Problem 106 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the following questions an Assertion (A) is given followed by a Reason. (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If \(f(x)=\left|\begin{array}{ccc}x+c_{1} & x+a & x+a \\ x+b & x+c_{2} & x+a \\ x+b & x+b & x+c_{3}\end{array}\right|\), then \(f(0)=\frac{b g(a)-a g(b)}{b-a}\), where \(g(x)=\left(c_{1}-x\right)\left(c_{2}-x\right)\) \(\left(c_{3}-x\right)\) Reason: \(f(x)\) is linear is \(x\).

Step-by-Step Solution

Answer

(C) Assertion(A) is True, Reason(R) is False.

Step 1: Understanding the Assertion

The assertion provides the function \( f(x) \) as a determinant of a 3x3 matrix. We need to check if \( f(0) \) equals \( \frac{b g(a) - a g(b)}{b-a} \) where \( g(x) = (c_1-x)(c_2-x)(c_3-x) \). First, evaluate \( f(0) \) by substituting \( x = 0 \) into the matrix.

Step 2: Evaluate \( f(x) \) for \( x = 0 \)

Substitute \( x = 0 \) into the determinant: \( f(0) = \begin{vmatrix} c_1 & a & a \ b & c_2 & a \ b & b & c_3 \ \end{vmatrix} \) and calculate this determinant using cofactor expansion or row/column operations.

Step 3: Analyze the Expression for \( f(0) \)

Using cofactor expansion along the first row of the matrix, compute the determinant: \[ f(0) = c_1 \begin{vmatrix} c_2 & a \ b & c_3 \ \end{vmatrix} - a \begin{vmatrix} b & a \ b & c_3 \ \end{vmatrix} + a \begin{vmatrix} b & c_2 \ b & b \ \end{vmatrix} \]Each 2x2 determinant can be simplified individually.

Step 4: Simplify the Determinant

Calculate the 2x2 determinants:1. \( \begin{vmatrix} c_2 & a \ b & c_3 \ \end{vmatrix} = (c_2 \cdot c_3) - (a \cdot b) \)2. \( \begin{vmatrix} b & a \ b & c_3 \ \end{vmatrix} = b \cdot (c_3 - a) \)3. \( \begin{vmatrix} b & c_2 \ b & b \ \end{vmatrix} = b \cdot (b - c_2) \)Substitute back into the expression for \( f(0) \) and simplify.

Step 5: Check the Given Expression

Verify if the simplified expression of \( f(0) \) matches \( \frac{b g(a) - a g(b)}{b-a} \). Consider what \( g(a) \) and \( g(b) \) are, and evaluate \( b g(a) \) and \( a g(b) \).

Step 6: Analyzing the Reason Statement

Analyze if the function \( f(x) \) is linear in \( x \). For this, observe how the determinant expression changes with respect to \( x \). Consider the linearity properties of determinants and confirm if standard linear transformations of matrix rows/columns yield a linear function.

Key Concepts

Linear Functions and Their Properties

A linear function is essentially a function that creates a straight line when plotted on a graph. Linear functions have the general form: \( f(x) = mx + b \) where \( m \) and \( b \) are constants. A unique characteristic of linear functions is that any change in input \( x \) results in a proportional change in the output \( f(x) \), which is determined by \( m \), the slope.
Understanding linear functions is critical for determining the behavior of more complex expressions by recognizing linear components within them.
When referring to determinants of matrices, especially in mathematical expressions where \( f(x) \) is discussed, linearity would imply that as \( x \) changes, the function exhibits changes in a linear manner. This means the highest power of \( x \) in the expression is 1.
In the context of this problem, the reason given claims that \( f(x) \) is linear in \( x \). This requires examining how the terms in the determinant change as \( x \) is varied. If the structure maintains a straight-line characterization when graphically interpreted, then the function can be classified as linear.

Cofactor Expansion of Determinants

Cofactor expansion is a method used to calculate the determinant of a matrix, especially helpful in solving larger matrices, like a 3x3 matrix. To compute the determinant via cofactor expansion, you choose any row or column and sum the products of each element of the row or column with its corresponding cofactor. A cofactor is calculated as the determinant of the minor matrix, which results when the row and column of that element are removed.
For example, in a matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i\end{bmatrix}\),
you can perform the cofactor expansion along the first row:
- The determinant \( |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \).

• Element \( a \) is multiplied by the determinant of its minor, \( \begin{bmatrix}e & fh & i\end{bmatrix} \).
• Element \( b \) is multiplied by the determinant of \( \begin{bmatrix}d & f \g & i\end{bmatrix} \) and subtracted because \( b \) is in an odd position (i.e., it follows a minus coefficient in pattern).
• Element \( c \) is multiplied by \( \begin{bmatrix}d & e \g & h\end{bmatrix} \) with a positive sign.

Cofactor expansion is reliable and can be adapted for any row or column that simplifies the computation aesthetically, or computationally in complexity.

The 3x3 Matrix

A 3x3 matrix is an arrangement of numbers in three rows and three columns. This is the smallest square matrix that allows interesting manipulation like determinants, which have practical applications in linear algebra, geometry, and applied mathematics.
3x3 matrices, such as:\[ \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33}\end{bmatrix} \],are commonly used in problems involving rotations of planes in three dimensions, transformations, and more.
The determinant of a 3x3 matrix is a critical value as it provides information on the matrix's invertibility, volume scale changes in transformations, and the potential consistency of linear equations.
The evaluation of a 3x3 determinant often involves using cofactor expansion, as showcased before, or by simplifying using row and column operations to form zeros. Each element of the determinant contributes to the product sum in which symmetry and patterns play an essential role in simpler calculations.