If lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-5}{1}=\frac{z-6}{-5}$ are mutually perpendicular,then $k$ is equal to

Prove that the lines $x=p y+q, z=r y+s$ and $x=p^{\prime} y+q^{\prime}, z=r^{\prime} y+s^{\prime}$ are perpendicular if $p p^{\prime}+r r^{\prime}+1=0$.

Find the acute angle between the line joining the points $(2, 1, -3)$ and $(-3, 1, 7)$ and the line parallel to $\frac{x - 1}{3} = \frac{y}{4} = \frac{z + 3}{5}$ passing through the point $(-1, 0, 4)$.

The lines $\overrightarrow{r} = (\hat{i} - \hat{j}) + \ell(2\hat{i} + \hat{k})$ and $\overrightarrow{r} = (2\hat{i} - \hat{j}) + m(\hat{i} + \hat{j} - \hat{k})$:

If the straight lines $\vec{r} = (1, 2, 3) + k(\lambda, 2, 3), k \in R$ and $\vec{r} = (2, 3, 1) + k(3, \lambda, 2), k \in R$ intersect at a point,then the integer $\lambda$ is equal to:

The point collinear with $(1, -2, -3)$ and $(2, 0, 0)$ among the following is