Let $L_1 = \frac{x-1}{-3} = \frac{y-2}{\frac{2 \lambda}{7}} = \frac{z-3}{2}$
and $L_2 = \frac{x-1}{-\frac{3 \lambda}{7}} = \frac{y-5}{1} = \frac{z-6}{-5}$
If the lines are perpendicular then
$-3 \times \frac{-3 \lambda}{7} + \frac{2\lambda}{7} \times 1 + 2 \times -5 = 0$
$\frac{9 \lambda}{7} + \frac{2 \lambda}{7} -10 = 0$
$\implies \frac{11 \lambda}{7} = 10$
$\implies \lambda = \frac{70}{11}$