$\text{ For which of the following ordered pairs }(\mu ,\delta ),\text{ the system of linear equations }$

$x+2y+3z=1, 3x+4y+5z=\mu and 4x+4y+4z=\delta$$is inconsistent?$

$\text{ Note }D=\left|\begin{array}{lll}3& 4& 5\\ 1& 2& 3\\ 4& 4& 4\end{array}\right|\left(R_3\to R_3-2R_1+3R_2\right)=\left|\begin{array}{lll}3& 4& 5\\ 1& 2& 3\\ 0& 0& 0\end{array}\right|=0$

$\begin{array}{l}\text{ Now let }{\mathrm{P}}_3=4\mathrm{x}+4\mathrm{y}+4\mathrm{z}-\delta =0\text{ . If the system has solutions it will have infinite }\\ \text{ solution, so }{P}_3=\alpha P_1+\beta P_2\end{array}$

$\begin{array}{l}\text{ Hence }3\alpha +\beta =4\text{ \& }4\alpha +2\beta =4\Rightarrow \alpha =2\&\beta =-2\\ \text{ So for infinite solution }2\mu -2=\delta \Rightarrow \text{ for }2\mu \ne \delta +2\end{array}$

system inconsistent