On the coordinate plane, there is a polygonal region $\Gamma$ (including boundary) as shown in the figure. If $k > 0$ , the line $7 x + 2 y = k$ and the two coordinate axes form a triangular region such that the polygonal region $\Gamma$ lies within this triangular region (including boundary), then the minimum positive real number $k =$ (7)(8).

If the possible values of random variable $X$ are $1 , 2 , 3 , 4$ , and the probability $P ( X = k )$ is proportional to $\frac { 1 } { k }$ , then the probability $P ( X = 3 )$ is $\frac { \text{(9)(11)} } { \text{(10)(11)} }$ . (Reduce to lowest terms)

A company has only three members: a manager, a secretary, and a salesperson. If only the secretary receives a 10\% salary increase, the company's total salary expenditure increases by 3\%; if only the salesperson receives a 20\% salary increase, the company's total salary expenditure increases by 4\%. If only the manager's salary is reduced by 15\%, then the company's total salary expenditure will decrease by (12).(13)\%.

On the coordinate plane, there is a trapezoid with four vertices at $A ( 0,0 ) , B ( 1,0 ) , P , Q$ , where the line passing through $P$ and $Q$ has equation $y = 2 x + 4$ . If the coordinates of point $Q$ are $( a , 2 a + 4 )$ , where $a \geq 0$ is a real number, then the area of trapezoid $A B P Q$ is (14)$a +$ (16). (Reduce to lowest terms)

In the early stages of an infectious disease outbreak, since most people have not been infected and have no antibodies, the total number of infected people usually grows exponentially. Under the premise that ``the initial number of infected people is $P _ { 0 }$ , and each infected person on average infects $r$ uninfected people per day'', the total number of people infected with the disease after $n$ days, $P _ { n }$ , can be expressed as
$$P _ { n } = P _ { 0 } ( 1 + r ) ^ { n } \text {, where } P _ { 0 } \geq 1 \text { and } r > 0 \text { . }$$
Answer the following questions:
(1) Given that $A = \frac { \log P _ { 5 } - \log P _ { 2 } } { 3 } , B = \frac { \log P _ { 8 } - \log P _ { 6 } } { 2 }$ , show that $A = B$ . (4 points)
(2) Given that a certain infectious disease in its early stages follows the above mathematical model and the total number of infected people increases tenfold every 16 days, find the value of $\frac { P _ { 20 } } { P _ { 17 } } \times \frac { P _ { 8 } } { P _ { 6 } } \times \frac { P _ { 5 } } { P _ { 2 } }$ . (5 points)
(3) Based on (2), find the value of $\frac { \log P _ { 20 } - \log P _ { 17 } } { 3 }$ . (4 points)

On the coordinate plane, two parallel lines $L _ { 1 } , L _ { 2 }$ both have slope 2 and are at distance 5 apart. Point $A ( 2 , - 1 )$ is a point on $L _ { 1 }$ in the fourth quadrant. Point $B$ is a point on $L _ { 2 }$ in the second quadrant with $\overline { A B } = 5$ . Line $L _ { 3 }$ has slope 3, passes through point $A$, and intersects $L _ { 2 }$ at point $C$. Answer the following questions:
(1) Find the slope of line $AB$ . (2 points)
(2) Find the vector $\overrightarrow { A B }$ . (4 points)
(3) Find the dot product $\overrightarrow { A B } \cdot \overrightarrow { A C }$ . (3 points)
(4) Find the vector $\overrightarrow { A C }$ . (4 points)

Which of the following matrices is equal to $\left[ \begin{array} { c c } - 1 & 0 \\ 1 & - 1 \end{array} \right] ^ { 5 }$ ?
(1) $\left[ \begin{array} { c c } - 1 & 0 \\ - 5 & - 1 \end{array} \right]$
(2) $\left[ \begin{array} { c c } 1 & 0 \\ - 5 & 1 \end{array} \right]$
(3) $\left[ \begin{array} { c c } - 1 & 5 \\ 0 & - 1 \end{array} \right]$
(4) $\left[ \begin{array} { l l } 1 & 0 \\ 5 & 1 \end{array} \right]$
(5) $\left[ \begin{array} { c c } - 1 & 0 \\ 5 & - 1 \end{array} \right]$

