Discriminant and Roots of Quadratic Equation — CAT Previous-Year Questions

Let k be the largest integer such that the equation (x - 1)² + 2kx + 11 = 0 has no real roots. If y is a positive real number, then the least possible value of k/4y + 9y is?

Let a, b and c be non-zero real numbers such that b² < 4ac, and f(x) = ax² + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

Suppose k is any integer such that the equation 2x² + kx + 5 = 0 has no real roots and the equation x² + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is

If r is a constant such that |x² – 4x - 13| = r has exactly three distinct real roots, then the value of r is?

Let m and n be positive integers, If x² + mx + 2n = 0 and x² + 2nx + m = 0 have real roots, then the smallest possible value of m + n is

Let A be a real number. Then the roots of the equation x² - 4x - log₂A = 0 are real and distinct if and only if

If a and b are integers such that 2x² − ax + 2 > 0 and x² − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

If f₁(x) = x² + 11x + n and f₂(x) = x, then the largest positive integer n for which the equation f₁(x) = f₂(x) has two distinct real roots, is :

If both a and b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax² + bx + 1 = 0 having real roots is:

The number of roots $\frac{A^2}{x}+\frac{B^2}{x-1}=1$ is