The equations $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$  have unique solution if 

Augmented matrix is 

$\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 10\\ 1& 2& \lambda & \mu \end{array}\right]$

$\begin{array}{c}{R}_2-R_1\\ R_3-R_1\\ ~\end{array}\left[\begin{array}{cccc}1& 1& 1& 6\\ 0& 1& 2& 4\\ 0& 1& \lambda -1& \mu -6\end{array}\right]$

$\begin{array}{c}{R}_3-R_2\\ ~\end{array}\left[\begin{array}{cccc}1& 1& 1& 6\\ 0& 1& 2& 4\\ 0& 0& \lambda -3& \mu -10\end{array}\right]$

$\therefore \text{ Given equations have unique solution }\Rightarrow \lambda -3\ne 0\Rightarrow \lambda \ne 3$