A graduating class has 8 students responsible for planning a class trip, divided into three groups A, B, and C, consisting of 3, 3, and 2 people respectively. Each of the 8 students will be assigned to one of the groups, and two students, A and B, must be in the same group. How many ways are there to divide the 8 students into groups?
(1) 140 ways
(2) 150 ways
(3) 160 ways
(4) 170 ways
(5) 180 ways

To understand whether IQ and brain volume are related, a small study used magnetic resonance imaging to measure the brain volume (in units of 10,000 pixels) of 5 people, along with their IQ listed in the table below:

Brain Volume $( X )$	90	95	91	88	106
$\mathrm { IQ } ( Y )$	90	100	112	80	103

It is known that the mean of $X$ in the table above is $\mu _ { X } = 94$ , the mean of $Y$ is $\mu _ { Y } = 97$ , and the correlation coefficient between brain volume ($X$) and IQ ($Y$) is $r _ { X , Y }$ . Based on the table above, determine which of the following options is most likely the value of $r _ { X , Y }$ ?
(1) $r _ { X , Y } \leq - 1$
(2) $- 1 < r _ { X , Y } < - 0.5$
(3) $r _ { X , Y } = 0$
(4) $0 < r _ { X , Y } < 0.5$
(5) $r _ { X , Y } \geq 1$

Let $f ( x )$ be a quadratic polynomial function with real coefficients such that $f ( x ) = 0$ has no real roots. Select the correct options.
(1) $f ( 0 ) > 0$
(2) $f ( 1 ) f ( 2 ) > 0$
(3) If $f ( x ) - 1 = 0$ has real roots, then $f ( x ) - 2 = 0$ has real roots
(4) If $f ( x ) - 1 = 0$ has a double root, then $f ( x ) - \frac { 1 } { 2 } = 0$ has no real roots
(5) If $f ( x ) - 1 = 0$ has two distinct real roots, then $f ( x ) - \frac { 1 } { 2 } = 0$ has real roots

A sequence $a _ { 1 } , a _ { 2 } , \cdots$ where the odd-indexed terms form a geometric sequence with common ratio $\frac { 1 } { 3 }$ , and the even-indexed terms form a geometric sequence with common ratio $\frac { 1 } { 2 }$ , with $a _ { 1 } = 3 , a _ { 2 } = 2$ . Select the correct options.
(1) $a _ { 4 } > a _ { 5 } > a _ { 6 } > a _ { 7 }$
(2) $\frac { a _ { 10 } } { a _ { 11 } } > 10$
(3) $\lim _ { n \rightarrow \infty } a _ { n } = 0$
(4) $\lim _ { n \rightarrow \infty } \frac { a _ { n + 1 } } { a _ { n } } = 0$
(5) $\sum _ { n = 1 } ^ { 100 } a _ { n } > 9$

There is a game involving moving a game piece on a number line. The way to move the piece is determined by rolling a fair die, with the following rules: (I) When the die shows 1 point, the piece does not move. (II) When the die shows 3 or 5 points, the piece moves left (negative direction) by ``that point number minus 1'' units. (III) When the die shows an even number, the piece moves right (positive direction) by ``half of that point number'' units. On the first die roll, the piece starts at the origin. From the second roll onwards, the piece starts from the position it was in after the previous roll. For example, if two die rolls result in 5 points and 2 points respectively, the piece first moves left 4 units to coordinate $-4$, then moves right 1 unit to coordinate $-3$. Select the correct options.
(1) After rolling the die once, the probability that the piece is at distance 2 from the origin is $\frac { 1 } { 2 }$
(2) After rolling the die once, the expected value of the piece's coordinate is 0
(3) After rolling the die twice, the piece's coordinate could be $-5$
(4) After rolling the die twice, given that the sum of the two rolls is odd, the probability that the piece's coordinate is positive is $\frac { 4 } { 9 }$
(5) After rolling the die three times, the probability that the piece is at the origin is $\left( \frac { 1 } { 6 } \right) ^ { 3 }$