Problem 43 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f_{j}=\sum_{i=0}^{2} a_{i j} x^{i}, j=1,2,3\) and if \(f_{j}^{\prime} f_{j}^{\prime \prime}\) denote \(\frac{d f_{j}}{d x}, \frac{d^{2} f_{j}}{d x^{2}}\) respectively, then \(g(x)=\left|\begin{array}{lll}f_{1} & f_{2} & f_{3} \\\ f_{1}^{\prime} & f_{2}^{\prime} & f_{3}^{\prime} \\ f_{1}^{\prime \prime} & f_{2}^{\prime \prime} & f_{3}^{\prime \prime}\end{array}\right|\) is (A) a cubic in \(x\) (B) a quadratic in \(x\) (C) linear in \(x\) (D) a constant

Step-by-Step Solution

Answer

(A) a cubic in \(x\).

Step 1: Understand the Given Functions

The function \( f_j = \sum_{i=0}^{2} a_{ij} x^{i} \) for \( j = 1, 2, 3 \) means each \( f_j \) is a quadratic polynomial in \( x \). This implies \( f_j = a_{0j} + a_{1j}x + a_{2j}x^2 \).

Step 2: Differentiate Each Function

Calculate the first and second derivatives of \( f_j \):- First derivative: \( f_j' = \frac{d}{dx}(a_{0j} + a_{1j}x + a_{2j}x^2) = a_{1j} + 2a_{2j} x \)- Second derivative: \( f_j'' = \frac{d^2}{dx^2}(a_{0j} + a_{1j}x + a_{2j}x^2) = 2a_{2j} \).

Step 3: Construct the Matrix

Construct the matrix for \( g(x) \):\[\begin{bmatrix} f_1 & f_2 & f_3 \ f_1' & f_2' & f_3' \ f_1'' & f_2'' & f_3''\end{bmatrix}= \begin{bmatrix} a_{01} + a_{11} x + a_{21} x^2 & a_{02} + a_{12} x + a_{22} x^2 & a_{03} + a_{13} x + a_{23} x^2 \ a_{11} + 2a_{21} x & a_{12} + 2a_{22} x & a_{13} + 2a_{23} x \ 2a_{21} & 2a_{22} & 2a_{23}\end{bmatrix}\]

Step 4: Evaluate the Determinant

The function \( g(x) \) is defined as the determinant of the constructed matrix. We have 3 rows and 3 columns, with each involving powers of \( x \) up to squared terms. Therefore, expanding this determinant will involve polynomial multiplication resulting in an overall degree of at most 3.

Step 5: Conclusion on the Nature of g(x)

Given that the highest terms in the polynomial come from the products of the maximum degrees in each row, \( g(x) \) will be a cubic polynomial due to terms involving \( x^3 \) appearing in the determinant calculation.

Key Concepts

Quadratic Polynomials

A quadratic polynomial is a mathematical expression of the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( x \) represents the variable. Quadratic polynomials have a degree of 2, which means the highest power of \( x \) is squared. These polynomials are important in various branches of mathematics, including algebra and calculus.

Quadratic polynomials can be expressed in different forms, such as:

• Standard form: \( ax^2 + bx + c \)
• Factored form: \( a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots of the polynomial.

In our exercise, each function \( f_j(x) \) is a quadratic polynomial, which means they can be expressed as \( a_{0j} + a_{1j}x + a_{2j}x^2 \) for \( j = 1, 2, 3 \). Understanding the structure of quadratic polynomials allows us to differentiate them efficiently, an essential step in calculating derivatives.

Derivatives

Derivatives are a fundamental concept in calculus that measure how a function changes as its input changes. For a given function \( f(x) \), the derivative \( f'(x) \) represents the rate of change or the slope of the function at a given point. When differentiating quadratic polynomials, you'll notice that the operation reduces the degree of the polynomial by one.

In our exercise, we needed to calculate both the first and second derivatives of the function \( f_j(x) = a_{0j} + a_{1j}x + a_{2j}x^2 \):

• First derivative: \( f_j'(x) = a_{1j} + 2a_{2j}x \). Here, the constant term \( a_{0j} \) disappears, and \( x^2 \) becomes \( 2x \).
• Second derivative: \( f_j''(x) = 2a_{2j} \). This derivative is constant, reflecting that the original function was a simple quadratic.

This differentiation process is crucial because it enables the construction of a matrix involving these derivatives, which is then used to determine the nature of \( g(x) \).

Matrix Algebra

Matrix algebra involves operations with matrices, which are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. Matrices are used in various fields of mathematics to solve systems of equations, transform geometric figures, and more.

In our exercise, we form a specific 3x3 matrix composed of the quadratic polynomials and their derivatives:

• The first row contains the original quadratic functions \( f_1, f_2, f_3 \).
• The second row contains their first derivatives \( f_1', f_2', f_3' \).
• The third row contains their second derivatives \( f_1'', f_2'', f_3'' \).

The challenge required evaluating the determinant of this matrix. The determinant is a special number calculated from a square matrix that provides valuable information about the matrix itself, such as whether it's invertible. In our case, evaluating the determinant of the matrix identifies the nature of \( g(x) \) in terms of its polynomial degree.

The steps involved in determinant calculation, such as pairing terms from each row according to specific rules, lead to a conclusion that \( g(x) \) is a cubic polynomial, owing to a maximum of degree 3.