1. Given the universal set $U = \{ 1,2,3,4,5,6 \}$, set $A = \{ 2,3,4 \}$, set $B = \{ 1,3,4,6 \}$, then set $A \cap C _ { U } B =$ (A) $\{ 3 \}$ (B) $\{ 2,5 \}$ (C) $\{ 1,4,6 \}$ (D) $\{ 2,3,5 \}$
2. Let variables $x , y$ satisfy the constraint conditions $x - 2y \geq 0$, $x \leq 2$, $y \geq 0$. Then the maximum value of the objective function $z = 3x + y$ is (A) 7 (B) 8 (C) 9 (D) 14
4. Let $x \in \mathbb{R}$. Then ``$1 < x < 2$'' is ``$|x - 2| < 1$'' a (A) sufficient but not necessary condition (B) necessary but not sufficient condition (C) necessary and sufficient condition (D) neither sufficient nor necessary condition
6. As shown in the figure, in circle $O$, $M, N$ are trisection points of chord $AB$. Chords $CD, CE$ pass through points $M, N$ respectively. If $CM = 3$, then the length of segment $NE$ is [Figure] (A) $\frac { 8 } { 3 }$ (B) 3 (C) $\frac { 10 } { 3 }$ (D) $\frac { 5 } { 2 }$
7. Given the function $f ( x ) = 2 ^ { | x - m | } - 1$ defined on $\mathbb{R}$ (where $m$ is a real number), let $a = f \left( \log _ { 0.5 } 3 \right)$, $b = f \left( \log _ { 2 } 5 \right)$, $c = f ( 2m )$. Then the size relationship of $a, b, c$ is (A) $a < b < c$ (B) $c < a < b$ (C) $a < c < b$ (D)
8. Given the function $f ( x ) = \begin{cases} 2 - | x | , & x \leq 2 \\ ( x - 2 ) ^ { 2 } , & x > 2 \end{cases}$, and function $g ( x ) = 3 - f ( 2 - x )$, then the number of intersections of the graphs of $y = f(x)$ and $y = g(x)$ is (A) 2 (B) 3 (C) 4 (D) 5
II. Fill-in-the-Blank Questions: This section has 6 questions, each worth 5 points, for a total of 30 points.
11. Given the function $f ( x ) = a x \ln x , x \in ( 0 , + \infty )$, where $a$ is a real number, and $f ^ { \prime } ( x )$ is the derivative of $f ( x )$. If $f ^ { \prime } ( 1 ) = 3$, then the value of $a$ is $\_\_\_\_$.
14. Given the function $f ( x ) = \sin \omega x + \cos \omega x ( \omega > 0 ) , x \in \mathbb{R}$. If the function $f ( x )$ is monotonically increasing on the interval $( - \omega , \omega )$, and the graph of $f ( x )$ is symmetric about the line $x = \omega$, then the value of $\omega$ is $\_\_\_\_$.
III. Solution Questions: This section has 6 questions, for a total of 80 points.
15. (13 points) Three table tennis associations have 27, 9, and 18 members respectively. Using stratified sampling, 6 athletes are selected from these three associations to participate in a competition. (I) Find the number of athletes to be selected from each of the three associations respectively; (II) The 6 selected athletes are numbered $A _ { 1 } , A _ { 2 } , A _ { 3 } , A _ { 4 } , A _ { 5 } , A _ { 6 }$ respectively. Two athletes are randomly selected from these 6 athletes to participate in a doubles match. (i) List all possible outcomes using the given numbering; (ii) Let event $A$ be ``at least one of the two athletes numbered $A _ { 5 }$ and $A _ { 6 }$ is selected''. Find the probability of event $A$ occurring.
16. (13 points) In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given that the area of $\triangle ABC$ is $3 \sqrt { 15 }$, $b - c = 2$, $\cos A = - \frac { 1 } { 4 }$. (I) Find the values of $a$ and $\sin C$; (II) Find the value of $\cos \left( 2 A + \frac { \pi } { 6 } \right)$.
18. Given that $\{ a _ { n } \}$ is a geometric sequence with all positive terms, $\{ b _ { n } \}$ is an arithmetic sequence, and $a _ { 1 } = b _ { 1 } = 1$, $b _ { 2 } + b _ { 3 } = 2 a _ { 3 }$, $a _ { 5 } - 3 b _ { 2 } = 7$. (1) Find the general term formulas for $\{ a _ { n } \}$ and $\{ b _ { n } \}$; (2) Let $c _ { n } = a _ { n } b _ { n } , n \in \mathbb{N} ^ { * }$. Find the sum of the first $n$ terms of the sequence $\{ c _ { n } \}$.
19. Given the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with upper vertex $B$, left focus $F$, and eccentricity $\frac { \sqrt { 5 } } { 5 }$. (1) Find the slope of line $BF$; (2) Let line $BF$ intersect the ellipse at point $P$ (where $P$ is different from $B$). A line passing through $B$ and perpendicular to $BF$ intersects the ellipse at point $Q$ (where $Q$ is different from $B$). Line $PQ$ intersects the $x$-axis at point $M$, and $|PM| = l|MQ|$. 1) Find the value of $l$; 2) If $|PM| \sin \angle BQP = \frac { 7 \sqrt { 5 } } { 9 }$, find the equation of the ellipse.
20. Given the function $f ( x ) = 4 x - x ^ { 4 } , x \in \mathbb{R}$. (1) Find the monotonicity of $f ( x )$; (2) Let $P$ be the intersection point of the curve $y = f ( x )$ and the positive $x$-axis. The tangent line to the curve at point $P$ is $y = g ( x )$. Prove that for any positive real number $x$, we have $f ( x ) \leq g ( x )$; (3) If the equation $f ( x ) = a$ (where $a$ is a real number) has two positive real roots $x _ { 1 } , x _ { 2 }$ with $x _ { 1 } < x _ { 2 }$, prove that $x _ { 2 } - x _ { 1 } < - \frac { a } { 3 } + 4 ^ { \frac { 1 } { 3 } }$.