The maximum value of

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:

  1. $\sqrt{7}$

  2. $\frac{3}{4}$

  3. $\sqrt{5}$

  4. 5


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. $\mathrm{R}_{1}$

$-(2 \sin 2 x-\cos 2 x)$

$\cos 2 x-2 \sin 2 x=f(x)$

$\left.\mathrm{f}(\mathrm{x})\right|_{\max }=\sqrt{1+4}=\sqrt{5}$

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