Problem 22 - 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

The degree of polynomial \( g(x) \) is 3, making \( g(x) \) a cubic in \( x \). (Answer: A)

Step 1: Understanding the Function Definitions

Given functions are polynomials of degree 2: \( f_j = a_{0j} + a_{1j}x + a_{2j}x^2 \) for \( j=1,2,3 \). This means that the degree of each \( f_j \) is 2.

Step 2: Derivative Calculations

First derivatives of \( f_j \) are: \( f'_j = a_{1j} + 2a_{2j}x \). These derivatives are of degree 1 in \( x \). Second derivatives of \( f_j \) are: \( f''_j = 2a_{2j} \), which are constants (degree 0).

Step 3: Forming the Determinant Matrix

We need the determinant of the matrix:\[\begin{vmatrix} f_1 & f_2 & f_3 \ f_1' & f_2' & f_3' \ f_1'' & f_2'' & f_3''\end{vmatrix}\]The rows correspond to the original polynomials, their first derivatives, and their second derivatives.

Step 4: Degree Analysis of Matrix Rows

Each row of the matrix has entries with degrees as follows:- First row (\(f_1, f_2, f_3\)): degree 2- Second row (\(f'_1, f'_2, f'_3\)): degree 1- Third row (\(f''_1, f''_2, f''_3\)): degree 0.This implies the overall degree of the determinant is dictated by the highest combined degree possible from these row combinations.

Step 5: Calculating the Determinant's Degree

For a 3x3 matrix with polynomials of degrees 2, 1, and 0 in its rows, the degree of the determinant is the sum of these degrees, which is \(2 + 1 + 0 = 3\). Therefore, the degree of the polynomial that is equal to the determinant is 3.

Key Concepts

Polynomial Degree

Understanding polynomial degree is crucial when analyzing mathematical functions. A polynomial is expressed as a sum of terms, each consisting of a coefficient multiplied by a variable raised to an exponent. The degree of a polynomial is the highest exponent on the variable. For instance, in the polynomial \(f(x) = a_0 + a_1x + a_2x^2\), the term with the highest exponent is \(a_2x^2\), so the degree is 2.
This degree tells us how the function behaves as \(x\) becomes very large. The degree determines the number of solutions a polynomial equation can have. For higher degree polynomials, this means more potential intersections with the x-axis.
In the case of the exercise, each polynomial \(f_j\) has a degree of 2. Recognizing the degree helps in further steps like derivative calculations and matrix formations.

Derivative Calculations

Derivatives represent the rate of change of a function. When working with polynomials, derivatives are calculated using power rules. For a polynomial \(f(x) = a_0 + a_1x + a_2x^2\), the first derivative is found by reducing each exponent by one and multiplying by the original exponent.
Thus, the first derivative of \(f(x)\) is \(f'(x) = a_1 + 2a_2x\), which is a linear function (degree 1). Calculating a derivative transforms a polynomial into one of a lower degree. The second derivative involves taking the derivative of the first derivative, resulting in a constant \(f''(x) = 2a_2\).
These derivative calculations are essential to form the matrix used in determinant analysis in the given problem.

Matrix Analysis

Matrix analysis is a powerful tool in linear algebra, often used for calculating determinants. In the given exercise, the matrix is assembled with the original polynomials and their first and second derivatives as given by:
\[\begin{vmatrix} f_1 & f_2 & f_3 \ f'_1 & f'_2 & f'_3 \ f''_1 & f''_2 & f''_3 \\end{vmatrix}\]
To find the determinant of this matrix, one must consider the degree of each entry. The top row entries have degree 2 (because they are the original polynomials), the second row entries have degree 1 (the first derivatives), and the bottom row entries degree 0 (the second derivatives).
The determinant's degree is calculated as the sum of the highest possible combination of the degrees from each row, giving an overall degree of 3 for the determinant. This analysis determines how the function behaves and is central to understanding its properties, confirming the result matches option (A) from the given choices.