<p>Let \(f(x) = \begin{vmatrix} 1 & 2x & (x-1) \\ 3x(x-1) & (x-1)(x-2) & x(x-1) \\ 1 & 2x & x-1 \\ 3x & x-2 & x \end{vmatrix}\)</p><p>Find \(f(50)\).</p>
Step-by-Step Solution
Key Concept: Factor out common terms and use row/column operations to create zeros, simplifying determinant evaluation.
<p><strong>Solution:</strong></p><p>Simplifying by factoring out \((x-1)\) from appropriate rows:</p><p>\(f(x) = (x-1)\begin{vmatrix} 1 & 2x & (x-1) \\ x & -1 & x \\ 2x & -2 & x \end{vmatrix}\)</p><p>Applying \(C_1 \to C_1 - C_3\) and \(C_2 \to C_2 - C_3\):</p><p>\(f(x) = (x-1)\begin{vmatrix} 0 & 0 & 1 \\ x & -1 & x \\ 2x & -2 & x \end{vmatrix}\)</p><p>The first row has two zeros, so expanding gives:</p><p>\(f(x) = (x-1)[-2x + 2x] = 0\)</p><p>∴ \(f(50) = 0\)</p>
Correct Answer: 0