The equation of the plane which is parallel to the $y$-axis and cuts off intercepts of length $2$ and $3$ from the $x$-axis and $z$-axis respectively is:

$A$ plane meets the coordinate axes at the points $A, B, C$ respectively in such a way that the centroid of $\triangle ABC$ is $(1, r, r^2)$ for some real $r$. If the plane passes through the point $(5, 5, -12)$, then $r=$

The plane $\frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 1$ cuts the axes at points $A, B,$ and $C$. Find the area of $\Delta ABC$.

$P_1$ and $P_2$ are two distinct and intersecting planes. Three non-collinear points lie on $P_1$ and another three non-collinear points lie on $P_2$ (none being on the line of intersection of the planes). Then the maximum number of tetrahedrons formed using these six points is:

If the plane $7x + 11y + 13z = 3003$ meets the coordinate axes at $A, B, C$,then the centroid of the $\triangle ABC$ is

The equations $x + y + z + 2 = 0$ and $x + y + z + 3 = 0$ together represent what in space?