If the minimum and the maximum values of the function $f:\left[\frac{\pi }{4},\frac{\pi }{2}\right]\to R$ defined by $f\left(\theta \right)=\left|\begin{array}{ccc}-{\sin }^2\theta & -1-{\sin }^2\theta & 1\\ -{\cos }^2\theta & -1-{\cos }^2\theta & 1\\ 12& 10& -2\end{array}\right|$are m and M respectively, then order pair  (m, M) is equal to

$\begin{array}{c}f\left(\theta \right)=\left|\begin{array}{ccc}-{\sin }^2\theta & -1-{\sin }^2\theta & 1\\ -{\cos }^2\theta & -1-{\cos }^2\theta & 1\\ 12& 10& -2 \end{array}\right|\\ R_1\to R_1+R_2+R_3\\ =\left|\begin{array}{ccc}11& 7& 0\\ -{\cos }^2\theta & -1-{\cos }^2\theta & 1\\ 12& 10& -2\end{array}\right| \\ R_3\to R_3-R_1\\ =\left|\begin{array}{ccc}11& 7& 0\\ -{\cos }^2\theta & -1-{\cos }^2\theta & 1\\ 1& 3& -2\end{array}\right|\\ C_1\to C_1-C_1-C_3\\ \\ =\left|\begin{array}{ccc}4& 7& 0\\ 0& -1-{\cos }^2\theta & 1\\ 0& 3& -2\end{array}\right|\\ =4\left(2+2{\cos }^2\theta -3\right)\\ =4\cos 2\theta \end{array}$

$2\theta \in \left[\frac{\pi }{2},\pi \right]\Rightarrow \cos 2\theta \in \left[-1,0\right]\Rightarrow 4\cos 2\theta \in \left[-4,0\right]$, m=-4 ,M=0