A) \[-2\]
B) \[-1\]
C) \[0\]
D) \[1\]
Correct Answer: C
Solution :
[c] \[\Delta =\left| \begin{matrix} {{\cos }^{2}}54{}^\circ & {{\cos }^{2}}36{}^\circ & \cot \,135{}^\circ \\ {{\sin }^{2}}53{}^\circ & \cot \,\,135{}^\circ & {{\sin }^{2}}37{}^\circ \\ \cot \,\,135{}^\circ & {{\cos }^{2}}25{}^\circ & {{\cos }^{2}}65{}^\circ \\ \end{matrix} \right|\] \[=\left| \begin{matrix} {{\cos }^{2}}54{}^\circ & {{\sin }^{2}}54{}^\circ & -1 \\ {{\cos }^{2}}37{}^\circ & -1 & {{\sin }^{2}}37{}^\circ \\ -1 & {{\cos }^{2}}25{}^\circ & {{\sin }^{2}}25{}^\circ \\ \end{matrix} \right|\] \[{{C}_{1}}\to {{C}_{1}}+{{C}_{2}}+{{C}_{3}}\] \[=\left| \begin{matrix} {{\cos }^{2}}54{}^\circ +{{\sin }^{2}}54{}^\circ -1 & {{\sin }^{2}}54{}^\circ & -1 \\ {{\cos }^{2}}37{}^\circ -1+{{\sin }^{2}}37{}^\circ & -1 & {{\sin }^{2}}37{}^\circ \\ -1+{{\cos }^{2}}25{}^\circ +{{\sin }^{2}}25{}^\circ & {{\cos }^{2}}25{}^\circ & {{\sin }^{2}}25{}^\circ \\ \end{matrix} \right|\] \[=\left| \begin{align} & \begin{matrix} 0 & {{\sin }^{2}}\,54{}^\circ -1 \\ \end{matrix} \\ & \begin{matrix} 0 & -1{{\sin }^{2}}37{}^\circ \\ \end{matrix} \\ & \begin{matrix} 0 & {{\cos }^{2}}25{}^\circ {{\sin }^{2}}25{}^\circ \\ \end{matrix} \\ \end{align} \right|=0\]
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