From 6, 8, 10, 12, select any three distinct numbers as the three sides of a triangle, and let $\theta$ be the largest interior angle of this triangle. Among all possible triangles formed, the minimum value of $\cos \theta$ is (Express as a fraction in lowest terms)

On the coordinate plane, a circle with radius 12 intersects the line $x + y = 0$ at two points, and the distance between these two points is 8. If this circle intersects the line $x + y = 24$ at points $P$ and $Q$, then the length of segment $\overline { P Q }$ is $\_\_\_\_$ (14)$\sqrt { (15) }$. (Express as a simplified radical)

Consider a trapezoid $ABCD$ where $\overline { AB }$ is parallel to $\overline { DC }$ . It is known that points $E$ and $F$ lie on diagonals $\overline { AC }$ and $\overline { BD }$ respectively, and $\overline { AB } = \frac { 2 } { 5 } \overline { DC }$ , $\overline { AE } = \frac { 3 } { 2 } \overline { EC }$ , $\overline { BF } = \frac { 2 } { 3 } \overline { FD }$ . If vector $\overrightarrow { FE }$ is expressed as $\alpha \overrightarrow { AC } + \beta \overrightarrow { AD }$ , then the real numbers $\alpha = \frac { \text{(16)} } { \text{(17)(18)} } , \beta = \frac { \text{(19)(20)} } { \text{(21)(22)} }$ . (Express as fractions in lowest terms)

In coordinate space, let $E$ be the plane passing through the three points $A ( 0 , - 1 , - 1 )$ , $B ( 1 , - 1 , - 2 )$ , $C ( 0,1,0 )$ . Assume $H$ is a point in space satisfying $\overrightarrow { AH } = \frac { 2 } { 3 } \overrightarrow { AB } - \frac { 1 } { 3 } \overrightarrow { AC } + 3 ( \overrightarrow { AB } \times \overrightarrow { AC } )$ . Based on the above, answer the following questions.
(1) Find the volume of tetrahedron $ABCH$ . (4 points) (Note: The volume of a tetrahedron is one-third of the base area times the height)
(2) Let $H ^ { \prime }$ be the symmetric point of point $H$ with respect to plane $E$ . Find the coordinates of $H ^ { \prime }$ . (4 points)
(3) Determine whether the projection of point $H ^ { \prime }$ onto plane $E$ lies inside $\triangle ABC$ . Explain your reasoning. (4 points) (Note: The interior of a triangle does not include the three sides of the triangle)

On the coordinate plane, let $\Gamma$ denote the graph of the polynomial function $y = x ^ { 3 } - 4 x ^ { 2 } + 5 x$ , and let $L$ denote the line $y = m x$ , where $m$ is a real number. Based on the above, answer the following questions.
(1) When $m = 2$ , find the $x$-coordinates of the three distinct intersection points of $\Gamma$ and $L$ in the range $x \geq 0$ . (2 points)
(2) Based on (1), find the area of the bounded region enclosed by $\Gamma$ and $L$ . (4 points)
(3) In the range $x \geq 0$ , if $\Gamma$ and $L$ have three distinct intersection points, then the maximum range of $m$ satisfying this condition is $a < m < b$ . Find the values of $a$ and $b$ . (6 points)

Let $x _ { 0 }$、$y _ { 0 }$ be positive real numbers. If the point $\left( 10 x _ { 0 } , 100 y _ { 0 } \right)$ on the coordinate plane lies on the graph of the function $y = 10 ^ { x }$ , then the point $\left( x _ { 0 } , \log y _ { 0 } \right)$ will lie on the graph of the line $y = a x + b$ , where $a$、$b$ are real numbers. What is the value of $2 a - b$?
(1) 4
(2) 9
(3) 15
(4) 18
(5) 22

A research team uses a certain rapid test reagent to understand the proportion of organisms in a protected area whose body toxin accumulation exceeds the standard due to environmental pollution. The test result of this reagent shows only two colors: red and yellow. Based on past experience, it is known that: if body toxin accumulation exceeds the standard, after testing with this reagent, $75\%$ shows red; if body toxin accumulation does not exceed the standard, after testing with this reagent, $95\%$ shows yellow. It is known that for a certain type of organism in this protected area, $7.8\%$ of the test results show red. Assuming the actual proportion of this type of organism with body toxin accumulation exceeding the standard is $p\%$ , select the correct option.
(1) $1 \leq p < 3$
(2) $3 \leq p < 5$
(3) $5 \leq p < 7$
(4) $7 \leq p < 9$
(5) $9 \leq p < 11$

Find the value of the limit $\lim _ { n \rightarrow \infty } \frac { 10 ^ { 10 } } { n ^ { 10 } } \left[ 1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + \cdots + ( 2 n ) ^ { 9 } \right]$ .
(1) $10 ^ { 9 }$
(2) $10 ^ { 9 } \times \left( 2 ^ { 10 } - 1 \right)$
(3) $2 ^ { 9 } \times \left( 10 ^ { 10 } - 1 \right)$
(4) $10 ^ { 9 } \times 2 ^ { 10 }$
(5) $2 ^ { 9 } \times 10 ^ { 10 }$

