11. In the polar coordinate system, the distance from the point $\left( 2 , \frac { \pi } { 3 } \right)$ to the line $\rho ( \cos \theta + \sqrt { 3 } \sin \theta ) = 6$ is $\_\_\_\_$.

12. Let variables $\mathrm { x } , \mathrm { y }$ satisfy the constraints $\left\{ \begin{array} { c } x + y \leq 4 \\ x - y \leq 2 \\ 3 x - y \geq 0 \end{array} \right.$. Then the maximum value of $3 x + y$ is $\_\_\_\_$.

As shown in the figure, the set of points consisting of $\triangle ABC$ and its interior is denoted as $D$. For any point $P(x, y)$ in $D$, the maximum value of $z = 2x + 3y$ is

267 students in the third year participated in the final examination. The ranking of Chinese, mathematics, and total scores of 37 students in a certain class in the entire grade is shown below. A, B, and C are three students in this class.\n\nBased on this examination score,\n(1) Between students A and B, the student whose Chinese score ranking is ahead of their total score ranking is\n(2) In the two subjects of Chinese and mathematics, the subject in which both students' score rankings are more advanced is

14. In the triangular prism $A B C - A _ { 1 } B _ { 1 } C _ { 1 }$, $\angle B A C = 90 ^ { \circ }$. Both the front view and side view are squares with side length 1, and the top view is an isosceles right triangle with legs of length 1. Let $M , N , P$ be the midpoints of edges $A B , B C , B _ { 1 } C _ { 1 }$ respectively. Then the volume of the triangular pyramid $P - A _ { 1 } M N$ is \_\_\_\_.

14. If real numbers $x , y$ satisfy $x ^ { 2 } + y ^ { 2 } \leq 1$ , then the minimum value of $| 2 x + y - 2 | + | 6 - x - 3 y |$ is $\_\_\_\_$ .
[Figure]
(Figure for Question 13)

15. A binary code is a string of digits $x _ { 1 } x _ { 2 } \cdots x _ { n }$ ($n \in \mathbb{N} ^ { * }$) composed of 0 and 1, where $x _ { k }$ ($k = 1,2, \cdots, n$) is called the $k$-th bit code element. Binary codes are commonly used in communication, but during the communication process [Figure]code element errors sometimes occur (that is, a code element changes from 0 to 1, or from 1 to 0).
A certain type of binary code $x _ { 1 } x _ { 2 } \cdots x _ { 7 }$ satisfies the following system of check equations: $\left\{ \begin{array} { l } x _ { 4 } \oplus x _ { 5 } \oplus x _ { 6 } \oplus x _ { 7 } = 0 , \\ x _ { 2 } \oplus x _ { 3 } \oplus x _ { 6 } \oplus x _ { 7 } = 0 , \\ x _ { 1 } \oplus x _ { 3 } \oplus x _ { 5 } \oplus x _ { 7 } = 0 , \end{array} \right.$ where the operation $\oplus$ is defined as: $0 \oplus 0 = 0$, $0 \oplus 1 = 1$, $1 \oplus 0 = 1$, $1 \oplus 1 = 0$. It is now known that such a binary code became 1101101 after a code element error occurred at the $k$-th position during transmission. Using the above system of check equations, we can determine that $k$ equals $\_\_\_\_$.

16. (This question is worth 12 points) This question has three optional parts I, II, and III. Please select any two to answer and write your solutions in the corresponding answer areas on the answer sheet. If you answer all three, only the first two will be graded. I (This question is worth 6 points) Elective 4-1: Geometric Proof As shown in Figure 5, in circle O, two chords AB and CD intersect at point E, with midpoints M and N respectively. The line MO intersects line CD at point F. Prove: (I) $\angle \mathrm { MEN } + \angle \mathrm { NOM } = 180 ^ { \circ }$; (II) $\mathrm { FE } \cdot \mathrm { FN } = \mathrm { FM } \cdot \mathrm { FO }$
[Figure]
Figure 5
II. (This question is worth 6 points) Elective 4-4: Coordinate Systems and Parametric Equations Given the line $l: \left\{ \begin{array} { l } x = 5 + \frac { \sqrt { 3 } } { 2 } t \\ y = \sqrt { 3 } + \frac { 1 } { 2 } t \end{array} \right.$ (where t is the parameter). With the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve C is $\rho = 2 \cos \theta$
(i) Convert the polar equation of curve C to rectangular coordinates; (II) Let the rectangular coordinates of point M be $( 5 , \sqrt { 3 } )$. The line $l$ intersects curve C at points $A$ and $B$. Find the value of $| M A | \cdot | M B |$ III. (This question is worth 6 points) Elective 4-5: Inequalities Let $\mathrm { a } > 0$, $\mathrm { b } > 0$, and $\mathrm { a } + \mathrm { b } = \frac { 1 } { a } + \frac { 1 } { b }$. Prove
(i) $\mathrm { a } + \mathrm { b } \geqslant 2$;
(ii) $\mathrm { a } ^ { 2 } + \mathrm { a } < 2$ and $\mathrm { b } ^ { 2 } + \mathrm { b } < 2$ cannot both be true.

17. (This question is worth 14 points) As shown in the figure, in the quadrangular pyramid $A - E F C B$, $\triangle A E F$ is an equilateral triangle, plane $A E F \perp$ plane $E F C B$, $E F / / B C$, $B C = 4$, $E F = 2 a$, $\angle E B C = \angle F C B = 60 ^ { \circ }$, and $O$ is the midpoint of $E F$. (I) Prove: $A O \perp B E$; (II) Find the cosine of the dihedral angle $F - A E - B$; (III) If $B E \perp$ plane $A O C$, find the value of $a$. [Figure]

As shown in the figure, in the triangular pyramid $E - ABC$, plane $EAB \perp$ plane $ABC$, triangle $EAB$ is equilateral, $AC \perp BC$, and $AC = BC = \sqrt { 2 }$. $O$ and $M$ are the midpoints of $AB$ and $EA$ respectively.\n(1) Prove that $EB \parallel$ plane $MOC$.\n(2) Prove that plane $MOC \perp$ plane $EAB$.\n(3) Find the volume of the triangular pyramid $E - ABC$.

19. (This question is worth 13 points) As shown in Figure 6, a frustum of a pyramid $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ {

19. (This question is worth 12 points). In the rectangular prism $A B C D - A _ { 1 } B _ { 1 } C _ { 1 } D _ { 1 }$, $A B = 16 , B C = 10 , A A _ { 1 } = 8$. Points $E$ and $F$ are on $A _ { 1 } B _ { 1 }$ and $D _ { 1 } C _ { 1 }$ respectively, with $A _ { 1 } E = D _ { 1 } F = 4$. A plane $\alpha$ passes through points $E$ and $F$ and intersects the faces of the rectangular prism, with the intersection lines forming a square. [Figure] (I) Draw this square in the figure (no need to explain the method or reasoning); (II) Find the ratio of the volumes of the two parts that plane $\alpha$ divides the rectangular prism into.

As shown in the figure for question (21), [content incomplete in source document]