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: - Mathematics MCQ AIEEE 2026 | ExamTest.live

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

Solution & Step-by-step Explanation

Let f (x) = x^{2} - 2 k x + k^{2} + k - 5 .Conditions for roots to be less than 5: D \geq 0 : (2 k)^{2} - 4 (k^{2} + k - 5) \geq 0 4 k^{2} - 4 k^{2} - 4 k + 20 \geq 0 ⟹ - 4 k \geq - 20 ⟹ k \leq 5 . - b / (2 a) < 5 : 2 k /2 < 5 ⟹ k < 5 . f (5) > 0 : 5^{2} - 2 k (5) + k^{2} + k - 5 > 0 25 - 10 k + k^{2} + k - 5 > 0 k^{2} - 9 k + 20 > 0 (k - 4) (k - 5) > 0 k \in (- \infty, 4) \cup (5, \infty) .Combining k \leq 5 and k \in (- \infty, 4) \cup (5, \infty), we get: k \in (- \infty, 4) .

Try it yourself before checking the explanation above.

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

(5, 6]

(6, \infty)

(- \infty, 4)

[4, 5]

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