Problem 22 - A Complete Resource Book in Mathematics for JEE Main 2017

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: Expressions for \(f_j(x)\)

Given \(f_j(x) = \sum_{i=0}^{2} a_{ij} x^i\), for each \(j=1,2,3\), we expand to forms: \(f_j(x) = a_{0j} + a_{1j}x + a_{2j}x^2\). This represents a quadratic polynomial in \(x\) with coefficients \(a_{ij}\).

Step 2: Calculate First Derivative

The first derivative \(f_j'(x)\) is obtained by differentiating \(f_j(x) = a_{0j} + a_{1j}x + a_{2j}x^2\). Thus, \(f_j'(x) = 0 + a_{1j} + 2a_{2j}x\) which simplifies to \(f_j'(x) = a_{1j} + 2a_{2j}x\).

Step 3: Calculate Second Derivative

The second derivative \(f_j''(x)\) is obtained from differentiating \(f_j'(x) = a_{1j} + 2a_{2j}x\). Thus, \(f_j''(x) = 2a_{2j}\). There is no \(x\) dependence in \(f_j''(x)\) as it's constant.

Step 4: Construct the Function \(g(x)\)

We construct \(g(x)\) as the determinant:\[g(x) = \begin{vmatrix} \ f_1 & f_2 & f_3 \ f_1' & f_2' & f_3' \ f_1'' & f_2'' & f_3'' \ \end{vmatrix}\]Substitute from earlier steps with polynomials and derivatives.

Step 5: Simplify the Determinant

The determinant involves evaluating terms in each row and then calculating the determinant in multi-variable terms of \(x\). Due to the nature of polynomial multiplication within the determinant, each element will impact the order of \(x\) present in the resultant calculation.

Step 6: Determine the Polynomial Degree

Compute the degree: - Row 1 \(f_j(x):\) degree 2- Row 2 \(f_j'(x):\) degree 1- Row 3 \(f_j''(x):\) degree 0.We use a structure from determinants: Summing degrees per each term in each row, total degree is \(2+1+0 = 3\).

Step 7: Conclude the Order

Since each row corresponds to decreasing polynomial degree when moving down, and highest degree results in the combined multiplicative form, the final polynomial from the determinant is cubic, due to maximum degree of 3 across weighted polynomial evaluation.

Key Concepts

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial of degree 2, also known as a quadratic polynomial, as seen in this problem is given by:

• \(f_j(x) = a_{0j} + a_{1j}x + a_{2j}x^2\)

The coefficients \( a_{0j}, a_{1j}, \) and \( a_{2j} \) determine the shape and position of the parabola described by the quadratic expression.
Polynomial functions are fundamental in algebra and calculus because they form the basis for polynomial equations and can describe a wide variety of real-world phenomena, from simple motion to complex financial models. Importantly, polynomials of higher degrees encompass more complex behaviors and features.

Derivative Calculations

Derivatives are used to determine the rate at which a function is changing at any given point. They are crucial for understanding and interpreting the behavior of polynomial functions, as they can give information about the slope and curvature of the graph.
To calculate the first derivative of the function \( f_j(x) \):

• You start with the function: \( f_j(x) = a_{0j} + a_{1j}x + a_{2j}x^2 \)
• The first derivative is found using the power rule, resulting in: \( f_j'(x) = a_{1j} + 2a_{2j}x \)

For the second derivative, which tells us about the concavity of the function:

• The second derivative is obtained by differentiating the first derivative: \( f_j''(x) = 2a_{2j} \)

This constant derivative indicates a consistent curvature, typical of a quadratic polynomial, where the concavity does not change.

Matrix Algebra

Matrix algebra is a powerful mathematical tool used for a variety of calculations, including those involving systems of linear equations and transformations. In this problem, the matrix determinant concept is employed.
The given function \( g(x) \) is defined by the determinant of a 3x3 matrix:

• Top row: contains the functions \( f_1, f_2, \) and \( f_3 \), each a polynomial of degree 2.
• Middle row: consists of their first derivatives \( f_1', f_2', \) and \( f_3' \), each of degree 1.
• Bottom row: consists of their second derivatives \( f_1'', f_2'', \) and \( f_3'' \), constants with degree 0.

The determinant of this matrix is calculated by taking specific elements from each row and traversing the structure of the determinant calculation. Because each function reduces the degree as you move from top to bottom, the determinant combines these changes, resulting in a polynomial function of degree 3, or cubic, as the highest degree terms from the first row dominate the overall result.