Matrices & Determinants
Determinant Simplification
Grade 12
Question:
<p>If \(f(x) = \begin{vmatrix} 3 & 3x & 3x^2 + 2a^2 \\ 3x & 3x^2 + 2a^2 & 3x^3 + 6a^2x \\ 3x^2 + 2a^3 & 3x^3 + 6a^2x & 3x^4 + 12a^2x^2 + 2a^4 \end{vmatrix}\), then which is true?</p>
<p>(a) \(f'(x) = 0\)</p>
<p>(b) \(y = f(x)\) is a straight line parallel to the \(x\)-axis</p>
<p>(c) \(\int_0^4 f(x)\,dx = 32a^4\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Observe that the determinant structure results in a function independent of $x$ after simplification.
<p>Expand the determinant using properties of row/column operations. The determinant reduces to a constant value independent of $x$, meaning $f(x)$ is a horizontal line.</p>
Correct Answer: B