# Evaluate :$\begin{vmatrix}y+z&z&y\\z&z+x&x\\y&x&x+y\end{vmatrix}$

$\begin{array}{1 1} 2xy \\ 4xy \\ 4xyz \\ None\;of\;these \end{array}$

Apply $C_1\rightarrow C_1-(C_2+C_3)$
$\begin{vmatrix}(y+z)-(z+y)&z&y\\z-(z+2x)&z+x&x\\y-(2x+y)&x&x+y\end{vmatrix}$
$\begin{vmatrix}0 &z &y\\-2x&z+x&x\\-2x&x&x+y\end{vmatrix}$
$-2x\begin{vmatrix}0 &z&y\\1 &z+x&x\\1&x&x+y\end{vmatrix}$
Apply $R_2\rightarrow R_2-R_3$
$\Rightarrow -2x\begin{vmatrix}0 &z&y\\0&z&-y\\1 &x&x+y\end{vmatrix}$
$\Rightarrow -2x(1)(-yz-yz)$
$\Rightarrow -2x(-2yz)$
$\Rightarrow 4xyz$
Hence (c) is the correct answer.