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CAT Concept of roots, sum of roots and product of roots - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Syllabus

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1 Questions around this concept.

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MEDIUM

Which of the following is quadratic expression for all $a\, \epsilon \, R$ .

$\left ( a^{2}-1 \right )x^{2}+ax+1$

$\left ( a^{2}+1 \right )x^{2}+x+3$

$\left ( a+2 \right )x^{2}+x-3$

$\left ( 2a-1 \right )x^{2}-x+5a$

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Concepts Covered - 1

Concept of roots, sum of roots and product of roots

Introduction

Quadratic equations are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. In this concept, we will understand the concept of zeroes, the sum of zeroes, and the product of zeroes of a quadratic equation.

Understanding Zeroes of a Quadratic Equation

A quadratic equation can have zero, one, or two real roots or solutions.

No real roots: If the quadratic equation does not intersect the x-axis, it has no real roots.
One real root: If the quadratic equation intersects the x-axis at only one point, it has one real root.
Two real roots: If the quadratic equation intersects the x-axis at two distinct points, it has two real roots.
Relationship between Coefficients and Zeroes

For a quadratic equation ax^2 + bx + c = 0, the sum of the zeroes is given by the formula: Sum of Zeroes = -b/a The product of the zeroes is given by the formula: Product of Zeroes = c/a

Examples from Previous Year Exams

Let's solve a few examples from previous year exams to understand the concept better. Example 1: If one root of the quadratic equation x^2 - 5x + k = 0 is 3, find the value of k. Solution: Using the sum of zeroes formula, we know that the sum of zeroes is equal to -b/a. Therefore, if one root is 3, the sum of zeroes is 3 + 3 = 6. We can write the equation as follows: x^2 - 5x + k = (x - 3)(x - p) = 0, where p is the other root. By comparing coefficients, we can derive the equation as 2p - 5 = 0. Solving for p, we get p = 5/2. Now, the product of zeroes is given by the formula c/a. Therefore, k = (3)(5/2) = 15/2.

Tips and Tricks
While solving quadratic equations, always consider the sum and product of zeroes to find relationships between the coefficients.
If the quadratic equation has two identical real roots, the sum of zeroes will be equal to two times the value of the root, and the product of zeroes will be equal to the square of the root.
If the quadratic equation has two distinct real roots, the sum of zeroes will be negative and the product of zeroes will be positive.
If the quadratic equation has no real roots, the sum and product of zeroes will be complex numbers.

By understanding the concept of zeroes, sum of zeroes, and product of zeroes in quadratic equations, you will be able to solve questions related to this topic with ease and accuracy in management entrance exams like CAT, MAT, XAT, SNAP, and others. Practise a variety of questions from previous year exams to strengthen your understanding and application of this concept.