4, 5]

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C

[5, 6]

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D

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Solution

The correct option is A


x2 - 2kx + k2 + k - 5 = 0

Roots are less than 5, D ≥ 0

4k2 - 4 (k2 + k - 5) ≥ 0 ............(i)

⇒ k ≤ 5

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

25-10k+k2+k-5>0

k2-9k+20>0 (k-5)(k-4)>0

⇒ k ∈ (-∞, 4) ∪ (5, ∞): -( 2k2 ) < 5 ⇒ k<5 .................(iii)

from (i), (ii) and (iii), k ∈ (-∞, 4)


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