Quadratic Equations and Inequalities: Parameter MCQ

Solve this quality-checked JEE multiple-choice question, then review the correct answer and explanation.

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ParameterHardQuestion 220014

Question

If both roots of x² − 2kx + k² + k − 5 = 0 are less than 5, then k ∈:

A
(−∞, 4)
Correct
B
(4, ∞)
C
(−∞, 5)
D
(5, ∞)

Correct answer

(−∞, 4)

Explanation

Conditions: D ≥ 0, vertex < 5, f(5) > 0. D: 4k²−4(k²+k−5) = −4k+20 ≥ 0 → k ≤ 5. Vertex x = k < 5. f(5) = 25−10k+k²+k−5 = k²−9k+20 = (k−4)(k−5) > 0 → k < 4 or k > 5. Combining: k < 4.

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