A) \[x=a\]
B) \[x=b\]
C) \[x=c\]
D) \[x\]has any value
Correct Answer: D
Solution :
Let \[\Delta =\left| \begin{matrix} x & a & b+c \\ x & b & c+a \\ x & c & a+b \\ \end{matrix} \right|\] Applying \[{{C}_{3}}\to {{C}_{2}}+{{C}_{3}}\] \[=x\left| \begin{matrix} 1 & a & a+b+c \\ 1 & b & a+b+c \\ 1 & c & a+b+c \\ \end{matrix} \right|\] \[=x(a+b+c)\left| \begin{matrix} 1 & a & 1 \\ 1 & b & 1 \\ 1 & c & 1 \\ \end{matrix} \right|\] \[=0\][\[\because \]two columns are identical] Note: If a determinant have any two rows or columns are identical, then values of determinant is zero.
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