The number of values of $k$ for which the points $A(-4, 9, k)$,$B(-1, 6, k)$,and $C(0, 7, 10)$ form a right-angled isosceles triangle is:

If the vertices of a triangle are $(1, 2, 3)$,$(2, 3, 1)$,and $(3, 1, 2)$,and if $H, G, S$,and $I$ respectively denote its orthocenter,centroid,circumcenter,and incenter,then $H+G+S+I$ is equal to:

If the orthocentre and the centroid of a triangle are $(-3,5,2)$ and $(3,3,4)$ respectively,then its circumcentre is

Tetrahedron $ABCD$ has side lengths $AB = CD = 12$. These edges are perpendicular to each other. Let $E$ and $F$ be the midpoints of $AB$ and $CD$ respectively. Given that $EF = 10$ and $EF$ is perpendicular to both $AB$ and $CD$,find the volume of the tetrahedron $ABCD$.

If $\alpha$ is the angle between any two diagonals of a cube and $\beta$ is the angle between a diagonal of a cube and a diagonal of its face,which intersects this diagonal of the cube,then $\cos \alpha + \cos^2 \beta =$

$A$ 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 $\sqrt{6}+\sqrt{13}+\sqrt{11}$