The value of $\begin{vmatrix} x & x+y & x+2y \\ x+2y & x & x+y \\ x+y & x+2y & x \end{vmatrix}$ is

$9y^2(x+y)$

The correct answer is Option (2) → $9y^2(x+y)$ ##

We have, $\begin{vmatrix} x & x+y & x+2y \\ x+2y & x & x+y \\ x+y & x+2y & x \end{vmatrix}$

On expanding along $R_1$, we get

$= x[x^2 - (x+2y)(x+y)] - (x+y)[x(x+2y) - (x+y)^2] + x+2y[(x+2y)^2 - x(x+y)]$

$= x[x^2 - x^2 - xy - 2xy - 2y^2] - (x+y)[x^2 + 2xy - x^2 - y^2 - 2xy] + (x+2y)[x^2 + 4y^2 + 4xy - x^2 - xy]$

$= x[-3xy - 2y^2] - (x+y)[-y^2] + (x+2y)[4y^2 + 3xy]$

$= -3x^2y - 2xy^2 + xy^2 + y^3 + 4xy^2 + 3x^2y + 8y^3 + 6xy^2$

$= 9y^3 + 9xy^2 = 9y^2(x+y)$