On a number line, there is the origin $O$ and three points $A ( - 2 ) , B ( 10 ) , C ( x )$, where $x$ is a real number. Given that the lengths of segments $\overline { B C } , \overline { A C } , \overline { O B }$ satisfy $\overline { B C } < \overline { A C } < \overline { O B }$, then the maximum range of $x$ is (8) $< x <$ (9).

Let matrix $A = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 1 & 0 \\ 0 & 6 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 } , B = \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { l l } 6 & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$, where $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right] ^ { - 1 }$ is the inverse matrix of $\left[ \begin{array} { c c } 1 & - 2 \\ 0 & 1 \end{array} \right]$. If $A + B = \left[ \begin{array} { l l } a & b \\ c & d \end{array} \right]$, then $a + b + c + d =$ (10)(11).

A biased coin has probability $\frac { 1 } { 3 }$ of showing heads and probability $\frac { 2 } { 3 }$ of showing tails. On a coordinate plane, a game piece moves to the next position based on the result of flipping this coin, according to the following rules: (I) If heads appears, the piece moves from its current position in the direction and distance of vector $( - 1,2 )$ to the next position; (II) If tails appears, the piece moves from its current position in the direction and distance of vector $( 1,0 )$ to the next position. For example: If the game piece is currently at coordinates $( 2,4 )$ and tails appears, the piece moves to coordinates $( 3,4 )$. Suppose the game piece starts at the origin $( 0,0 )$ and, according to the above rules, flips the coin 6 times consecutively, with each flip being independent. After 6 moves, the game piece is most likely to stop at coordinates (12), (13)).

On a coordinate plane, there are two points $A ( - 3,4 ) , B ( 3,2 )$ and a line $L$. Points $A$ and $B$ are on opposite sides of line $L$, and $\vec { n } = ( 4 , - 3 )$ is a normal vector to line $L$. The distance from point $A$ to line $L$ is 5 times the distance from point $B$ to line $L$. Based on the above, answer the following questions.
(1) Find the dot product of vector $\overrightarrow { A B }$ and vector $\vec { n }$. (4 points)
(2) Find the equation of line $L$. (4 points)
(3) Point $P$ is on line $L$ and $\overline { P A } = \overline { P B }$. Find the coordinates of point $P$. (4 points)

A manufacturer produces two types of electric vehicles, Type A and Type B. The costs for producing these two types involve three categories: battery, motor, and others. The costs for each category are shown in the table below (unit: 10,000 yuan):

	Battery Cost	Motor Cost	Other Cost
Type A	56	26	48
Type B	40	20	56

The selling price formula for the two types of electric vehicles is the sum of ``$x$ times the battery cost'', ``$y$ times the motor cost'', and ``$\frac { x + y } { 2 }$ times the other cost'', that is,
Selling Price $=$ Battery Cost $\times x +$ Motor Cost $\times y +$ Other Cost $\times \frac { x + y } { 2 }$ where the multipliers $x, y$ must satisfy ``$1 \leq x \leq 2, 1 \leq y \leq 2$, and the selling prices of both Type A and Type B electric vehicles do not exceed 200 (10,000 yuan)''. To differentiate its products, the manufacturer wants to maximize the price difference between Type A and Type B electric vehicles. Based on the above information, answer the following questions.
(1) Write the selling prices of Type A and Type B electric vehicles (in terms of $x$ and $y$), and explain why ``the selling price of Type A electric vehicles is always higher than that of Type B electric vehicles''. (4 points)
(2) On a coordinate plane, draw the feasible region of $(x, y)$ satisfying the conditions in the problem, and shade the region with diagonal lines. (4 points)
(3) Find the values of multipliers $x$ and $y$ that maximize the price difference between Type A and Type B electric vehicles. What is the maximum price difference in units of 10,000 yuan? (6 points)

Given that a real-coefficient quadratic polynomial function $f ( x )$ satisfies $f ( - 1 ) = k , f ( 1 ) = 9 k , f ( 3 ) = - 15 k$, where $k > 0$. Let the $x$-coordinate of the vertex of the graph of $y = f ( x )$ be $a$. Select the correct option.
(1) $a \leq - 1$
(2) $- 1 < a < 1$
(3) $a = 1$
(4) $1 < a < 3$
(5) $3 \leq a$

A company holds a year-end lottery. Each person randomly draws two cards from six cards numbered 1 to 6. Assume each card has an equal chance of being drawn, and the rules are as follows: (I) If the sum of the numbers on the two cards is odd, the person wins 100 yuan and the lottery ends; (II) If the sum is even, the two cards are discarded, and two cards are randomly drawn from the remaining four cards. If the sum of their numbers is odd, the person wins 50 yuan; otherwise, there is no prize and the lottery ends. According to the above rules, what is the expected value of the prize money for each person participating in this lottery?
(1) 50
(2) 70
(3) 72
(4) 80
(5) 100

Let $a = \log _ { 2 } 8 , ~ b = \log _ { 3 } 1 , ~ c = \log _ { 0.5 } 8$. Select the correct options.
(1) $b = 0$
(2) $a + b + c > 0$
(3) $a > b > c$
(4) $a ^ { 2 } > b ^ { 2 } > c ^ { 2 }$
(5) $2 ^ { a } > 3 ^ { b } > \left( \frac { 1 } { 2 } \right) ^ { c }$

A convenience store packages three building block models (A, B, C) and five character figurines ($a, b, c, d, e$), totaling eight toys, into two bags for sale. Each bag contains four toys, and the packaging follows these principles: (I) A and $a$ must be in the same bag. (II) Each bag must contain at least one building block model. (III) $d$ and $e$ must be in different bags. Based on the above, select the correct options.
(1) Each bag must contain at least two character figurines
(2) B and C must be in different bags
(3) If B and $d$ are in the same bag, then C and $e$ must be in the same bag
(4) If B and $d$ are in different bags, then $b$ and $c$ must be in different bags
(5) If $b$ and $c$ are in different bags, then B and C must be in the same bag

Given a real number sequence $\left\langle a _ { n } \right\rangle$ satisfying $a _ { 1 } = 1 , a _ { n + 1 } = \frac { 2 n + 1 } { 2 n - 1 } a _ { n } , n$ is a positive integer. Select the correct options.
(1) $a _ { 2 } = 3$
(2) $a _ { 4 } = 9$
(3) $\left\langle a _ { n } \right\rangle$ is a geometric sequence
(4) $\sum _ { n = 1 } ^ { 20 } a _ { n } = 400$
(5) $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n } = 2$

A person's probability of hitting a dart each time is $\frac { 1 } { 2 }$, and the results of each dart throw are independent. From the following options, select the events with probability $\frac { 1 } { 2 }$.
(1) Throwing darts 2 times consecutively, hitting exactly 1 time
(2) Throwing darts 4 times consecutively, hitting exactly 2 times
(3) Throwing darts 4 times consecutively, the total number of hits is odd
(4) Throwing darts 6 times consecutively, given that the first throw misses, the second throw hits
(5) Throwing darts 6 times consecutively, given that exactly 1 hit in the first 2 throws, exactly 2 hits in the last 4 throws