CAT Quadratic & Polynomials questions (2017–2025) — 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

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

A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is

Let r and c be real numbers. If r and −r are roots of 5x³+cx²−10x+9=0, then c equals

The number of integer solutions of the equation (x²−10)^x2−3x−10=1 is

If 3x+2|y|+y=7 and x+|x|+3y=1, then x+2y is

Suppose one of the roots of the equation a x² − b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c3 then |a| equals

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

The number of pairs of integers(x,y) satisfying x ≥ y ≥ −20 and 2x + 5y = 99 is

The number of integers that satisfy the equality (x² − 5x + 7)^x + 1 = 1 is

In how many ways can a pair of integers (x , a) be chosen such that x² − 2 | x | + | a − 2 | = 0 ?

Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is

The number of distinct real roots of the equation (x + 1/x)² − 3(x + 1/x) + 2 = 0 equals

How many distinct positive integer-valued solutions exist to the equation (x² − 7x + 11)^{x2 − 13x + 42} = 1?

Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if

What is the largest positive integer such that n²+7n+12/n²−n−12 is also positive integer?

The quadratic equation x² + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b² + c?

The number of solutions of the equation |x|(6x² + 1) = 5x² is

The product of the distinct roots of ∣x² − x − 6∣ = x + 2 is

The minimum possible value of the sum of the squares of the roots of the equation x² + (a + 3)x − (a + 5) = 0 is