Question:
The maximum value of
$f(x)=\left|\begin{array}{ccc}\sin ^{2} x & 1+\cos ^{2} x & \cos 2 x \\ 1+\sin ^{2} x & \cos ^{2} x & \cos 2 x \\ \sin ^{2} x & \cos ^{2} x & \sin 2 x\end{array}\right|, x \in R$ is:
Correct Option: , 3
Solution:
$\mathrm{C}_{1}+\mathrm{C}_{2} \rightarrow \mathrm{C}_{1}$
$\left|\begin{array}{ccc}2 & 1+\cos ^{2} x & \cos 2 x \\ 2 & \cos ^{2} x & \cos 2 x \\ 1 & \cos ^{2} x & \sin 2 x\end{array}\right|$
$\mathrm{R}_{1}-\mathrm{R}_{2} \rightarrow \mathrm{R}_{1}$
$\left|\begin{array}{ccc}0 & 1 & 0 \\ 2 & \cos ^{2} x & \cos 2 x \\ 1 & \cos ^{2} x & \sin 2 x\end{array}\right|$
Open w.r.t. $\mathrm{R}_{1}$
$-(2 \sin 2 x-\cos 2 x)$
$\cos 2 x-2 \sin 2 x=f(x)$
$\left.\mathrm{f}(\mathrm{x})\right|_{\max }=\sqrt{1+4}=\sqrt{5}$