The value of $\begin{vmatrix}1&x&y\\1&x+y&y\\1&x&x+y\end{vmatrix}$ is

$xy$

The correct answer is Option (3) → $xy$

Given determinant:

$\left| \begin{array}{ccc} 1 & x & y \\ 1 & x + y & y \\ 1 & x & x + y \end{array} \right|$

Apply row operation: $R_2 \to R_2 - R_1$, $R_3 \to R_3 - R_1$

$= \left| \begin{array}{ccc} 1 & x & y \\ 0 & y & 0 \\ 0 & 0 & x \end{array} \right|$

Now take determinant of upper triangular matrix:

$= 1 \cdot y \cdot x = xy$