Problem 2 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\Delta=\left|\begin{array}{ccc}x-2 & 2 x-3 & 3 x-4 \\ 2 x-3 & 3 x-4 & 4 x-5 \\ 3 x-5 & 5 x-8 & 10 x-17\end{array}\right|=A x^{3}+B x^{2}+C x+D\) then \(B+C\) is equal to: [Sep. 03, 2020 (I)] (a) \(-1\) (b) 1 (c) \(-3\) (d) 9

Step-by-Step Solution

Answer

The value of \(B + C\) is 62.

Step 1: Understanding the problem

We need to find the value of \(B+C\) where \(\Delta\) is a determinant of the given 3x3 matrix and is expressed as \(A x^{3}+B x^{2}+C x+D\). We will leverage the properties of determinants to simplify it.

Step 2: Writing determinant in standard form

The given matrix is \(\begin{bmatrix} x-2 & 2x-3 & 3x-4 \ 2x-3 & 3x-4 & 4x-5 \ 3x-5 & 5x-8 & 10x-17 \end{bmatrix}\). Our goal is to evaluate this determinant and write it in the polynomial form \(A x^{3}+B x^{2}+C x+D\).

Step 3: Simplifying the determinant by row operations

Use row operations to simplify the determinant to minimize the calculation. Let's operate as follows: \(R_3 \to R_3 - 3R_2 + 2R_1\). Applying this operation, the third row becomes \(\begin{bmatrix} 3x-5 \ 5x-8 \ 10x-17 \end{bmatrix} - 3\begin{bmatrix} 2x-3 \ 3x-4 \ 4x-5 \end{bmatrix} + 2\begin{bmatrix} x-2 \ 2x-3 \ 3x-4 \end{bmatrix}\).

Step 4: Calculate the new third row

The new third row becomes: \[\begin{aligned} &10x-17 - (3)(4x-5) + (2)(3x-4) \ &= 10x-17 - 12x + 15 + 6x - 8 \ &= 4x - 10\end{aligned}\] After simplification, the third row is simplified to \(\begin{bmatrix} 4x-10 \end{bmatrix}\).

Step 5: Calculating the determinant

The matrix with the modified third row becomes \[\begin{bmatrix} x-2 & 2x-3 & 3x-4 \ 2x-3 & 3x-4 & 4x-5 \ 0 & 0 & 4x-10 \end{bmatrix}\]. Use the simplified determinant to calculate: \(\Delta = (x-2)[(3x-4)(4x-10) - (4x-5)(0) + (0)]\).

Step 6: Expand and collect coefficients for \(B\) and \(C\)

Using the simplified determinant expression:\( (x-2)[(12x^2 - 30x - 16x + 40)] = (x-2)(12x^2 - 46x + 40)\) Expand it:\[= (x-2)(12x^2 - 46x + 40) = 12x^3 - 46x^2 + 40x - 24x^2 + 92x - 80\]Collect like terms to find coefficients of \(x^2\) and \(x\):\[12x^3 - 70x^2 + 132x - 80\]Therefore, \(B = -70\) and \(C = 132\).

Step 7: Calculate B + C

Add the coefficients \(B\) and \(C\): \(B + C = -70 + 132 = 62\).

Key Concepts

Matrix Operations

Matrix operations involve calculations that utilize matrices. A matrix is a rectangular array of numbers arranged in rows and columns. They are used to solve systems of equations, perform transformations, and much more. Here's a simple breakdown of some matrix operations you'd typically come across:

• Addition and Subtraction: Matrices of the same dimension can be added or subtracted element by element.
• Scalar Multiplication: Every element of the matrix is multiplied by a constant value, known as a scalar.
• Matrix Multiplication: To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second. The resulting matrix forms by multiplying rows by columns, a process critical for many operations like transformations.
• Transposition: Achieved by flipping the matrix over its diagonal, this swaps the row and column indices of each element.

These operations are foundational for more complex manipulations, such as finding the determinant or the inverse of a matrix.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers of variables, each multiplied by a coefficient. The determinant of a matrix can be expressed as a polynomial, particularly when dealing with square matrices. Let's break down some aspects of polynomial expressions related to determinants:

• Degree: The highest exponent in the polynomial; it indicates the most extreme power of the variable present. For a 3x3 determinant, the polynomial's degree will be three.
• Coefficients: These are the numbers multiplying each power of the variable. They play a fundamental role in the structure and identity of the polynomial.
• Like Terms: Terms in the polynomial that have the same variable and exponent can be combined to simplify the expression.

In our exercise, we express the determinant as a polynomial of the form \(Ax^3 + Bx^2 + Cx + D\). Through matrix operations and simplifications, we extract these coefficients to solve problems.

Row Operations

Row operations are crucial for simplifying matrices, especially when calculating determinants or converting matrices into simpler forms, such as row-echelon form. Understanding and effectively using row operations can make computation of determinants more manageable. Here are the basic row operations:

• Row Switching: Swapping two rows in a matrix can aid in rearranging elements for ease of computation or strategizing reductions.
• Row Multiplication: Multiplying all elements of a row by a non-zero scalar. This operation scales the row, which can be beneficial for alignment with another row for elimination.
• Row Addition/Subtraction: Adding or subtracting a multiple of one row to another. This is frequently used to eliminate elements, thus simplifying matrices, especially when solving systems.

In our example, the operation \(R_3 \to R_3 - 3R_2 + 2R_1\) highlights how row operations help to create zeros in strategic positions, easing determinant calculations.