Problem 10 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

if \(\left|\begin{array}{ccc}x^{2}+x & x+1 & x-2 \\ 2 x^{2}+3 x-1 & 3 x & 3 x-3 \\ x^{2}+2 x+3 & 2 x-1 & 2 x-1\end{array}\right|=a x-12\), then ' \(a\) ' is equal to: [Online April 11, 2015] (a) 24 (b) \(-12\) (c) \(-24\) (d) 12

Step-by-Step Solution

Answer

The value of 'a' is 12.

Step 1: Calculate Determinant Using First Row

To find the determinant of the given matrix \( \begin{bmatrix} x^2+x & x+1 & x-2 \ 2x^2+3x-1 & 3x & 3x-3 \ x^2+2x+3 & 2x-1 & 2x-1 \end{bmatrix} \), we will use the first row for expansion. The formula for the determinant is \( a(ei-fh) - b(di-fg) + c(dh-eg) \). Substitute values from the matrix to expand the determinant as follows:

Step 2: Expand the Determinant

Let's expand the terms for the determinant: 1. \( (x^2 + x)((3x)(2x - 1) - (3x - 3)(2x - 1)) 2. - (x+1)((2x^2 + 3x - 1)(2x - 1) - (3x - 3)(x^2 + 2x + 3)) 3. + (x-2)((2x^2 + 3x - 1)(2x - 1) - (3x)(x^2 + 2x + 3)) \).

Step 3: Simplify Each Term

Calculate and simplify each of the three terms separately:1. Compute the result of \((3x)(2x - 1) - (3x - 3)(2x - 1)\) and multiply by \((x^2 + x)\).2. Compute \((2x^2 + 3x - 1)(2x - 1) - (3x - 3)(x^2 + 2x + 3 )\) and multiply by \(-(x+1)\).3. Compute \((2x^2 + 3x - 1)(2x - 1) - (3x)(x^2 + 2x + 3)\) and multiply by \((x-2)\).

Step 4: Combine All Terms Together

Add the simplified results of the expressions from each step. Combine like terms carefully to simplify the full determinant to a form \( ax - 12 \). The goal is to find the coefficient \( a \).

Step 5: Compare and Solve for 'a'

Set the determinant expression equal to \( ax - 12 \) and solve for the coefficient \( a \). Simplifying should give a direct comparison after canceling out the constant terms and matching coefficients of \( x \).

Key Concepts

Matrix Expansion

When working with matrices, the process of expansion, specifically row or column expansion, is essential when calculating the determinant. To perform matrix expansion, you focus on a specific row or column and systematically break down the determinant calculation into manageable chunks.

The rule for the 3x3 matrix expansion is often expressed as:

• (a(ei-fh) - b(di-fg) + c(dh-eg))

Here, each variable represents the matrix elements. Choosing the first row, you take each element and multiply it by the corresponding 2x2 determinant formed by skipping the element's row and column. This method reduces complex determinants into simpler calculations.

This systematic approach allows you to handle larger matrices by breaking them down into bite-sized, solvable problems using minor determinants.

Determinant Properties

Understanding the properties of determinants can greatly simplify matrix calculations. Several key properties are useful:

• Switching Rows or Columns: Swapping the positions of any two rows or columns of a matrix changes the sign of the determinant.
• Zero Rows or Columns: If a matrix has a row or column of all zeros, its determinant is zero.
• Multiplicative Constants: Multiplying a row or column by a scalar will multiply the determinant by the same scalar.
• Adding Rows or Columns: Adding a multiple of one row or column to another does not change the determinant.

These properties help to simplify calculations, as seen in our example. By following these principles, the expansion becomes more intuitive, reducing the potential for errors. Familiarity with these rules can streamline finding the determinant and understanding its algebraic structure.

Algebraic Simplification

The process of algebraic simplification involves reducing complex expressions into simpler forms. In our determinant problem, it's crucial after each matrix expansion step.

Each expanded determinant term results in polynomial expressions involving products and sums. Careful algebraic manipulation helps in simplifying these:

• Distribute Terms: Ensure all components are expanded thoroughly.
• Combine Like Terms: Group and simplify similar terms to reduce complexity.
• Cancel Terms: Identify and remove terms that counteract each other.

In essence, algebraic simplification ensures the expanded determinant results are manageable and accurate. This simplification lays the groundwork for comparing polynomial coefficients, crucial for solving for constants like 'a' in our problem.