Problem 119 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the roots of \(10 x^{3}-c x^{2}-54 x-27=0\) are in harmonic progression, then the roots are (A) \(\frac{-3}{5}\) (B) \(\frac{-3}{2}\) (C) 3 (D) \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The root is \(\frac{-3}{5}\).

Step 1: Define Harmonic Progression

In harmonic progression (HP), if the roots are \( a \), \( b \), and \( c \), then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in arithmetic progression (AP). This implies \( \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \).

Step 2: Relate Roots to Equation Coefficients

For a cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), the sum of the reciprocals of its roots is given by \( \frac{-b}{a} \), and the sum of the products of its roots taken two at a time is \( \frac{c}{a} \). Set the polynomial: \( 10x^3 - cx^2 - 54x - 27 = 0 \).

Step 3: Calculate the Sum of Roots

The sum of the roots \( a + b + c = \frac{-b}{a} = \frac{c}{10} \).

Step 4: Formulate Harmonic Mean Condition

From the harmonic progression condition, formulate the relationship: \( \frac{1}{a} + \frac{1}{c} = \frac{2}{b} \). Using this, if \( a, b, c \) are in HP, \( 2 = \frac{a+c}{ac}b \). From the equation \( ac = \frac{54}{10} = 5.4 \).

Step 5: Set Up Equations from Coefficients

Using Vieta's formulas: 1. \( a + b + c = \frac{c}{10} \)2. \( ab+bc+ca = -\frac{54}{10} = -5.4 \)3. Relationship from harmonic sequence \( 2bc = ab+ac \).

Step 6: Solve Quadratic from Vieta and HP Condition

By solving these equations, particularly using the relationship \( ab + ac + bc = -5.4 \), along with the HP conditions, set up a quadratic equation for the common root. Use trial with options provided.

Step 7: Confirm with Given Options

By trial and substitution based on the provided options, we try each value to see if it fits the derived conditions from Vieta's and HP conditions. For example, substituting \( -\frac{3}{5} \) into the equation should satisfy all conditions.

Key Concepts

Cubic Polynomial

A cubic polynomial is an equation of the form \( ax^3 + bx^2 + cx + d = 0 \). Because it is of degree 3, it can have up to three real roots. The coefficients \( a \), \( b \), \( c \), and \( d \) determine the characteristics and the specific roots of the polynomial.
One way to solve a cubic polynomial is by finding its roots, which can be real or complex numbers that satisfy the equation when substituted for \( x \). This is commonly achieved by using algebraic techniques, trial and error, or numerical methods for more complex scenarios.
In our example, the polynomial \( 10x^3 - cx^2 - 54x - 27 = 0 \) has the key feature that its roots are in harmonic progression, a condition that imposes additional constraints on what the roots are like and how they relate to each other.

Roots of Equations

Finding the roots of polynomial equations involves determining the values of \( x \) that satisfy the equation \( P(x) = 0 \). For cubic equations, we are interested in up to three roots.
The roots can be calculated directly if the polynomial can be factored, or by using advanced methods like the cubic formula, synthetic division, or numerical approximations for more complex cubic polynomials.
In the context of harmonic progression, as given in our problem, if the roots are \( a, b, c \) and they are in harmonic progression, we must use the relationship of their reciprocals being in arithmetic progression. This adds another layer of complexity to solving for the roots, particularly if the roots cannot be easily factored out by inspection or simple algebraic manipulation.

Vieta's Formulas

Vieta's formulas establish a relationship between the coefficients of a polynomial equation and its roots. For cubic polynomials like \( ax^3 + bx^2 + cx + d=0 \), Vieta's formulas state the following:

• The sum of the roots, \( a + b + c \), is equal to \( -\frac{b}{a} \).
• The sum of the products of the roots taken two at a time, \( ab + ac + bc \), is \( \frac{c}{a} \).
• The product of the roots, \( abc \), is obtained as \( -\frac{d}{a} \).

In our case, Vieta's formulas help to mathematically frame the problem within the constraints imposed by harmonic progression. Specifically, since \( ab + ac + bc = -\frac{54}{10} \), these values are crucial for setting up and solving the system of equations formed by HP conditions and Vieta's relationships, thus allowing us to either confirm our roots or iterate through possible solutions given in the problem statement.