In coordinate space, let $C$ be the circle formed by the intersection of the sphere $S : x^2 + y^2 + z^2 = 4$ and the plane $\alpha : y - \sqrt{3}z = 2$. For point $\mathrm{A}(0, 2, 0)$ on circle $C$, let $\mathrm{P}$ and $\mathrm{Q}$ be the endpoints of a diameter of circle $C$ such that $\overline{\mathrm{AP}} = \overline{\mathrm{AQ}}$. Let $\mathrm{R}$ be another point where the line passing through $\mathrm{P}$ and perpendicular to plane $\alpha$ meets sphere $S$. If the area of triangle $\mathrm{ARQ}$ is $s$, find the value of $s^2$. [4 points]

In coordinate space, triangle ABC satisfies the following conditions. (가) The area of triangle ABC is 6. (나) The area of the orthogonal projection of triangle ABC onto the $yz$-plane is 3.
What is the maximum area of the orthogonal projection of triangle ABC onto the plane $x - 2 y + 2 z = 1$? [4 points]
(1) $2 \sqrt { 6 } + 1$
(2) $2 \sqrt { 2 } + 3$
(3) $3 \sqrt { 5 } - 1$
(4) $2 \sqrt { 5 } + 1$
(5) $3 \sqrt { 6 } - 2$

In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points]
(1) $\sqrt{43}$
(2) $\sqrt{47}$
(3) $\sqrt{51}$
(4) $\sqrt{55}$
(5) $\sqrt{59}$

Three small colorful starfish are located on the sea floor and are represented in the model by the points $P , Q$ and $R$. The photographer moves for his shots from the location described in the model by the point $K$, parallel to the sea floor. The camera lens points perpendicular to the sea floor and has a cone-shaped field of view with an opening angle of $90 ^ { \circ }$.
Assess whether the photographer can reach a location in this way where he can see all three starfish simultaneously in the camera's field of view.

Show that the edge length of the cube is 12.

The points $( 4,7 , - 1 ) , ( 1,2 , - 1 ) , ( - 1 , - 2 , - 1 )$ and $( 2,3 , - 1 )$ in $\mathbb { R } ^ { 3 }$ are the vertices of a
(A) rectangle which is not a square.
(B) rhombus.
(C) parallelogram which is not a rectangle.
(D) trapezium which is not a parallelogram.

There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. Let $Q$ be a point on $\overline{FC}$ such that $\overrightarrow{AQ}$ is parallel to $\overrightarrow{DF}$. Using the fact that $\triangle ABC$ and $\triangle ACQ$ are congruent triangles, prove that if a horizontal plane $W$ lies between $A$ and $P$ and is at distance $x$ from $A$, then the rectangular region formed by the intersection of $W$ with this wooden block has area $20x + \frac{4}{9}x^2$. (Non-multiple choice question, 4 points)

Below is shown a structure made with two identical rectangular prisms with edge lengths of 2, 3, and 4 units. These prisms are placed adjacent to each other as shown in the figure.
According to this, what is the length of the line segment AB connecting vertices A and B in units?
A) $6 \sqrt { 2 }$
B) $8 \sqrt { 3 }$
C) $5 \sqrt { 5 }$
D) 7
E) 9