Let a, b, c, p, q be real numbers. Let α and β be the two zeros of the quadratic x² + 2px + q = 0, and let α and 1/β be the two zeros of the quadratic x² + 2bx + c = 0, where β² ∈ (-1, 0, 1). Consider the following statements:

Statement-1: (p² - q)(b² - ac) ≥ 0
Statement-2: b ≠ p or c ≠ q

Question

Let a, b, c, p, q be real numbers. Let α and β be the two zeros of the quadratic x² + 2px + q = 0, and let α and 1/β be the two zeros of the quadratic x² + 2bx + c = 0, where β² ∈ (-1, 0, 1). Consider the following statements:

Statement-1: (p² - q)(b² - ac) ≥ 0
Statement-2: b ≠ p or c ≠ qa

Accepted Answer

Statement-1 is true, Statement-2 is false. Statement-1 is true because it reflects a relationship between the coefficients and roots of the quadratics, indicating a certain condition on the discriminants. However, Statement-2 is false as it incorrectly suggests a necessary condition that does not hold true given the established relationships between the variables.

Solution

Related JEE Main Maths questions