Let $A(3, 0, -1)$,$B(2, 10, 6)$,and $C(1, 2, 1)$ be the vertices of a triangle and $M$ be the midpoint of $AC$. If $G$ divides $BM$ in the ratio $2 : 1$,then $\cos(\angle GOA)$ ($O$ being the origin) is equal to

Find the reflection of the point $P(2, -1, 3)$ in the plane $3x - 2y - z = 9$.

$A(2,3,-4), B(-3,3,-2), C(-1,4,2)$ and $D(3,5,1)$ are the vertices of a tetrahedron. If $E, F, G$ are the centroids of its faces containing the point $A$, then the centroid of the triangle $EFG$ is

The incenter of the triangle $ABC$,whose vertices are $A(0,2,1)$,$B(-2,0,0)$,and $C(-2,0,2)$ is

$A$ straight line drawn from the point $P(1,3,2)$,parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$,intersects the plane $L_1: x-y+3z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2: 2x-y+z=-4$ at the point $R$. Then which of the following statements is(are) $TRUE$?
$(A)$ The length of the line segment $PQ$ is $\sqrt{6}$
$(B)$ The coordinates of $R$ are $(1,6,0)$
$(C)$ The centroid of the triangle $PQR$ is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
$(D)$ The perimeter of the triangle $PQR$ is $\sqrt{6}+\sqrt{13}+\sqrt{11}$

The incentre of the triangle whose vertices are $P(0,3,0)$,$Q(0,0,4)$,and $R(0,3,4)$ is