The value of $\begin{vmatrix} a-b & b+c & a \\ b-a & c+a & b \\ c-a & a+b & c \end{vmatrix}$ is

None of these

The correct answer is Option (4) → None of these ##

We have, $\begin{vmatrix} a-b & b+c & a \\ b-a & c+a & b \\ c-a & a+b & c \end{vmatrix}$

On expanding along $R_1$, we get

$(a-b)[c(c+a) - b(a+b)] - (b+c)[c(b-a) - b(c-a)] + a[(a+b)(b-a) - (c+a)(c-a)]$

$= (a-b)[c^2 + ac - ba - b^2] - (b+c)[cb - ac - bc + ab] + a[ab - a^2 + b^2 - ab - c^2 + a^2 + ca - ac]$

$= (a-b)[a(c-b) + (c^2 - b^2)] - (b+c)[ab - ac] + a[b^2 - c^2]$

$= (a-b)[(c-b)(a+c+b)] - a(b+c)(b-c) + a(b+c)(b-c)$

$= (c-b)(a-b)(a+b+c)$