CAT Quadratic & Polynomials questions (2024–2025) — actual Quant questions with solutions

The equations 3x² − 5x + p = 0 and 2x² − 2x + q = 0 have one common root. The sum of the other roots of these two equations is

If 9 (x² + 2x – 3) − 4(3 (x² + 2x − 2) ) + 27 = 0, then the product of all possible values of x is _____

If m and n are integers such that (m + 2n)(2m + n) = 27, then the maximum possible value of 2m − 3n is ___

Let a n be the nth term of a decreasing infinite geometric progression. If a₁ + a₂ + a₃ = 52 and a₁a₂ + a₂a₃ + a₃a₁ = 624, then the sum of this geometric progression is

The number of non-negative integer values of k for which the quadratic equation x² – 5x + k = 0 has only integer roots, is

Functions + Quadratic

For any non-zero real x, let f(x) + 2 f(1/x) = 3x. Find the sum of all possible values of x for which f(x) = 3.

The roots \( \alpha, \beta \) of the equation \( 3x^2 + \lambda x - 1 = 0 \) satisfy \[ \frac{1}{\alpha^2} + \frac{1}{\beta^2} = 15. \] The value of \( (\alpha^3 + \beta^3)^2 \) is:

Indices + Quadratic

MathJax Exponent Equation The sum of all real values of \( k \) for which \( \left( \frac{1}{8} \right)^k \times \left( \frac{1}{32768} \right)^{\frac{1}{3}} = \frac{1}{8} \times \left( \frac{1}{32768} \right)^{\frac{1}{k}} \), is:

If the equations \( x^2 + mx + 9 = 0 \), \( x^2 + nx + 17 = 0 \), and \( x^2 + (m + n)x + 35 = 0 \) have a common negative root, then the value of \( (2m + 3n) \) is

Roots α, β of 3x² + λx − 1 = 0 satisfy 1/α² + 1/β² = 15. Then (α³ + β³)² equals:

If the equations x² + mx + 9 = 0, x² + nx + 17 = 0, and x² + (m + n)x + 35 = 0 have a common negative root, then the value of 2m + 3n is