A) \[-(a+b)\]
B) \[-(b+c)\]
C) \[-a\]
D) \[-(a+b+c)\]
Correct Answer: D
Solution :
\[\left| \,\begin{matrix} x+a & b & c \\ b & x+c & a \\ c & a & x+b \\ \end{matrix}\, \right|=0\] \[\Rightarrow \] \[(x+a+b+c)\,\left| \,\begin{matrix} 1 & b & c \\ 1 & x+c & a \\ 1 & a & x+b \\ \end{matrix}\, \right|=0\] , \[=a\{{{a}^{2}}+ab-2{{a}^{2}}-ab]=-{{a}^{3}}=i\] \[\Rightarrow \] \[x=-(a+b+c)\] is one of the root of the equation.
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