A) 2
B) 4
C) 6
D) 8
Correct Answer: C
Solution :
[c] \[f(x)=\left| \begin{matrix} 1+{{\sin }^{2}}x & {{\cos }^{2}}x & 4\sin 2x \\ {{\sin }^{2}}x & 1+{{\cos }^{2}}x & 4\sin 2x \\ {{\sin }^{2}}x & {{\cos }^{2}}x & 1+4\sin 2x \\ \end{matrix} \right|\] Applying \[{{C}_{1}}\to {{C}_{1}}+{{C}_{2}}\] \[=\left| \begin{matrix} 2 & {{\cos }^{2}}\theta & 4\sin 2x \\ 2 & 1+{{\cos }^{2}}\theta & 4\sin 2x \\ 1 & {{\cos }^{2}}\theta & 1+4\sin 2x \\ \end{matrix} \right|\] (Applying \[{{R}_{2}}\to {{R}_{2}}-{{R}_{1}}\] and \[{{R}_{3}}\to {{R}_{3}}-{{R}_{1}}\]) \[=\left| \begin{matrix} 2 & {{\cos }^{2}}\theta & 4\sin 2x \\ 0 & 1 & 0 \\ -1 & 0 & 1 \\ \end{matrix} \right|\] \[f(x)=2+4\sin 2x\] \[\therefore \,\,\,-1\le \sin 2x\le 1,\] maximum value of \[sin\text{ }2x=1\] Thus, maximum value of \[f(x)=2+4=6\]
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