Matrices & Determinants
Differentiation of Determinants
Grade 12
Question:
<p>If \(f'(x) = \begin{vmatrix} mx & mx-p & mx+p \\ n & n+p & n-p \\ mx+2n & mx+2n+p & mx+2n-p \end{vmatrix}\), then \(y = f(x)\) represents</p>
<p>(1) a straight line parallel to \(x\)-axis</p>
<p>(2) a straight line parallel to \(y\)-axis</p>
<p>(3) parabola</p>
<p>(4) a straight line with negative slope</p>
Step-by-Step Solution
Key Concept: Observe that each column differs by a constant amount (the middle column is the first plus p, the third is the first minus p). Apply column operations: C₂ - C₁ and C₃ - C₁ to factor out p, then expand to find f'(x) is independent of x.
<p><strong>Step 1:</strong> Apply column operations C₂ → C₂ - C₁ and C₃ → C₃ - C₁:</p><p>$$f'(x) = \begin{vmatrix} mx & -p & p \\ n & p & -p \\ mx+2n & p & -p \end{vmatrix}$$</p><p><strong>Step 2:</strong> Factor out p from C₂ and -p from C₃ (giving (-p²) factor):</p><p>$$f'(x) = -p^2 \begin{vmatrix} mx & 1 & -1 \\ n & 1 & 1 \\ mx+2n & 1 & 1 \end{vmatrix}$$</p><p><strong>Step 3:</strong> Notice C₂ = C₃ (the last two columns are identical), so the determinant equals 0.</p><p><strong>Step 4:</strong> Therefore f'(x) = 0 for all x, which means f(x) = c (a constant).</p><p><strong>Step 5:</strong> A constant function y = f(x) is represented by a horizontal straight line parallel to the x-axis.</p><p>∴ Answer: A (y = f(x) represents a straight line parallel to x-axis, or a constant function)</p>
Correct Answer: A