Matrices & Determinants
Determinant Evaluation
Grade None
Question:
<p>If <span class="math">\begin{vmatrix} (x+1) & (x+1)^2 & (x+1)^3 \\ (x+2) & (x+2)^2 & (x+2)^3 \\ (x+3) & (x+3)^2 & (x+3)^3 \end{vmatrix}</span> is expressed as a polynomial in \(x\), then the term independent of \(x\) is:</p>
<p>(a) 0</p>
<p>(b) 2</p>
<p>(c) 12</p>
<p>(d) 16</p>
Step-by-Step Solution
Key Concept: Recognize this as a Vandermonde determinant structure and factor out terms systematically to find the constant term.
<p><strong>Solution:</strong> The determinant can be factored as a product of differences. Factor out common terms from each row and apply determinant properties for Vandermonde-type determinants. The constant term (independent of $x$) evaluates to 12.</p>
Correct Answer: C