Two spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 81 , x ^ { 2 } + ( y - 5 ) ^ { 2 } + z ^ { 2 } = 56$ are denoted by $S _ { 1 } , S _ { 2 }$ respectively. Let P be a point on the circle formed by the intersection of the two spheres $S _ { 1 } , S _ { 2 }$, and let $\mathrm { P } ^ { \prime }$ be the orthogonal projection of point P onto the $xy$-plane. Let Q and R be the points where the sphere $S _ { 1 }$ intersects the $y$-axis. Find the maximum volume of the tetrahedron $\mathrm { PQP } ^ { \prime } \mathrm { R }$. [4 points]

In coordinate space, one face ABC of a regular tetrahedron ABCD lies on the plane $2 x - y + z = 4$, and the vertex D lies on the plane $x + y + z = 3$. When the centroid of triangle ABC has coordinates $( 1,1,3 )$, what is the length of one edge of the regular tetrahedron ABCD? [4 points]
(1) $2 \sqrt { 2 }$
(2) 3
(3) $2 \sqrt { 3 }$
(4) 4
(5) $3 \sqrt { 2 }$

In coordinate space, there is a sphere $$S : ( x - 2 ) ^ { 2 } + ( y - \sqrt { 5 } ) ^ { 2 } + ( z - 5 ) ^ { 2 } = 25$$ with center $\mathrm { C } ( 2 , \sqrt { 5 } , 5 )$ passing through point $\mathrm { P } ( 0,0,1 )$. For a point Q moving on the circle formed by the intersection of sphere $S$ and plane OPC, and a point R moving on sphere $S$, let $\mathrm { Q } _ { 1 }$ and $\mathrm { R } _ { 1 }$ be the orthogonal projections of points $\mathrm { Q }$ and $\mathrm { R }$ onto the xy-plane respectively.
For two points $\mathrm { Q } , \mathrm { R }$ that maximize the area of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$, the area of the orthogonal projection of triangle $\mathrm { OQ } _ { 1 } \mathrm { R } _ { 1 }$ onto plane PQR is $\frac { q } { p } \sqrt { 6 }$. Find the value of $p + q$. [4 points]

In rectangular parallelepiped $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = B C = 2$, and the angle between $A C _ { 1 }$ and plane $B B _ { 1 } C _ { 1 } C$ is $30 ^ { \circ }$. Then the volume of the rectangular parallelepiped is
A. 8
B. $6 \sqrt { 2 }$
C. $8 \sqrt { 2 }$
D. $8 \sqrt { 3 }$

A regular tetrahedron has all its vertices on a sphere of radius $R$. Then the length of each edge of the tetrahedron is
(a) $( \sqrt{2} / \sqrt{3} ) R$
(b) $( \sqrt{3} / 2 ) R$
(c) $( 4 / 3 ) R$
(d) $( 2 \sqrt{2} / \sqrt{3} ) R$