Problem 38 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ (a+\lambda)^{2} & (b+\lambda)^{2} & (c+\lambda)^{2} \\ (a-\lambda)^{2} & (b-\lambda)^{2} & (c-\lambda)^{2}\end{array}\right|=k \lambda\left|\begin{array}{ccc}a^{2} & b^{2} & c^{2} \\ a & b & c \\ 1 & 1 & 1\end{array}\right|, \lambda \neq 0\) then \(\mathrm{k}\) is equal to: \(\quad\) [Online April 12, 2014] (a) \(4 \lambda\) abc (b) \(-4 \lambda a b c\) (c) \(4 \lambda^{2}\) (d) \(-4 \lambda^{2}\)

Step-by-Step Solution

Answer

\(k = 4\lambda\)

Step 1: Express the Determinants

Let's express the given matrices within the determinants on both sides of the equation. We have two determinants: 1. \( \begin{vmatrix} a^2 & b^2 & c^2 \ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2 \end{vmatrix}\) 2. \( \begin{vmatrix} a^2 & b^2 & c^2 \ a & b & c \ 1 & 1 & 1 \end{vmatrix}\)

Step 2: Expand the Determinants

Expand the determinant on the left side. Notice that it involves quadratic expressions of both \((a+\lambda)^2\) and \((a-\lambda)^2\): - \((a+\lambda)^2 = a^2 + 2a\lambda + \lambda^2\)- \((a-\lambda)^2 = a^2 - 2a\lambda + \lambda^2\)Similar expressions hold for \(b\) and \(c\). Notice symmetry in the problem which might allow simplification.

Step 3: Apply Row Operations

Use properties of determinants to simplify: the determinant is \[\begin{vmatrix} a^2 & b^2 & c^2 \ a^2 + 2a\lambda + \lambda^2 & b^2 + 2b\lambda + \lambda^2 & c^2 + 2c\lambda + \lambda^2 \ a^2 - 2a\lambda + \lambda^2 & b^2 - 2b\lambda + \lambda^2 & c^2 - 2c\lambda + \lambda^2 \end{vmatrix}\]Subtract the first row from the next two rows to simplify the expressions.

Step 4: Simplify the Expressions

After row operations, simplify:- The second row becomes \([2a\lambda, 2b\lambda, 2c\lambda]\)- The third row becomes \([-2a\lambda, -2b\lambda, -2c\lambda]\)Thus, the determinant simplifies to:\(4\lambda^2 \begin{vmatrix} a^2 & b^2 & c^2 \ a & b & c \ 1 & 1 & 1 \end{vmatrix}\)

Step 5: Compare and Solve for \(k\)

Now compare with the right side of the given equation, which is \(k \lambda \begin{vmatrix} a^2 & b^2 & c^2 \ a & b & c \ 1 & 1 & 1 \end{vmatrix}\)Thus, solving for \(k\) involves equating the two expressions:\[4\lambda^2 \begin{vmatrix} a^2 & b^2 & c^2 \ a & b & c \ 1 & 1 & 1 \end{vmatrix} = k \lambda \begin{vmatrix} a^2 & b^2 & c^2 \ a & b & c \ 1 & 1 & 1 \end{vmatrix}\]This implies \(k = 4 \lambda\).

Key Concepts

Row Operations

Working with matrices often requires us to manipulate rows for simplification. This process is known as row operations. Row operations are especially useful when dealing with determinants, which require reduction to a simpler form for easier computation. There are three primary types of row operations: exchanging two rows, multiplying a row by a non-zero scalar, and adding or subtracting the multiple of one row to another row.

The purpose of using row operations, as demonstrated in the solution, was to make specific rows easier to calculate by subtracting the first row from the subsequent rows. This step resulted in simpler expressions and highlighted a pattern of zeros that helped to factor out common terms like \(4\lambda^2\). When simplified, these operations allow us to relate two determinants in their simplest possible form for easy comparison and computation.

Matrix Simplification

In matrix simplification, the goal is to transform a complex matrix into a simpler one while maintaining the properties necessary for subsequent calculations, such as determinants. By this process, we aim to reduce calculations associated with determinants or other matrix functions.

In the given exercise, matrix simplification was achieved by the use of row operations to create zeroes and common factors. This simplification allowed us to express the determinant in terms of a simpler form and involved reducing complex quadratic expressions into simpler linear combinations, which are easier to handle. The result of this simplification process was crucial in transforming the left-side determinant into a scalar multiple of the right-side determinant, enabling a direct comparison.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the form \(ax^2 + bx + c\) where \(a, b,\) and \(c\) are constants and \(x\) is an unknown. These expressions appear in a variety of mathematical contexts, including within matrices like in the current exercise.

When expanding determinants, noticing quadratic expressions can be key to simplification. In this particular problem, the quadratic expressions such as \((a+\lambda)^2\) and \((a-\lambda)^2\) are expanded to expose symmetry or relationships that can be used to simplify calculations. Transforming quadratic expressions into more straightforward terms often involves factoring, completing the square, or simple expansion. Recognizing when and how to use each method is crucial for integrating broader mathematical concepts, such as matrix operations and determinants, into simpler, more manageable forms.