Problem 107 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let for \(\mathrm{i}=1,2,3, \mathrm{p}_{\mathrm{i}}(\mathrm{x})\) be a polynomial of degree \(2 \mathrm{in} \mathrm{x}, \mathrm{p}_{\mathrm{i}}^{\prime}(\mathrm{x})\) and \(\mathrm{p}^{\prime \prime} \cdot(\mathrm{x})\) be the first and second order derivatives of \(\mathrm{p}_{\mathrm{i}}(\mathrm{x})\) respectively. Let, \(\mathrm{A}(\mathrm{x})=\left[\begin{array}{lll}\mathrm{p}_{1}(\mathrm{x}) & \mathrm{p}_{1}^{\prime}(\mathrm{x}) & \mathrm{p}_{1}^{\prime \prime}(\mathrm{x}) \\ \mathrm{p}_{2}(\mathrm{x}) & \mathrm{p}_{2}^{\prime}(\mathrm{x}) & \mathrm{p}_{2}^{\prime}(\mathrm{x}) \\\ \mathrm{p}_{3}(\mathrm{x}) & \mathrm{p}_{3}^{\prime}(\mathrm{x}) & \mathrm{p}_{3}^{\prime \prime}(\mathrm{x})\end{array}\right]\) and \(\mathrm{B}(\mathrm{x})=[\mathrm{A}(\mathrm{x})]^{\mathrm{T}} \mathrm{A}(\mathrm{x})\). Then determinant of \(\mathrm{B}(\mathrm{x}):\) (a) is a polynomial of degree 6 in \(x\). (b) is a polynomial of degree 3 in \(x\). (c) is a polynomial of degree 2 in \(x\). (d) does not depend on \(\mathrm{x}\).

Step-by-Step Solution

Answer

(a) is a polynomial of degree 6 in \(x\).

Step 1: Identify the Polynomial Degrees

Each polynomial \(p_i(x)\) is of degree 2. The first derivative \(p_i'(x)\) is therefore of degree 1, and the second derivative \(p_i''(x)\) is of degree 0 since differentiating twice reduces the degree by two.

Step 2: Understand Matrix \(A(x)\) and \(B(x)\) Setup

The matrix \(A(x)\) is a 3x3 matrix consisting of the polynomials, their first derivatives, and their second derivatives. Matrix \(B(x) = [A(x)]^T A(x)\) is calculated by multiplying the transpose of \(A(x)\) by \(A(x)\), resulting in another 3x3 matrix.

Step 3: Assess Degree of Matrix Elements

Some key elements of final matrix \(B(x)\) will result from polynomial multiplication: since \(p_i(x)\) is the highest degree row/column, \((p_i(x))^2\) gives degree 4 terms, cross terms like \(p_i(x)p_i'(x)\) give degree 3 terms, overall degree of 4.

Step 4: Calculating the Determinant's Degree

Calculating determinants involves multiplying various combinations of elements. With a dominant degree of 4 across these involved multiplication combinations, the determinant of \(B(x)\) will have productions mostly degree of 4, which results in an overall degree of 6 for the determinant.

Step 5: Verify If Degree Matches Any Options

Among the options provided, the only one acknowledging a degree of 6 is 'a'. Thus, the determinant of \(B(x)\), having been verified mathematically, is in match with the given choice.

Key Concepts

Matrix Algebra

Matrix algebra involves mathematical operations on matrices, which are rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Understanding these operations is fundamental when dealing with polynomial matrices, like in our exercise.

• Matrix Transposition: This is a basic operation where the matrix rows become columns and vice versa. For a matrix \( A(x) \), its transpose \( [A(x)]^T \) switches its rows and columns.
• Matrix Multiplication: When we multiply matrices, such as \( [A(x)]^T A(x) \) in the exercise, it involves combining row elements from the first matrix with column elements from the second matrix. Each element in the resulting matrix is determined by summing these products.

Matrix algebra is key in transforming polynomial expressions and derivatives into a format that allows further operations, which in this case, ultimately leads to calculating the matrix determinant.

Derivatives

Derivatives represent how a function changes as its input changes. They are central to calculus and are utilized in this exercise to analyze changes in polynomial functions.

• First Derivative \( p_i'(x) \): This represents the slope or rate of change of the polynomial \( p_i(x) \). Since our polynomial is of degree 2, its first derivative is of degree 1.
• Second Derivative \( p_i''(x) \): This describes how the slope changes and is the derivative of the first derivative. For a degree 2 polynomial, the second derivative is constant (degree 0).

Derivatives help in decomposing polynomials for further matrix computations, showing immediate variations in function values and aiding in the construction of matrices like \( A(x) \) in our problem.

Determinants

The determinant of a matrix is a special number that provides important information about the matrix. It can tell us, for instance, about the invertibility of the matrix.

• Matrix Complications: When calculating a determinant, we essentially sum the products of elements along its "main diagonals," taking signs into account based on permutations.
• Degree Impact: The degree of the determinant in our case results from the degrees of polynomial matrix elements. For matrix \( B(x) \), we derive products that include degrees up to 6 after calculating its determinant.

Determinants are crucial in our exercise, as they consolidate the polynomial and derivatives to show the highest degree in the resulting determinant, which aids in selecting the correct answer option.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It's a critical factor in the given exercise.

• Original Polynomial: Each \( p_i(x) \) in our matrix has a maximum degree of 2.
• Derived Expressions: Derivatives lower the polynomial degree. So, \( p_i'(x) \) has a degree of 1 and \( p_i''(x) \) a degree of 0.
• Matrix Products: When matrices are multiplied, like \( [A(x)]^T A(x) \), the degree of resulting polynomial expressions tends to increase, leading to a maximum degree of 6 in our determinant.

Understanding the degree of a polynomial helps in predicting the degree outcome in various mathematical operations, such as determining the influence of higher-degree terms in multiplication and their eventual impact on the determinant's degree.