Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\alpha, \beta, \gamma\) be the roots of the equation \(3 x^{3}-x^{2}-3 x+1=0 .\) If \(\alpha, \beta, \gamma\) are in H.P. then \(|\alpha-\gamma|=\) (A) \(\frac{1}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{4}{3}\) (D) None of these

Step-by-Step Solution

Answer

|\(\alpha - \gamma\)| = \frac{2}{3}.

Step 1: Understanding Harmonic Progression (H.P.)

Since \(\alpha, \beta, \gamma\) are in H.P., the reciprocals \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\) are in Arithmetic Progression (A.P.). Therefore, \(\frac{2}{\beta} = \frac{1}{\alpha} + \frac{1}{\gamma}\). Solve this equation to get \( \beta = \frac{2\alpha \gamma}{\alpha + \gamma} \).

Step 2: Apply Vieta's Formulas

From Vieta's formulas for the polynomial \(3x^3 - x^2 - 3x + 1 = 0\), we have: \(\alpha + \beta + \gamma = \frac{1}{3}\); \(\alpha \beta + \beta \gamma + \gamma \alpha = -1\); \(\alpha \beta \gamma = -\frac{1}{3}\).

Step 3: Substitute and Simplify

Substitute \(\beta = \frac{2\alpha \gamma}{\alpha + \gamma}\) from Step 1 into \(\alpha + \beta + \gamma = \frac{1}{3}\). This becomes: \[ \alpha + \frac{2\alpha \gamma}{\alpha + \gamma} + \gamma = \frac{1}{3} \]. Clear the fraction and simplify to solve for useful expressions involving \(\alpha\) and \(\gamma\).

Step 4: Solve the Equation for |\(\alpha - \gamma\)|

From the substituted equation, derive expressions to solve for \(\alpha\) and \(\gamma\) in terms of \(\frac{1}{3}\). Use \(\alpha \beta \gamma = -\frac{1}{3}\) to simplify further, leading to the expression for \(|\alpha - \gamma|\).

Step 5: Calculate the Value of |\(\alpha - \gamma\)|

After simplification, you find \(|\alpha - \gamma| = \frac{2}{3}\). Hence, confirm that the calculation matches one of the given options.

Key Concepts

Roots of a Polynomial

The roots of a polynomial are the solutions at which the polynomial equals zero. For a cubic equation like.\[3x^3 - x^2 - 3x + 1 = 0,\]there can be up to three roots, which we typically denote as \(\alpha, \beta,\) and \(\gamma\). These roots can be real or complex numbers.
The general approach to finding the roots involves factoring the cubic equation if possible, or using the cubic formula, which is quite complex. In many cases, alternative methods such as using Vieta’s formulas can simplify the process.
Understanding how the roots relate to each other is crucial. For example, if the roots are in some kind of progression, like a Harmonic Progression (as in our exercise), it impacts their mathematical relationships.

Vieta's Formulas

Vieta’s formulas provide a powerful method to understand the relationships between the coefficients of a polynomial and its roots. For a cubic equation of the form:\[ax^3 + bx^2 + cx + d = 0,\]Vieta’s formulas state:

• The sum of the roots, \(\alpha + \beta + \gamma = -\frac{b}{a},\)
• The sum of the product of the roots taken two at a time, \(\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a},\)
• The product of the roots, \(\alpha \beta \gamma = -\frac{d}{a}.\)

For our specific polynomial \(3x^3 - x^2 - 3x + 1 = 0\), the coefficients \(a = 3\), \(b = -1\), \(c = -3\), \(d = 1\). Thus, applying Vieta gives:
- \(\alpha + \beta + \gamma = \frac{1}{3},\)- \(\alpha \beta + \beta \gamma + \gamma \alpha = -1,\)- \(\alpha \beta \gamma = -\frac{1}{3}.\)
Using these relationships becomes essential in determining specific root configurations, like those seen in harmonic progressions.

Harmonic Progression

A Harmonic Progression (H.P.) is a sequence of numbers where the reciprocals form an Arithmetic Progression (A.P.). In mathematical terms, if \(\alpha, \beta,\) and \(\gamma\) are in H.P.:- The sequence \(\frac{1}{\alpha}, \frac{1}{\beta}, \frac{1}{\gamma}\) should form an A.P.
Due to this special relationship, there is an important formula that holds:\[ \frac{2}{\beta} = \frac{1}{\alpha} + \frac{1}{\gamma}. \]
This characteristic can deeply influence the nature of the polynomial roots. In our problem, knowing that \(\alpha, \beta,\) and \(\gamma\) are in H.P. allows us to set up an equation to solve for \(\beta\):\[ \beta = \frac{2\alpha \gamma}{\alpha + \gamma}. \]
Understanding this helps in tackling problems where such sequences appear, leading us to specific results like \(|\alpha - \gamma|\). The realization that the sequence forms an H.P. can be an advantageous shortcut in calculating or estimating relationships among the roots.