Problem 51 - A Complete Resource Book in Mathematics for JEE Main

Question

Given that \(\alpha, \gamma\) are roots of the equation \(A x^{2}-4 x+1=0\) and \(\beta, \delta\) are roots of the equation \(B x^{2}-6 x+1=0\). If \(\alpha, \beta, \gamma\) and \(\delta\) are in H.P., then (A) \(A=5\) (B) \(A=-3\) (C) \(B=8\) (D) \(B=-8\)

Step-by-Step Solution

Answer

The valid value appears to be optionized for illustration since there's no unique solution based on the provided conditions.

Step 1: Understand the Problem Statement

You are given two quadratic equations with roots \(\alpha, \gamma\) and \(\beta, \delta\) respectively. These roots are arranged in a Harmonic Progression (H.P.). You are asked to find the values of \(A\) and \(B\) based on the options provided.

Step 2: Recognize Harmonic Progression Property

In a Harmonic Progression, the reciprocals of the numbers form an Arithmetic Progression (A.P.). This means that \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}, \frac{1}{\delta}\) must be in an A.P. This gives us relationships between these reciprocals.

Step 3: Identify Relationships Using A.P. Property

If four numbers \(a, b, c, d\) are in Arithmetic Progression, then \(b - a = c - b = d - c\). Applying this to reciprocals, we get relations among \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}, \frac{1}{\delta}\).

Key Concepts

Harmonic Progression

A harmonic progression (H.P.) is a sequence of numbers where the reciprocals of the numbers form an arithmetic progression (A.P.). This means if you have numbers like \( x_1, x_2, x_3, \ldots \), they are in harmonic progression if \( \frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \ldots \) are in an arithmetic progression.
Understanding harmonic progressions starts by grasping arithmetic progressions, where the difference between consecutive terms is constant. This relationship, when applied to reciprocals, ensures the sequence aligns harmonically.
Here’s a quick way to analyze harmonic progressions:

• Transform the sequence into its reciprocals.
• Verify if the reciprocals form an A.P.
• Use properties of arithmetic sequences to glean insights about the original sequence.

This concept is crucial in complex algebra problems and can simplify the understanding of difficult equations.

Roots of a Quadratic Equation

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). The solutions to these equations are called roots, commonly denoted as \( \alpha \) and \( \beta \) for one equation, and \( \gamma \) and \( \delta \) for another.
To find the roots, we often use the quadratic formula, \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which computes the roots based on the coefficients \( a, b, \) and \( c \). These roots are central to exploring relationships in polynomial equations and identifying sequences such as harmonic progressions.
The sum of the roots, noted as \( \alpha + \beta = -\frac{b}{a} \), and the product of the roots, \( \alpha \beta = \frac{c}{a} \), are key properties that facilitate the analysis of the equations. These relationships can be pivotal when considering more complex interactions, like those found within harmonic progressions in sequences.

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers with a constant difference between consecutive terms. For example, in the sequence \( 2, 4, 6, 8, \ldots \), the difference between each term is \( 2 \). This consistent difference is known as the 'common difference'.
To establish if a sequence is in arithmetic progression:

• Calculate the difference between consecutive numbers.
• Check if this difference is the same throughout the sequence.

The formula for the \( n \)-th term in an arithmetic sequence is given by \( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.
This concept plays a fundamental role in solving problems involving harmonic progressions because understanding the reciprocal relationship allows us to convert a seemingly complex problem into a simpler one using well-known properties of arithmetic sequences.