If the matrix

Given: The matrix $A=\left[\begin{array}{ccc}1 & 3 & x+2 \\ 2 & 4 & 8 \\ 3 & 5 & 10\end{array}\right]$ is singular

$A$ is singular $\Rightarrow|A|=0$ Thus,

$\left|\begin{array}{ccc}1 & 3 & \mathrm{x}+2 \\ 2 & 4 & 8 \\ 3 & 5 & 10\end{array}\right|=0$

$\Rightarrow 1(40-40)-3(20-24)+(\mathrm{x}+2)(10-12)=0$

$\Rightarrow 1(0)-3(-4)+(\mathrm{x}+2)(-2)=0$

$\Rightarrow 12-2 \mathrm{x}-4=0$

$\Rightarrow 8-2 \mathrm{x}=0$

$\Rightarrow 2 \mathrm{x}=8$

$\Rightarrow \mathrm{x}=4$

Hence, $x=\underline{4}$.