A) \[0\]
B) \[abc\]
C) \[ab+bc+ca\]
D) \[2abc\]
Correct Answer: A
Solution :
[a] Given, \[\left| \begin{matrix} a & \cot \,\,A/2 & \lambda \\ b & \cot \,\,B/2 & \mu \\ c & \cot \,\,c/2 & \gamma \\ \end{matrix} \right|=0\] \[\Rightarrow \left| \begin{matrix} a & \frac{s(s-a)}{\Delta } & \lambda \\ b & \frac{s(s-b)}{\Delta } & \mu \\ c & \frac{s(s-c)}{\Delta } & \gamma \\ \end{matrix} \right|=0\] \[\Rightarrow \frac{1}{r}\left| \begin{matrix} a & s-a & \lambda \\ b & s-b & \mu \\ c & s-c & \gamma \\ \end{matrix} \right|=0\] Apply \[{{C}_{2}}\to {{C}_{2}}+{{C}_{1}}\] \[\Rightarrow \frac{1}{r}\left| \begin{matrix} a & s & \lambda \\ b & s & \mu \\ c & s & \gamma \\ \end{matrix} \right|=0,\] where \[r=\frac{\Delta }{s}\] \[\Rightarrow \frac{\Delta }{{{r}^{2}}}\left| \begin{matrix} a & 1 & \lambda \\ b & 1 & \mu \\ c & 1 & \gamma \\ \end{matrix} \right|=0\] Apply \[{{R}_{1}}\to {{R}_{1}}-{{R}_{2}},{{R}_{2}}\to {{R}_{2}}-{{R}_{3}}\] \[\Rightarrow \frac{\Delta }{{{r}^{2}}}\left| \begin{matrix} a-b & 0 & \lambda -\mu \\ b-c & 0 & \mu -\gamma \\ c & 1 & \gamma \\ \end{matrix} \right|=0\] \[\Rightarrow \frac{\Delta }{{{r}^{2}}}[(b-c)(\lambda -\mu )-(\mu -\gamma )(a-b)]=0\] \[\Rightarrow b(\lambda -\mu )-c(\lambda -\mu )-a(\mu -\gamma )+b(\mu -\gamma )=0\] \[\Rightarrow -a(\mu -\gamma )+b(\lambda -\mu +\mu -\gamma )-c(\lambda -\mu )=0\] \[\Rightarrow -a(\mu -\gamma )+b(\lambda -\gamma )-c(\lambda -\mu )=0\] \[\Rightarrow a(\mu -\gamma )+b(\gamma -\lambda )+c(\lambda -\mu )=0\]
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