Mathematics - Determinant Question with Solution | TestHub

$\begin{array}{rl}& \left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 4x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 4x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 4x\end{array}\right|,x\in R\\ & R_2\to R_2-R_1\&R_3\to R_3\to R_1\\ & f(x)\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4\sin 4x\\ -1& 1& 0\\ -1& 0& 1\end{array}\right|\end{array}$

Expand about ${\mathrm{R}}_1$, use get
$f(x)=2+4\sin 4x$
$\therefore \mathrm{M}=$ max value of $\mathrm{f}(\mathrm{x})=6$
$m=\min$ value of $f(x)=-2$
$\therefore {\mathrm{m}}^4-{\mathrm{M}}^4=1280$