JEE Main 2024 Mathematics question

For 0 < c < b < a, let (a + b − 2c)x² + (b + c − 2a)x + (c + a − 2b) = 0 and α ≠ 1 be one of its root. Then, among the two statements (I) If α ∈ (−1, 0), then b cannot be the geometric mean of a and c (II) If α ∈ (0, 1), then b may be the geometric mean of a and c

(A)Only (I) is true
(B)Only (II) is true
(C)Neither (I) nor (II) is true
(D)Both (I) and (II) are trueCorrect

Answer: (D)

Solution

The sum of the coefficients is zero, so x = 1 is a root, and the coefficient of x², (a − c) + (b − c), is positive. If α ∈ (−1, 0), then f(0)f(−1) < 0, which gives c + a − 2b < 0, i.e. b > (a + c)/2 (b

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Question ID VK-017436