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CAT Quadratic Equations and Polynomials questions with solutions

Roots, discriminant and polynomial equations. 18 questions from our 2016–2025 CAT pattern papers. Each has the answer and a full solution.

Also practise: Quadratic Equations and Polynomials for JEE Main · Quadratic Equations and Polynomials for JEE Advanced · Quadratic Equations and Polynomials for NDA

CAT 2025

If α and β are the roots of x² − 7x + 12 = 0, what is the value of α² + β²?CAT 2025 Slot 1 · medium
The roots of the equation x² − 14x + k = 0 differ by 4. Find the value of k.CAT 2025 Slot 3 · medium

CAT 2022

The two roots of the quadratic equation x² − 11x + b = 0 are positive integers that differ by 3. Find the value of b.CAT 2022 Slot 1 · medium
The sum of the roots of a quadratic equation is 10 and the product of the roots is 21. Find the sum of the squares of the roots.CAT 2022 Slot 2 · medium

CAT 2021

If α and β are the roots of x² − 6x + k = 0 and α² + β² = 20, what is the value of α³ + β³?CAT 2021 Slot 1 · easy
If α and β are the roots of x² − 6x + 4 = 0, then √α + √β equalsCAT 2021 Slot 2 · medium
If the roots of the equation x² − 7x + k = 0 differ by 3, what is the value of k?CAT 2021 Slot 3 · easy

CAT 2020

Α and β are the roots of x² − 7x + k = 0. If α² + β² = 29, find the value of α³ + β³.CAT 2020 Slot 2 · easy
If α and β are the roots of x² − px + q = 0 such that α + β = 10 and α² + β² = 58, what is the value of q?CAT 2020 Slot 3 · easy

CAT 2019

The roots of the equation x² − 11x + k = 0 differ by 5. What is the value of k?CAT 2019 Slot 1 · easy

CAT 2018

The roots of the equation x² − 15x + k = 0 are in the ratio 2 : 3.CAT 2018 Slot 1 · easy
Real numbers a, b and c satisfy a + b + c = 0 and a² + b² + c² = 6. Find the value of a⁴ + b⁴ + c⁴.CAT 2018 Slot 1 · hard
If the roots of x² − 5x + k = 0 differ by 3, what is the value of k?CAT 2018 Slot 2 · easy

CAT 2017

The roots of the quadratic equation x² − 11x + k = 0 differ by 3.CAT 2017 Slot 1 · easy
When a polynomial p(x) is divided by (x − 1) the remainder is 5, and when it is divided by (x + 2) the remainder is −4. What is the remainder when…CAT 2017 Slot 1 · medium

CAT 2016

The roots of the equation x² − 7x + k = 0 differ by 3. What is the value of k?CAT 2016 Slot 1 · medium
If the roots of the equation x² − px + q = 0 are two consecutive integers, then p² − 4q equals:CAT 2016 Slot 2 · easy
What is the remainder when x¹⁰⁰ is divided by x² − 3x + 2?CAT 2016 Slot 2 · hard