Matrices & Determinants
Trigonometric Determinants
nta_pyq_2025_apr
Grade 12
Question:
Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x) = \begin{vmatrix}1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x\end{vmatrix}$, $x \in \mathbb{R}$. Then $M^4 - m^4$ is equal to:
Step-by-Step Solution
Key Concept: Apply row operations $R_2 \to R_2 - R_1$ and $R_3 \to R_3 - R_1$ to simplify the determinant, then expand.
After $R_2 \to R_2 - R_1$, $R_3 \to R_3 - R_1$: $f(x) = 2 + 4\sin 4x$. Max $M = 6$ (when $\sin 4x = 1$), min $m = -2$ (when $\sin 4x = -1$). $M^4 - m^4 = 1296 - 16 = 1280$.
Correct Answer: 1280