If the minimum and the maximum values of the function
$f:\left[\frac{\pi}{4}, \frac{\pi}{2}\right] \rightarrow \mathrm{R}$, defined by
$f(\theta)=\left|\begin{array}{ccc}-\sin ^{2} \theta & -1-\sin ^{2} \theta & 1 \\ -\cos ^{2} \theta & -1-\cos ^{2} \theta & 1 \\ 12 & 10 & -2\end{array}\right|$ are $m$ and $M$
respectively, then the ordered pair $(m, M)$ is equal to :
Correct Option: , 2
Applying $C_{2} \rightarrow C_{2}-C_{1}$
$f(\theta)=\left|\begin{array}{rrc}-\sin ^{2} \theta & -1 & 1 \\ -\cos ^{2} \theta & -1 & 1 \\ 12 & -2 & -2\end{array}\right|$
$=4\left(\cos ^{2} \theta-\sin ^{2} \theta\right)$
$=4 \cos 2 \theta, \theta \in\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$
Max. $f(\theta)=M=0$
Min. $f(\theta)=m=-4$
So, $(m, M)=(-4,0)$
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