### Related Formula sin^2 x + cos^2 x = 1$\sin^2 x + \cos^2 x = 1$ -1 le sin 4x le 1$-1 \le \sin 4x \le 1$ ### Core Logic Apply the row operations R_2 to R_2 - R_1$R_2 \to R_2 - R_1$ and R_3 to R_3 - R_1$R_3 \to R_3 - R_1$ to simplify the determinant: f(x) = left| beginarrayc c c 1 + sin^ 2 x & cos^ 2 x & 4 sin 4 x \\ -1 & 1 & 0 \\ -1 & 0 & 1 endarray right|$f(x) = \left| \begin{array}{c c c} 1 + \sin^ {2} x & \cos^ {2} x & 4 \sin 4 x \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array} \right|$ ### Step 1: Expand the Determinant Expanding along the first row: f(x) = (1 + sin^2 x)(1 - 0) - cos^2 x(-1 - 0) + 4sin 4x(0 - (-1))$f(x) = (1 + \sin^2 x)(1 - 0) - \cos^2 x(-1 - 0) + 4\sin 4x(0 - (-1))$ f(x) = 1 + sin^2 x + cos^2 x + 4sin 4x$f(x) = 1 + \sin^2 x + \cos^2 x + 4\sin 4x$ Since sin^2 x + cos^2 x = 1$\sin^2 x + \cos^2 x = 1$, we get: f(x) = 2 + 4sin 4x$f(x) = 2 + 4\sin 4x$ ### Step 2: Find Maximum and Minimum Values The range of sin 4x$\sin 4x$ is [-1, 1]$[-1, 1]$. M = 2 + 4(1) = 6$M = 2 + 4(1) = 6$ m = 2 + 4(-1) = -2$m = 2 + 4(-1) = -2$ ### Step 3: Calculate M^4 - m^4$M^4 - m^4$ M^4 - m^4 = 6^4 - (-2)^4 = 1296 - 16 = 1280$M^4 - m^4 = 6^4 - (-2)^4 = 1296 - 16 = 1280$ ### Pattern Recognition Look for repeated structures or cyclic additions in rows. Subtracting rows quickly creates zeros, reducing complex trigonometric matrices into elementary algebraic expressions. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices and Determinants Class 11 Mathematics: Trigonometric Functions