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(A) Given the determinant $f(x) = \left| \begin{array}{ccc} 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

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$1280$

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(A) Given the determinant $f(x) = \left| \begin{array}{ccc} 1+\sin^2 x & \cos^2 x & 4\sin 4x \ \sin^2 x & 1+\cos^2 x & 4\sin 4x \ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{array} ight|$.Applying row operations $R_2 ightarrow R_2 - R_1$ and $R_3 ightarrow R_3 - R_1$:$f(x) = \left| \begin{array}{ccc} 1+\sin^2 x & \cos^2 x & 4\sin 4x \ -1 & 1 & 0 \ -1 & 0 & 1 \end{array} ight|$.Expanding along the first row $(R_1)$:$f(x) = (1+\sin^2 x)(1 - 0) - (\cos^2 x)(-1 - 0) + (4\sin 4x)(0 - (-1))$.$f(x) = 1 + \sin^2 x + \cos^2 x + 4\sin 4x$.Since $\sin^2 x + \cos^2 x = 1$,we get $f(x) = 1 + 1 + 4\sin 4x = 2 + 4\sin 4x$.The maximum value $M$ occurs when $\sin 4x = 1$,so $M = 2 + 4(1) = 6$.The minimum value $m$ occurs when $\sin 4x = -1$,so $m = 2 + 4(-1) = -2$.Therefore,$M^4 - m^4 = 6^4 - (-2)^4 = 1296 - 16 = 1280$.

Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x) = \left| \begin{array}{ccc} 1+\sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1+\cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1+4\sin 4x \end{array} \right|$,$x \in R$. Then $M^4 - m^4$ is equal to:

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