CAT Nature of roots MCQ - Practice Questions & Answers

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If $a+b\neq 2c$ and $a,b,c\; \epsilon \; Q$ then the number of rational roots of equation $\left ( a+b-2c \right )x^{2}+\left ( b+c-2a \right )x+\left ( c+a-2b \right )= 0$ equals:

$0$

$1$

$2$

Can't be determined.

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Nature of roots

Study Notes: Nature of Roots

Introduction to Quadratic Equations

A quadratic equation is an equation of the form $\mathrm{ax^2 + bx + c = 0,}$ where a, b, and c are coefficients.
Quadratic equations have a degree of 2, indicating that the highest power of the variable is 2.
The solutions of a quadratic equation are called roots.
Quadratic equations can have zero, one, or two distinct real roots, based on the discriminant.

Discriminant

The discriminant of a quadratic equation is defined as $\mathrm{b^2 - 4ac.}$
The value of the discriminant determines the nature of the roots.
If the discriminant is positive, the quadratic equation has two distinct real roots.
If the discriminant is zero, the quadratic equation has one real root.
If the discriminant is negative, the quadratic equation has two complex conjugate roots.

Type of Roots

Real and Different Roots: If the discriminant is positive, the roots are real and different.
Real and Equal Roots: If the discriminant is zero, the roots are real and equal.
Complex Roots: If the discriminant is negative, the roots are complex conjugates, i.e. in the form of a + bi.

Examples of Nature of Roots

Example 1:

Solve the equation $\mathrm{2x^2 + 5x + 2 = 0}$ and determine the nature of its roots.
Using the quadratic formula, $\mathrm{x = (-b \pm \sqrt(b^2 - 4ac))/(2a),}$
Here, $\mathrm{a = 2, b = 5, and c = 2.}$
Calculating the discriminant: $\mathrm{b^2 - 4ac = 5^2 - 4(2)(2) = 25 - 16 = 9.}$
As the discriminant is positive $\mathrm{(9 > 0),}$ the equation has two distinct real roots.

Example 2:

Solve the equation $\mathrm{3x^2 - 6x + 3 = 0}$ and determine the nature of its roots.
Using the quadratic formula, $\mathrm{ x = (-b \pm \sqrt(b^2 - 4ac))/(2a),}$
Here, a = 3, b = -6, and c = 3.
Calculating the discriminant: $\mathrm{b^2 - 4ac = (-6)^2 - 4(3)(3) = 36 - 36 = 0.}$
As the discriminant is zero (0 = 0), the equation has one real root.

Example 3:

Solve the equation $\mathrm{x^2 + 4 = 0}$ and determine the nature of its roots.
Using the quadratic formula, $\mathrm{x = (-b \pm \sqrt(b^2 - 4ac))/(2a),}$
Here, $\mathrm{a = 1, b = 0, and\: c = 4.}$
Calculating the discriminant: $\mathrm{b^2 - 4ac = 0 - 4(1)(4) = -16.}$
As the discriminant is negative $\mathrm{(-16 < 0),}$ the equation has two complex conjugate roots.

Tips and Tricks for Nature of Roots

Remember that the discriminant is the key to determine the nature of roots.
For quick mental calculation, if the discriminant is a perfect square, the roots are rational.
If the discriminant is a non-perfect square, the roots are irrational.
When solving multiple-choice questions, plugging the solution options into the equation can help determine the nature of roots.
Practice solving previous year's questions to gain familiarity with different scenarios and types of root nature.