The perpendicular distance from the origin to the plane containing the points $A(1, -2, 1)$, $B(2, -1, -3)$ and $C(0, 1, 5)$ is (in units)

$P$ is a fixed point $(a, a, a)$ on a line through the origin equally inclined to the axes. Then,any plane through $P$ perpendicular to $OP$ makes intercepts on the axes,the sum of whose reciprocals is equal to:

If $\alpha$ is the acute angle between the planes $P_1$ and $P_2$,where the combined equation of the planes $P_1$ and $P_2$ is $2x^2 - 6y^2 - 12z^2 + 18yz + 2zx + xy = 0$,then the value of $\cos \alpha$ is:

If the plane $2x + 3y + z = 6$ cuts the coordinate axes at $A$, $B$, and $C$, then the volume of the tetrahedron $OABC$ (where $O$ is the origin) in cubic units is:

Find the Cartesian equation of the following plane: $\vec{r} \cdot(\hat{i}+\hat{j}-\hat{k})=2$

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $(-\lambda^2, 1, 1), (1, -\lambda^2, 1)$ and $(1, 1, -\lambda^2)$ also passes through the point $(-1, -1, 1)$. Then $S$ is equal to