Problem 36 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\alpha, \beta \neq 0\), and \(f(n)=\alpha^{n}+\beta^{n}\) and \(\left|\begin{array}{ccc}3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\\ 1+f(2) & 1+f(3) & 1+f(4)\end{array}\right|=K(1-\alpha)^{2}(1-\beta)^{2}(\alpha-\beta)^{2}\), then \(K\) is equal to: [2014] (a) 1 (b) \(-1\) (c) \(\alpha \beta\) (d) \(\frac{1}{\alpha \beta}\)

Step-by-Step Solution

Answer

The value of \( K \) is \(-1\).

Step 1: Understand the Given Function

We have the function definition as \( f(n) = \alpha^n + \beta^n \). This represents a sum of the power of two terms, \( \alpha \) and \( \beta \), both of which are non-zero.

Step 2: Matrix Entry Calculation

Each element in the given matrix is calculated by adding 1 to \( f(n) \). Hence, the matrix elements are: \[ \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \ 1+f(1) & 1+f(2) & 1+f(3) \ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \].

Step 3: Calculate f(n) for Required Terms

Calculate \( f(1), f(2), f(3), \) and \( f(4) \):- \( f(1) = \alpha^1 + \beta^1 = \alpha + \beta \)- \( f(2) = \alpha^2 + \beta^2 \)- \( f(3) = \alpha^3 + \beta^3 \)- \( f(4) = \alpha^4 + \beta^4 \)

Step 4: Determinant of the Matrix

The determinant \( D \) of the matrix is given by a special form: \[(1+f(1))\times((1+f(2))(1+f(4))-(1+f(3))^2) - (1+f(1))\times((1+f(2))(1+f(3))-(1+f(3))(1+f(2))) + (1+f(2))\times(1+f(2)-(1+f(1)))\].

Step 5: Substituting f(n) Values

Substitute the calculated values of \( f(n) \) into the determinant equation and expand. Recognize that this determinant should simplify down when expressed as \( K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2 \).

Step 6: Identify Symmetrical Polynomial Forms

Observe if expressions such as \( (1-\alpha), (1-\beta), \) and \( (\alpha-\beta) \) can be factors in the expanded determinant. Recognizing determinants of such forms can help isolate \( K \).

Step 7: Simplification and Matching

After expansion and simplification of the determinant using symmetrical polynomial theory, match it to \( K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2 \). Identify terms to solve for \( K \), often using polynomial identity properties.

Key Concepts

Symmetric Polynomials

Symmetric polynomials are an essential concept in algebra, crucial for understanding equations involving multiple variables. A polynomial is called symmetric if its value remains unchanged under any permutation of its variables. For instance, if you take a polynomial in two variables, \( x \) and \( y \), like \( x^2 + y^2 + 2xy \), and swap \( x \) and \( y \), the polynomial still looks the same.

In the context of this exercise, we see terms such as \( (1-\alpha), (1-\beta), \) and \( (\alpha-\beta) \) within the determinant expression. The reason these expressions are highlighted in the solution process is due to their relation to symmetric polynomials. They represent differences between variables, which are fundamental in building symmetric polynomial expressions.

This symmetry helps simplify and generalize approaches when dealing with higher-degree polynomials and systems, such as those serving as the building blocks of the matrix determinant mentioned in this problem.

Matrix Determinants

Matrix determinants are numbers that can help us understand various properties of matrices, such as whether a matrix is invertible, or how it transforms space. For a 3x3 matrix like the one in our problem, the determinant can be calculated using a specific formula involving a cross multiplication process.

In our case, we have a special formulation derived from the entries in the matrix. It becomes crucial to unfold patterns like these because they can lead us to simplified forms—and eventually link to properties or patterns significant to symmetric polynomials. Understanding these relations ties into our broader mathematical toolkit.

• If the determinant equals zero, the matrix is said to be singular and does not have an inverse.
• The particular form \( D = K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2 \) ensures that the determinant has terms indicating symmetry among \( \alpha \) and \( \beta \).

With the right substitutions and pattern recognition, we can solve for \( K \), which often aligns with symmetric polynomial concepts, intersecting these disciplines efficiently.

Polynomial Functions

Polynomial functions, such as the one defined in the problem with \( f(n) = \alpha^n + \beta^n \), are foundational in both theoretical and applied mathematics. The function \( f(n) \) represents a sum of powers where each term grows exponentially with \( n \).

Understanding \( f(n) \) and its relation to symmetric polynomials allowed us to express the determinant in a simplified form. The polynomials here aren't just basic expressions—they help reveal deeper relationships within the matrix construct.

Analyzing polynomial functions relates closely with symmetric properties:

• These functions become remarkably useful in diagonalizing matrices, directly influencing their determinant values.
• The polynomials can help identify eigenvalues, which are closely tied to determinants.

Overall, the connectivity between polynomial functions and matrix determinants broadens when introducing the notion of symmetry, thus enriching the mathematical structure and offering a more comprehensive understanding of the given problem.