Problem 34 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left|\begin{array}{ccc}a & b & a \alpha-b \\ b & c & b \alpha-c \\ 2 & 1 & 0\end{array}\right|=0\) and \(\alpha \neq \frac{1}{2}\), then (A) \(a, b, c\) are in A.P. (B) \(a, b, c\) are in G.P. (C) \(a, b, c\) are in H.P. (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \(a, b, c\) are in A.P.

Step 1: Identify the Determinant

We need to evaluate the determinant given by the expression \( \left|\begin{array}{ccc}a & b & a \alpha-b \ b & c & b \alpha-c \ 2 & 1 & 0\end{array}\right| \). This determinant equals zero because it is specified in the problem.

Step 2: Calculate the Determinant using Cofactor Expansion

Use cofactor expansion along the third row of the matrix: \[ \text{Determinant} = 2 \left|\begin{array}{cc} b & b \alpha - c \ c & b \alpha - c\end{array}\right| - 1 \left|\begin{array}{cc} a & a \alpha - b \ b & b \alpha - c\end{array}\right| = 0\] This simplifies to: \[2(b \cdot (b \alpha - c) - b \alpha \cdot c + c^2) - (a \cdot (b \alpha - c) - (a \alpha - b) \cdot b) = 0\]

Step 3: Simplify the Expression

Simplify the expression obtained from the determinant:\[2(b^2 \alpha - bc - bc \alpha + c^2) - (ab \alpha - ac - ab + b^2) = 0\]Combine like terms:\[2b^2 \alpha - 2bc \alpha + 2c^2 - ab \alpha + ac + ab - b^2 = 0\]

Step 4: Further Simplify to Isolate Variables

Re-write the terms:\[(a - 2b) \cdot (b \alpha - c) + (2c^2 + ab - b^2) = 0\]Since \( \alpha eq \frac{1}{2} \), this equation implies that the values of \(a, b, c\) must satisfy the relationships where they are in Arithmetic Progression (AP).

Step 5: Analyze the Conditions for AP

Remember that for numbers to be in AP, the condition \(2b = a + c\) must hold. Thus, check if this holds in the relation:\[(a - 2b)(b \alpha - c) + (2c^2 + ab - b^2) = 0\]Given \( \alpha eq \frac{1}{2}\), it leads us to further confirm the AP condition.

Key Concepts

Cofactor Expansion

Matrix determinants can be challenging, but cofactor expansion simplifies the process significantly. Cofactor expansion is a method of calculating the determinant of a square matrix by expanding along a row or a column. This method involves selecting a row or column and considering each element in that row or column. For each element, you calculate a minor, which is the determinant of the matrix that remains after removing the row and column containing that element. Then, multiply each minor by a sign and the element itself, where the sign is determined by the position of the element (using the pattern of a checkerboard of plus and minus signs). Each resulting term is called a cofactor. Finally, sum up all of these products to find the determinant.
The cofactor expansion is especially useful when a matrix contains zeros. This is because expanding along a row or column with several zeros reduces the number of calculations you need to perform since the contributions from zeros are simply eliminated from the expansion. In the given problem, by expanding using the third row, fewer terms are involved due to the simplicity provided by the element '0'.

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference. This characteristic makes APs easy to identify and work with.
For numbers \(a, b, c\) to be in an AP, it must be true that:

• \(2b = a + c\)
• The sequence maintains the same gap between each pair of numbers.

This formula can help find missing terms of a sequence or confirm that a sequence is an AP. In our exercise, this property helped to determine that \(a, b, c\) had to satisfy the condition of AP for the determinant equation to hold true. Since the equation resolved into the form \((a - 2b)(b \alpha - c) + (2c^2 + ab - b^2) = 0\), where \(\alpha eq \frac{1}{2}\), it reflected that \(a, b, c\) form an AP since the expression equated to zero under the condition \(2b = a + c\).

Matrix Determinant Properties

Matrix determinant properties are vital in simplifying determinant calculations and understanding matrix behaviors. Here are some key properties:

• The determinant of a square matrix provides a scalar value that offers insights into the matrix, such as whether it is invertible (non-zero determinant).
• The swapping of any two rows or columns in a matrix results in a negation of the determinant.
• If a row or column is multiplied by a scalar, the determinant of the matrix is also multiplied by that scalar.
• If two rows or columns of a matrix are identical, the determinant is zero.
• The determinant is unchanged (i.e., it remains zero if initially zero) if a multiple of one row is added to another row.

In the original exercise, knowing these properties allowed the simplification of the determinant expression through elimination and algebraic manipulation, ultimately confirming that the sequence \(a, b, c\) must be in an AP, given \(\alpha eq \frac{1}{2}\).