Matrices & Determinants
Differentiation of Determinant
nta_pyq_2024_jan
Grade 12
Question:
If $f(x)=\begin{vmatrix}2\cos^4 x&2\sin^4 x&3+\sin^2 2x\\3+2\cos^4 x&2\sin^4 x&\sin^2 2x\\2\cos^4 x&3+2\sin^4 x&\sin^2 2x\end{vmatrix}$, then $\frac{1}{5}f'(0)$ is equal to
Step-by-Step Solution
Key Concept: Apply row operations $R_2\to R_2-R_1$, $R_3\to R_3-R_1$ to simplify the determinant. The resulting $f(x)$ will be a constant, so $f'(x)=0$ for all $x$.
$R_2\to R_2-R_1$, $R_3\to R_3-R_1$: $\begin{vmatrix}2\cos^4x&2\sin^4x&3+\sin^2 2x\\3&0&-3\\0&3&-3\end{vmatrix}$. Expanding: $f(x)=45$ (constant). $f'(x)=0\Rightarrow\frac{1}{5}f'(0)=0$.
Correct Answer: 1