Problem 61 - A Complete Resource Book in Mathematics for JEE Main

Question

\(a, b, c\) are three distinct real numbers, which are in G.P. and \(a+b+c=x b\). Then, (A) \(x<-1\) or \(x>3\) (B) \(-1

Step-by-Step Solution

Answer

(A) \(x<-1\) or \(x>3\).

Step 1: Understanding G.P.

Since \(a, b, c\) are in a geometric progression (G.P.), we can express them as \(a = ar^0 = a\), \(b = ar^1 = ar\), and \(c = ar^2\), where \(r\) is the common ratio. Given they are distinct, \(r eq 1\) and \(a eq 0\).

Step 2: Set Up the Sum Equation

According to the problem, \(a + b + c = x b\). Substituting the G.P. terms, we have \(a + ar + ar^2 = x (ar)\). Simplifying, we obtain \(a(1 + r + r^2) = arx\).

Step 3: Simplifying the Equation

Cancel out the \(a\) from both sides (assuming \(a eq 0\)), we get \(1 + r + r^2 = rx\). This results in the quadratic equation \(r^2 + r(1-x) + 1 = 0\).

Step 4: Calculate the Discriminant

For the quadratic \(r^2 + (1-x)r + 1 = 0\) to have real roots, its discriminant must be non-negative. The discriminant is \((1-x)^2 - 4 \cdot 1 \cdot 1 = (1-x)^2 - 4\).

Step 5: Solve the Discriminant Inequality

Set the discriminant \((1-x)^2 - 4 \geq 0\). Solving \((1-x)^2 \geq 4\) gives two cases: \(1-x \geq 2\) or \(1-x \leq -2\). This results in \(x \leq -1\) or \(x \geq 3\).

Step 6: Final Inequality Condition

The inequality \(x \leq -1\) or \(x \geq 3\) implies that the permissible values of \(x\) satisfy either being less than or equal to -1 or greater than or equal to 3.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. It is commonly written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the unknown variable.

The reason such equations are called "quadratic" is because "quad" represents "square", stemming from the squared term \(x^2\). Quadratic equations are used in various mathematical problems and applications, including physics and engineering, whenever the relationship between two variables is parabolic.

To solve a quadratic equation, one can use several methods:

• Factoring: Breaking down the quadratic into a product of two linear expressions, if possible.
• Using the Quadratic Formula: Utilizing the famous formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), which provides solutions for any quadratic equation.
• Completing the Square: Rewriting the equation in a form that allows you to solve for \(x\) by taking the square root of both sides.

Quadratic equations are fundamental in algebra and help to form the basis for many mathematical concepts studied later on.

Discriminant

The discriminant is a critical component of a quadratic equation, represented by the expression \(b^2 - 4ac\). It is part of the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), located under the square root.

The discriminant determines the nature and number of the roots of the quadratic equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, no real roots exist; instead, there are two complex conjugate roots.

Understanding the discriminant helps in predicting the type of solutions before solving the equation. In many problems, analyzing the discriminant provides a quick insight into whether solutions exist within the real number system or not.

Real Roots

Real roots refer to the solutions of an equation that lie within the set of real numbers. For quadratic equations, the concept of real roots is intertwined with the discriminant.

As previously discussed, a quadratic equation can have two, one, or no real roots, depending on the value of its discriminant:

• Two distinct real roots when \(b^2 - 4ac > 0\).
• A single real root when \(b^2 - 4ac = 0\).
• No real roots when \(b^2 - 4ac < 0\), as the solutions involve imaginary numbers.

The real roots are the \(x\)-values where the quadratic touches or crosses the x-axis on a graph. These points are sometimes called the zeros of the function and play a vital role in understanding the parabola's behavior and intersections with the axis. Recognizing whether or not real roots exist is essential in graphing and solving real-world problems.