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Question
If\[f\left( x \right) = \begin{vmatrix}0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0\end{vmatrix}\]
Options
f(a) = 0
f(b) = 0
f(0) = 0
f(1) = 0
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Solution
Let \[f\left( x \right) = \begin{vmatrix}0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0\end{vmatrix}\]
Now,
\[f\left( a \right) = \begin{vmatrix}0 & a - a & a - b \\ a + a & 0 & a - c \\ a + b & a + c & 0\end{vmatrix}\]
\[ = \begin{vmatrix}0 & 0 & a - b \\ 2a & 0 & a - c \\ a + b & a + c & 0\end{vmatrix}\]
\[ = \left( a - b \right)\left( 2 a^2 + 2ac \right) \neq 0\]
\[f\left( b \right) = \begin{vmatrix}0 & b - a & b - b \\ b + a & 0 & b - c \\ b + b & b + c & 0\end{vmatrix}\]
\[ = \begin{vmatrix}0 & b - a & 0 \\ b + a & 0 & b - c \\ 2a & b + c & 0\end{vmatrix}\]
\[ = \left( b - a \right)\left( 2ab - 2ac \right) \neq 0\]
\[f\left( 0 \right) = \begin{vmatrix}0 & 0 - a & 0 - b \\ 0 + a & 0 & 0 - c \\ 0 + b & 0 + c & 0\end{vmatrix}\]
\[ = \begin{vmatrix}0 & - a & - b \\ a & 0 & - c \\ b & c & 0\end{vmatrix}\]
\[ = a(bc) - b(ac) = 0\]
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