Matrices & Determinants
Differentiation of Determinant — Evaluating at a Point
nta_pyq_2024_jan
Grade 12
Question:
If $f(x)=\begin{vmatrix}x^3&2x^2+1&1+3x\\3x^2+2&2x&x^3+6\\x^3-x&4&x^2-2\end{vmatrix}$ for all $x\in\mathbb{R}$, then $2f(0)+f'(0)$ is equal to
Step-by-Step Solution
Key Concept: Evaluate $f(0)$ directly by substituting $x=0$. For $f'(0)$, use the formula for differentiating a determinant (differentiate each row separately and sum). Evaluate each at $x=0$.
$f(0)=\begin{vmatrix}0&1&1\\2&0&6\\0&4&-2\end{vmatrix}=1\cdot(-4-0)-1\cdot(8-0)... =12$. Differentiating row by row and evaluating at $x=0$: $f'(0)=24-6=18$. $2f(0)+f'(0)=24+18=42$.
Correct Answer: 3