Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
Let $f(x) = \begin{vmatrix} x \cos x & 2x \sin x & x \tan x \\ 1 & 2x & 1 \end{vmatrix}$, then $\lim_{x \to 0} \frac{f(x)}{x^2} =$
Step-by-Step Solution
Key Concept: Factor out $x$ from the first row and use determinant properties to simplify before computing the limit.
Given $f(x) = \begin{vmatrix} x\cos x & 2x\sin x & x\tan x \\ 1 & x & 1 \\ 1 & 2x & 1 \end{vmatrix}$, rewrite as $f(x) = \begin{vmatrix} \cos x & 2\sin x & \tan x \\ 1 & 1 & 1 \\ 1 & 2 & 1 \end{vmatrix}$. Evaluate $\lim_{x \to 0} \frac{f(x)}{x^2}$ using $R_1 - R_2$ to get $\begin{vmatrix} 1 & 2 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 1 \end{vmatrix} = -1$.
Correct Answer: 3