Find the value of $\lambda$ such that the line $\frac{x-2}{6}=\frac{y-1}{\lambda }=\frac{z+5}{-4}$ is perpendicular to the plane $3x-y-2z=7$.

Let the given line is

$\frac{x-2}{6}=\frac{y-1}{\lambda }=\frac{z+5}{-4}$

Direction ratios are $6,\lambda ,-4$

And the given plane

$3x-y-2z=7$

Its direction ratios are

$3,-1,-2$

And, the given line and plane are perpendicular

then,

$6\left(3\right)+\lambda (-1)+(-4)(-2)=0$

$18-\lambda +8=0$

$\lambda =26$

Hence,

the value of $\lambda$ is $26$.