A) \[uv+vw+wu=0\]
B) \[uv+vw+wu=3\]
C) \[uv+vw+wu=2\]
D) \[uv+ww+wu=1\]
Correct Answer: A
Solution :
\[\left| \begin{matrix} 3{{u}^{2}} & 2{{u}^{3}} & 1 \\ 3{{v}^{2}} & 2{{v}^{3}} & 1 \\ 3{{w}^{2}} & 2{{w}^{3}} & 1 \\ \end{matrix} \right|=0\] \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\]and\[{{R}_{2}}\to {{R}_{2}}-{{R}_{3}}\] \[\left| \begin{matrix} {{u}^{2}}-{{v}^{2}} & {{u}^{3}}-{{v}^{3}} & 0 \\ {{v}^{2}}-{{w}^{2}} & {{v}^{3}}-{{w}^{3}} & 0 \\ {{w}^{2}} & {{w}^{3}} & 1 \\ \end{matrix} \right|=0\] \[\Rightarrow \] \[\left| \begin{matrix} u+v & {{u}^{2}}+{{v}^{2}}+vu & 0 \\ v+w & {{v}^{2}}+{{w}^{2}}+vw & 0 \\ {{w}^{2}} & {{w}^{3}} & 1 \\ \end{matrix} \right|=0\] \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}}\] \[\left| \begin{matrix} u-w & ({{u}^{2}}-{{w}^{2}})+v(u-w) & 0 \\ v+w & {{v}^{2}}+{{w}^{2}}+vw & 0 \\ {{w}^{2}} & {{w}^{3}} & 0 \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[\left| \begin{matrix} 1 & u+w+v & 0 \\ v+w & {{v}^{2}}+{{w}^{2}}+vw & 0 \\ {{w}^{2}} & {{w}^{3}} & 0 \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[({{v}^{2}}+{{w}^{2}}+vw)-(v+w)[(v+w)+u]=0\] \[\Rightarrow \] \[{{v}^{2}}+{{w}^{2}}+vw={{(v+w)}^{2}}+u(v+w)\] \[\Rightarrow \] \[uv+vw+wu=0\]
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