if $$a, b, c$$ are three unequal numbers and $$\left| \begin{array}{ccc}0 & x - a & x - b \\x + a & 0 & x - c \\x + b & x + c & 0 \\\end{array}\right| = 0$$ then $$x$$ is​

Correct option is B

Solution:

$$\textbf{Given:} \\\begin{vmatrix}0 & x - a & x - b \\x + a & 0 & x - c \\x + b & x + c & 0\end{vmatrix} = 0\\[10pt]\textbf{Expanding along the first row:} \\\Delta = 0 \cdot \begin{vmatrix}0 & x - c \\x + c & 0\end{vmatrix}- (x - a) \cdot \begin{vmatrix}x + a & x - c \\x + b & 0\end{vmatrix}+ (x - b) \cdot \begin{vmatrix}x + a & 0 \\x + b & x + c\end{vmatrix}\\[10pt]= - (x - a)\left[ - (x + b)(x - c) \right] + (x - b)(x + a)(x + c)\\[10pt]= (x - a)(x + b)(x - c) + (x - b)(x + a)(x + c)\\[10pt]\textbf{Try } x = 0:\\= (-a)(b)(-c) + (-b)(a)(c) = abc - abc = 0\\[10pt]\boxed{x = 0}$$

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