Question

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

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

Answer 1

Answer: (C)

$C_1+C_2\to C_1$
$\left|\begin{array}{ccc}2& 1+{\cos }^{2}x& \cos 2x\\ 2& {\cos }^{2}x& \cos 2x\\ 1& {\cos }^{2}x& \sin 2x\end{array}\right|$
$R_1-R_2\to R_1$
$\left|\begin{array}{ccc}0& 1& 0\\ 2& {\cos }^{2}x& \cos 2x\\ 1& {\cos }^{2}x& \sin 2x\end{array}\right|$
Open w.r.t. $R_1$
$-(2\sin 2x-\cos 2x)$
$\cos 2x-2\sin 2x=f(x)$
${f(x)|}_{\max }=\sqrt{1+4}=\sqrt{5}$