Question

Let $f\left(x\right)=\left|\begin{array}{ccc}1+si{n}^{2}x& co{s}^{2}x& 4sin2x\\ si{n}^{2}x& 1+co{s}^{2}x& 4sin2x\\ si{n}^{2}x& co{s}^{2}x& 1+4sin2x\end{array}\right|$, then the maximum value of $f\left(x\right)=$

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (C)

6

$f\left(x\right)=\left|\begin{array}{ccc}1+{\sin }^{2}x& {\cos }^{2}x& 4{\sin }^{2}x\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$

$R_1\to R_1-R_2$

$f\left(x\right)=\left|\begin{array}{ccc}1& -1& 0\\ {\sin }^{2}x& 1+{\cos }^{2}x& 4\sin 2x\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$

$R_2\to R_2-R_3$

$f\left(x\right)=\left|\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ {\sin }^{2}x& {\cos }^{2}x& 1+4\sin 2x\end{array}\right|$

$1+4\sin 2x+{\cos }^{2}x+{\sin }^{2}x$

$\Rightarrow 2+4\sin 2x$

MAximum value is attained when $\sin 2x$ is max

$\Rightarrow \sin 2x=1$

Maximum value$=2+4=6$