A plane passing through the line of intersection of the given $2$ planes is $\left(x-y+z-2\right)+\mu \left(\lambda x+3z+5\right)=0$
$\Rightarrow x\left(1+\lambda \mu \right)-y+z\left(1+3\mu \right)+5\mu -2=0$
Since the given line $\frac{x-1}{2}=\frac{y}{-1}=\frac{z}{2}$ must lie in this plane
$\Rightarrow 2\left(1+\lambda \mu \right)+\left(-1\right)\left(-1\right)+\left(1+3\mu \right)2=0$
$\Rightarrow 2+2\lambda \mu +1+2+6\mu =0$
$\Rightarrow 2\lambda \mu +6\mu +5=0\dots ..\left(\mathrm{i}\right)$
The point $\left(1,0,0\right)$ must satisfy the equation of the plane
$\Rightarrow 1+\lambda \mu -0+0+5\mu -2=0$
$\Rightarrow \lambda \mu +5\mu -1=0\dots \dots \left(\mathrm{i}\mathrm{i}\right)$
Solving $\left(\mathrm{i}\right)$ and $\left(\mathrm{i}\mathrm{i}\right)$ we get,

$\mu =\frac{7}{4}, \lambda =\frac{-31}{7}$

$\Rightarrow 7\lambda =-31$