An electronics company has several hundred employees with two types of meal arrangements: bringing meals from home or eating out. Long-term surveys have found that: if an employee brings meals from home on a given day, then $10\%$ will switch to eating out the next day; if an employee eats out on a given day, then $20\%$ will switch to bringing meals from home the next day. Let $x _ { 0 }$、$y _ { 0 }$ respectively represent the proportion of employees bringing meals from home and eating out today relative to the total number of employees, where $x _ { 0 }$、$y _ { 0 }$ are both positive, and $x _ { n }$、$y _ { n }$ respectively represent the proportion of employees bringing meals from home and eating out after $n$ days relative to the total number of employees. Given that the number of employees in the company remains unchanged, select the correct options.
(1) $y _ { 1 } = 0.9 y _ { 0 } + 0.2 x _ { 0 }$
(2) $\left[ \begin{array} { l } x _ { n + 1 } \\ y _ { n + 1 } \end{array} \right] = \left[ \begin{array} { l l } 0.9 & 0.2 \\ 0.1 & 0.8 \end{array} \right] \left[ \begin{array} { l } x _ { n } \\ y _ { n } \end{array} \right]$
(3) If $\frac { x _ { 0 } } { y _ { 0 } } = \frac { 2 } { 1 }$ , then $\frac { x _ { n } } { y _ { n } } = \frac { 2 } { 1 }$ holds for any positive integer $n$
(4) If $y _ { 0 } > x _ { 0 }$ , then $y _ { 1 } > x _ { 1 }$
(5) If $x _ { 0 } > y _ { 0 }$ , then $x _ { 0 } > x _ { 1 }$

Assume $f ( x )$ is a fifth-degree polynomial with real coefficients, and the remainder when $f ( x )$ is divided by $x ^ { n } - 1$ is $r _ { n } ( x )$ , where $n$ is a positive integer. Select the correct options.
(1) $r _ { 1 } ( x ) = f ( 1 )$
(2) $r _ { 2 } ( x )$ is a first-degree polynomial with real coefficients
(3) The remainder when $r _ { 4 } ( x )$ is divided by $x ^ { 2 } - 1$ equals $r _ { 2 } ( x )$
(4) $r _ { 5 } ( x ) = r _ { 6 } ( x )$
(5) If $f ( - x ) = - f ( x )$ , then $r _ { 3 } ( - x ) = - r _ { 3 } ( x )$

A scratch-off lottery game with 12 boxes labeled 1 to 12. Each game involves tossing a fair coin four times to determine which boxes to scratch. The rules are as follows: (I) On the first coin toss, if heads, scratch box 1; if tails, scratch box 3. (II) On the second, third, and fourth coin tosses, if heads, the number of the box to scratch is the number of the previous box plus 1; if tails, the number of the box to scratch is the number of the previous box plus 3, and so on. Example: If the results of four coin tosses are ``heads, tails, tails, heads'' in order, then boxes numbered 1, 4, 7, and 8 will be scratched. Let $p _ { m }$ denote the probability that box $m$ is scratched in each game. Select the correct options.
(1) $p _ { 2 } = \frac { 1 } { 4 }$
(2) $p _ { 3 } = \frac { 1 } { 2 }$
(3) $p _ { 4 } = \frac { 1 } { 2 } p _ { 1 } + \frac { 1 } { 2 } p _ { 3 }$
(4) $p _ { 8 } > p _ { 10 }$
(5) Given that box 4 is scratched, the probability that box 3 is scratched is $\frac { 1 } { 2 }$

Let $F ( x )$ be a polynomial with real coefficients and $F ^ { \prime } ( x ) = f ( x )$ . It is known that $f ^ { \prime } ( x ) > x ^ { 2 } + 1.1$ holds for all real numbers $x$. Select the correct options.
(1) $f ^ { \prime } ( x )$ is an increasing function
(2) $f ( x )$ is an increasing function
(3) $F ( x )$ is an increasing function
(4) $[ f ( x ) ] ^ { 2 }$ is an increasing function
(5) $f ( f ( x ) )$ is an increasing function

Let $z _ { 1 }$、$z _ { 2 }$、$z _ { 3 }$、$z _ { 4 }$ be four distinct complex numbers whose corresponding points on the complex plane can be connected in order to form a parallelogram. Which of the following options must be real numbers?
(1) $\left( z _ { 1 } - z _ { 3 } \right) \left( z _ { 2 } - z _ { 4 } \right)$
(2) $z _ { 1 } - z _ { 2 } + z _ { 3 } - z _ { 4 }$
(3) $z _ { 1 } + z _ { 2 } + z _ { 3 } + z _ { 4 }$
(4) $\frac { z _ { 1 } - z _ { 2 } } { z _ { 3 } - z _ { 4 } }$
(5) $\left( \frac { z _ { 2 } - z _ { 4 } } { z _ { 1 } - z _ { 3 } } \right) ^ { 2 }$