Problem 48 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\alpha, \beta\) be the roots of the equation \(a x^{2}+b x+c=0\). Let \(s_{n}=\alpha^{n}+\beta^{n}\) for \(n \geq 1\). Then, the value of the determinant \(\left|\begin{array}{ccc}3 & 1+s_{1} & 1+s_{2} \\ 1+s_{1} & 1+s_{2} & 1+s_{3} \\ 1+s_{2} & 1+s_{3} & 1+s_{4}\end{array}\right|\) is (A) \(\frac{(a+b+c)\left(b^{2}-4 a c\right)}{a^{4}}\) (B) \(\frac{(a+b+c)^{2}\left(b^{2}-4 a c\right)}{a^{4}}\) (C) \(\frac{(a+b+c)^{2}\left(b^{2}-4 a c\right)}{a^{2}}\) (D) None of these

Step-by-Step Solution

Answer

The value of the determinant is option B: \(\frac{(a+b+c)^{2}(b^{2}-4ac)}{a^{4}}\)."

Step 1: Identifying Polynomial Properties

First, recall that for a quadratic polynomial \( ax^2 + bx + c = 0 \) with roots \( \alpha \) and \( \beta \), we have the relationships: \( \alpha + \beta = -\frac{b}{a} \) and \( \alpha\beta = \frac{c}{a} \).

Step 2: Recurrence Relation for Series

Using the properties of roots, establish the recurrence for \( s_n = \alpha^n + \beta^n \): \( s_n = -\frac{b}{a}s_{n-1} - \frac{c}{a}s_{n-2} \). Start by computing \( s_1 = \alpha + \beta = -\frac{b}{a} \) and \( s_2 = \alpha^2 + \beta^2 = (-\frac{b}{a})^2 - 2\frac{c}{a} \) using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \).

Step 3: Compute Required Powers

Use the recurrence to find \( s_3 \) and \( s_4 \). Following the relation: \( s_3 = -\frac{b}{a}s_2 - \frac{c}{a}s_1 \) and \( s_4 = -\frac{b}{a}s_3 - \frac{c}{a}s_2 \).

Step 4: Setting Up the Determinant

Form the matrix from the problem statement using the computed \( s_1, s_2, s_3, \) and \( s_4 \): \[\begin{pmatrix}3 & 1+s_1 & 1+s_2 \1+s_1 & 1+s_2 & 1+s_3 \1+s_2 & 1+s_3 & 1+s_4\end{pmatrix}.\]

Step 5: Evaluating the Determinant

Calculate the determinant \( D \) using cofactor expansion or row reduction. Simplify the expression by substituting the values obtained for \( s_1, s_2, s_3, \) and \( s_4 \).

Key Concepts

Algebraic Identities

Algebraic identities are simplified equations that can be used to relate the terms of polynomials, aiding in simplifying calculations.
When working with quadratic equations, certain identities are particularly useful.
For example, the relationships for the roots of a quadratic polynomial, such as \( ax^2 + bx + c = 0 \), are established through algebraic identities like:

• The sum of roots, \( \alpha + \beta = -\frac{b}{a} \).
• The product of roots, \( \alpha\beta = \frac{c}{a} \).

These identities are derived from the polynomial equation and hold true for any quadratic equation.
In the context of determining powers of the roots, the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \) is applied to simplify calculations.
By understanding and applying these identities correctly, complex algebraic expressions can be simplified significantly, enabling students to solve related problems more easily.

Recurrence Relations

Recurrence relations are very useful when dealing with sequences.
They represent how each term in a sequence is related to its predecessors.
In the given exercise, the sequence \( s_n = \alpha^n + \beta^n \) is defined by a recurrence relation:

• \( s_n = -\frac{b}{a}s_{n-1} - \frac{c}{a}s_{n-2} \)

This recurrence essentially helps in finding future sequence values based on previous terms. Knowing \( s_1 \) and \( s_2 \) allows us to calculate \( s_3 \), \( s_4 \), and so on.
The key is to understand these relations. They map out how each new term depends on the previous ones, making them powerful tools in sequence analysis.
In mathematics, recurrence relations simplify recursive calculations, which could otherwise involve cumbersome computations.

Quadratic Polynomial Properties

Quadratic polynomials have properties defined by their degrees which are precisely 2.
These properties are crucial because they allow us to derive specific characteristics and behaviors about the polynomial.
For example, the polynomial \( ax^2 + bx + c = 0 \) has inherent properties like:

• Having exactly two roots, which can be real or complex.
• The parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
• Possessing a symmetry axis, \( x = -\frac{b}{2a} \).

Understanding these properties can aid students in determining solutions and interpreting results geometrically.
Furthermore, these properties assist in deriving expressions involving the roots, like summation and multiplication, and are instrumental in many algebraic transformations.

Matrix Theory

Matrix theory involves the study of matrices, which are rectangular arrays of numbers arranged in rows and columns.
Matrices can represent and solve complex systems of equations, making them powerful tools in algebra and various applications.
In the given exercise, a 3x3 matrix is used to calculate a determinant, which helps solving polynomial-related problems:

• The determinant represents a scalar value that provides a lot of information about the matrix, like if it is invertible.
• For a 3x3 matrix, calculating the determinant can be done by using cofactors or row reduction techniques.
• Determinants can be used to find volumes in three-dimensional spaces, which is a fascinating application in geometry.

Understanding the structure and calculation of determinants makes matrix algebra applicable in solving various mathematical and real-world problems.
It also aids in deriving insights about properties of the system described by the matrix.