Question

The maximum value of
$f(x)=\begin{vmatrix}\sin ^{2} x & 1+\cos ^{2} x & \cos 2 x \\ 1+\sin ^{2} x & \cos ^{2} x & \cos 2 x \\ \sin ^{2} x & \cos ^{2} x & \sin 2 x\end{vmatrix} x \in R$ is

• A. $\sqrt{7}$
• B. $\frac{3}{4}$
• C. $\sqrt{5}$
• D. 5

Answer 1

Answer: (C)

$C _{1}+ C _{2} \rightarrow C _{1}$
$\begin{vmatrix}2 & 1+\cos ^{2} x & \cos 2 x \\ 2 & \cos ^{2} x & \cos 2 x \\ 1 & \cos ^{2} x & \sin 2 x\end{vmatrix}$
$R _{1}- R _{2} \rightarrow R _{1}$
$\begin{vmatrix}0 & 1 & 0 \\ 2 & \cos ^{2} x & \cos 2 x \\ 1 & \cos ^{2} x & \sin 2 x\end{vmatrix}$
Open w.r.t. $R _{1}$
$-(2 \sin 2 x-\cos 2 x)$
$\cos 2 x-2 \sin 2 x=f(x)$
$\left.f(x)\right|_{\max }=\sqrt{1+4}=\sqrt{5}$