$\text{ If the two lines }\mathrm{x}+2\mathrm{y}+\mathrm{z}-5=0=\mathrm{x}+\mathrm{y}+\mathrm{z}-4\text{ and }2\mathrm{x}-\mathrm{y}+\mathrm{z}-3=0=\mathrm{x}-\mathrm{z}+\mathrm{k}\text{ are co- planer}$

$\text{ and the equation of the plane containing the two lines is }\mathrm{ax}+b\text{y}+\mathrm{cz}=7\text{ then }$$\text{ the value of }\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{k}\text{ equals }$

Solving first three planes P.I = (1,1,2) which also lines on 4th plane $\Rightarrow 1-2+k=0\Rightarrow k=1$

$\left.\begin{array}{l}{\mathrm{L}}_1\text{ drs }:1,0,-1\\ {\mathrm{L}}_2\text{ drs: }1,3,1\end{array}\right\}\text{ Plane normal drs: }3,-2,3$

$\text{ Equation of plane }3x-2y+3z=7$

$\left.\begin{array}{l}\mathrm{a}=3\\ \mathrm{b}=-2\\ \mathrm{c}=3\end{array}\right\}\text{ also }\mathrm{k}=1\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{k}=5$