As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions. (가) The sphere $S$ is tangent to both the cylinder and the cone. (나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$.
When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]

In triangle ABC, $\overline { \mathrm { AB } } = 1$, $\angle \mathrm { A } = \theta$, and $\angle \mathrm { B } = 2 \theta$. Point D on side AB is chosen so that $\angle \mathrm { ACD } = 2 \angle \mathrm { BCD }$. When $\lim _ { \theta \rightarrow + 0 } \frac { \overline { \mathrm { CD } } } { \theta } = a$, find the value of $27 a ^ { 2 }$. (Given that $0 < \theta < \frac { \pi } { 4 }$.) [4 points]

19. As shown in the figure, in the triangular pyramid P–ABC, $\mathrm { PA } \perp$ plane $\mathrm { ABC } , PA = 1 , AB = 1 , AC = 2 , \angle B A C = 60 ^ { \circ }$.
(1) Find the volume of the triangular pyramid P–ABC;
(2) Prove: There exists a point M on the line segment PC such that $\mathrm { AC } \perp \mathrm { BM }$, and find the value of $\frac { P M } { M C }$. [Figure]

Given a regular triangular frustum $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$ with volume $\frac { 52 } { 3 }$, $A B = 6$, $A _ { 1 } B _ { 1 } = 2$, then the tangent of the angle between $A _ { 1 } A$ and plane $A B C$ is ( )
A. $\frac { 1 } { 2 }$
B. 1
C. 2
D. 3

The angle of elevation of the top $P$ of a vertical tower $P Q$ of height 10 from a point $A$ on the horizontal ground is $45 ^ { \circ }$. Let $R$ be a point on $A Q$ and from a point $B$, vertically above $R$, the angle of elevation of $P$ is $60 ^ { \circ }$. If $\angle B A Q = 30 ^ { \circ } , A B = d$ and the area of the trapezium $P Q R B$ is $\alpha$, then the ordered pair ( $d , \alpha$ ) is
(1) $( 10 ( \sqrt { 3 } - 1 ) , 25 )$
(2) $\left( 10 ( \sqrt { 3 } - 1 ) , \frac { 25 } { 2 } \right)$
(3) $( 10 ( \sqrt { 3 } + 1 ) , 25 )$
(4) $\left( 10 ( \sqrt { 3 } + 1 ) , \frac { 25 } { 2 } \right)$

A rectangular frame with side lengths 30 and 40 units is hung on a wall with nails at four corners as shown in Figure 1, with side AB parallel to the ground and at height $h$ units from the ground. Then, except for the nail at corner A, the other nails loosen and fall, and the frame rotating around corner A comes to rest as shown in Figure 2 with all corners touching the wall when corner C touches the ground.
If the heights of corners B and D from the ground are equal in this equilibrium position, what is $h$ in units?
A) 42
B) 48
C) 54
D) 60
E) 64