If the equation of the plane passing through the point $(3,2,5)$ and perpendicular to the planes $2x-3y+5z=7$ and $5x+2y-3z=11$ is $x+by+cz+d=0$,then $2b+3c+d=$

The Cartesian equation of the plane,passing through the points $(3,1,1)$,$(1,2,3)$ and $(-1,4,2)$,is

If the angle between the planes $\vec{r} \cdot(m \hat{i}-\hat{j}+2 \hat{k})+3=0$ and $\vec{r} \cdot(2 \hat{i}-m \hat{j}+\hat{k})-5=0$ is $\frac{\pi}{3}$,then $m=$

The Cartesian equation of the plane passing through the point $(3, -2, -1)$ and parallel to the vectors $\vec{b} = \hat{i} - 2\hat{j} + 4\hat{k}$ and $\vec{c} = 3\hat{i} + 2\hat{j} - 5\hat{k}$ is:

$A$ variable plane is at a distance of $6$ units from the origin. If it meets the coordinate axes in $A, B$, and $C$, then the equation of the locus of the centroid of the $\triangle ABC$ is

If the plane $\frac{x}{2}+\frac{y}{3}+\frac{z}{6}=1$ cuts the coordinate axes at points $A, B, C$ respectively,then the area of the triangle $ABC$ is