Question

If $\Delta(x)= \begin{vmatrix} x - 2 & (x - 1)^2 & x^3 \\ x - 1 & x^2 & (x + 1)^3 \\ x & (x + 1)^2 & (x + 2)^3 \end{vmatrix}$, then coefficient of $x$ in $\Delta(x)$ is

• A. 2
• B. -2
• C. 3
• D. -4

Answer 1

The Correct Option is B

Given determinant: $$ \Delta(x) = \begin{vmatrix} x - 2 & (x - 1)^2 & x^3 \\ x - 1 & x^2 & (x + 1)^3 \\ x & (x + 1)^2 & (x + 2)^3 \end{vmatrix} $$ We are asked to find the coefficient of \(x\) in the expansion of \(\Delta(x)\). Step 1: Expand the determinant. We can start by expanding the determinant along the first row: $$ \Delta(x) = (x - 2) \cdot \begin{vmatrix} x^2 & (x + 1)^3 \\ (x + 1)^2 & (x + 2)^3 \end{vmatrix} - (x - 1)^2 \cdot \begin{vmatrix} x - 1 & (x + 1)^3 \\ x & (x + 2)^3 \end{vmatrix} + x^3 \cdot \begin{vmatrix} x - 1 & x^2 \\ x & (x + 1)^2 \end{vmatrix} $$ Step 2: Calculate each 2x2 determinant. We will now calculate the 2x2 minors individually. 1. For the first 2x2 determinant: $$ \begin{vmatrix} x^2 & (x + 1)^3 \\ (x + 1)^2 & (x + 2)^3 \end{vmatrix} = x^2 \cdot (x + 2)^3 - (x + 1)^3 \cdot (x + 1)^2 $$ This simplifies to: $$ x^2 \cdot (x + 2)^3 - (x + 1)^5 $$ 2. For the second 2x2 determinant: $$ \begin{vmatrix} x - 1 & (x + 1)^3 \\ x & (x + 2)^3 \end{vmatrix} = (x - 1) \cdot (x + 2)^3 - x \cdot (x + 1)^3 $$ This simplifies to: $$ (x - 1) \cdot (x + 2)^3 - x \cdot (x + 1)^3 $$ 3. For the third 2x2 determinant: $$ \begin{vmatrix} x - 1 & x^2 \\ x & (x + 1)^2 \end{vmatrix} = (x - 1) \cdot (x + 1)^2 - x \cdot x^2 $$ This simplifies to: $$ (x - 1) \cdot (x + 1)^2 - x^3 $$ Step 3: Focus on the coefficient of \(x\). To determine the coefficient of \(x\) in \(\Delta(x)\), we need to focus only on the terms that contribute to \(x\) when expanded. For each of the terms: - The first term: \( (x - 2) \cdot \left[ x^2 \cdot (x + 2)^3 - (x + 1)^5 \right] \) contributes the coefficient of \(x\). - The second term: \( -(x - 1)^2 \cdot \left[ (x - 1) \cdot (x + 2)^3 - x \cdot (x + 1)^3 \right] \) contributes the coefficient of \(x\). - The third term: \( x^3 \cdot \left[ (x - 1) \cdot (x + 1)^2 - x^3 \right] \) contributes the coefficient of \(x\). Through expansion and simplification of these terms, the coefficient of \(x\) in the entire expression is found to be: $$ -2 $$ Conclusion: The coefficient of \(x\) in \(\Delta(x)\) is \(-2\). Thus, the correct answer is \(-2\).