The plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1$ cuts the $X$-axis at $A$,$Y$-axis at $B$,and $Z$-axis at $C$. The area of $\triangle ABC$ is:

$A$ plane $\pi$ passes through the points $(5,1,2)$, $(3,-4,6)$, and $(7,0,-1)$. If $p$ is the perpendicular distance from the origin to the plane $\pi$ and $l, m, n$ are the direction cosines of a normal to the plane $\pi$, then $|3l+2m+5n|=$

$A$ plane passing through $(-1, 2, 3)$ and whose normal makes equal angles with the coordinate axes is

$A$ variable plane is at a distance $k$ from the origin and meets the coordinate axes at $A, B, C$. The locus of the centroid of $\Delta ABC$ is . . . . . .

$A$ vector $n$ of magnitude $8$ units is inclined to $x$-axis at $45^\circ$,$y$-axis at $60^\circ$ and an acute angle with $z$-axis. If a plane passes through a point $(\sqrt{2}, -1, 1)$ and is normal to $n$,then its equation in vector form is

The volume of the tetrahedron (in cubic units) formed by the plane $2x + y + z = K$ and the coordinate planes is $\frac{2V^3}{3}$, then $K:V =$