For two sets
$$A = \left\{ x \left\lvert \, \frac { ( x - 2 ) ^ { 2 } } { x - 4 } \leq 0 \right. \right\} , \quad B = \left\{ x \mid x ^ { 2 } - 8 x + a \leq 0 \right\}$$
When $A \cup B = \{ x \mid x \leq 5 \}$, what is the value of the constant $a$? [3 points]
(1) 7
(2) 10
(3) 12
(4) 15
(5) 16

1. Given sets $A = \{ x \mid - 1 < x < 2 \} , B = \{ x \mid 0 < x < 3 \}$, then $A \cup B =$
A. $( - 1,3 )$
B. $( - 1,0 )$
C. $( 0,2 )$
D. $( 2,3 )$

1. Let set $A = \{ \mathrm { x } / ( \mathrm { x } + 1 ) ( \mathrm { x } - 2 ) < 0 \}$, set $B = \{ \mathrm { x } / 1 < \mathrm { x } < 3 \}$, then $A \cup B =$
A. $\{ X / - 1 < X < 3 \}$ B. $\{ X / - 1 < X < 1 \}$ C. $\{ X / 1 < X < 2 \}$ D. $\{ X / 2 < X < 3 \}$

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) $\{ 2,3,5 \}$

1. Given sets $\mathrm { P } = \left\{ x \mid x ^ { 2 } - 2 x \geq 3 \right\} , \mathrm { Q } = \{ x \mid 2 < x < 4 \}$ , then $\mathrm { P } \cap \mathrm { Q } =$ $\_\_\_\_$

1. Given sets $P = \left\{ x \mid x ^ { 2 } - 2 x \geq 0 \right\} , Q = \{ x \mid 1 < x \leq 2 \}$ , then $\left( \complement _ { \mathbb{R} } P \right) \cap Q =$
A. $[0,1)$
B. $( 0,2 ]$
C. $( 1,2 )$
D. $[ 1,2 ]$

Let $A = \left\{ x \mid x ^ { 2 } - 4 x + 3 < 0 \right\} , B = \{ x \mid 2 x - 3 > 0 \}$, then $A \cap B =$
(A) $\left( - 3 , - \frac { 3 } { 2 } \right)$
(B) $\left( - 3 , \frac { 3 } { 2 } \right)$
(C) $\left( 1 , \frac { 3 } { 2 } \right)$
(D) $\left( \frac { 3 } { 2 } , 3 \right)$

Given sets $A = \{ x \mid x < 2 \}$, $B = \{ x \mid 3 - 2x > 0 \}$, then
A. $A \cap B = \{ x \mid x \leq \frac{3}{2} \}$
B. $A \cap B = \varnothing$
C. $A \cup B = \left\{ x \left\lvert \, x < \frac{3}{2} \right. \right\}$
D. $A \cup B = \mathbf{R}$

Given the set $A = \left\{ x \mid x ^ { 2 } - x - 2 > 0 \right\}$, then $\mathrm { C } _ { \mathrm { R } } A =$
A. $\{ x \mid - 1 < x < 2 \}$
B. $\{ x \mid - 1 \leqslant x \leqslant 2 \}$
C. $\{ x \mid x < - 1 \} \cup \{ x \mid x > 2 \}$
D. $\{ x \mid x \leqslant - 1 \} \cup \{ x \mid x \geqslant 2 \}$

Let $A = \left\{ x \mid x ^ { 2 } - 5 x + 6 > 0 \right\} , B = \{ x \mid x - 1 < 0 \}$, then $A \cap B =$
A． $( - \infty , 1 )$
B． $( - 2,1 )$
C． $( - 3 , - 1 )$
D． $( 3 , + \infty )$

Given sets $A = \{ - 1,0,1,2 \} , B = \left\{ x \mid x ^ { 2 } \leqslant 1 \right\}$ , then $A \cap B =$
A. $\{ - 1,0,1 \}$
B. $\{ 0,1 \}$
C. $\{ - 1,1 \}$
D. $\{ 0,1,2 \}$

5. Keep the answer sheet clean, do not fold it, do not tear or wrinkle it, and do not use correction fluid, correction tape, or scrapers.

I. Multiple Choice Questions: 12 questions in total, 5 points each, 60 points total. For each question, only one of the four options is correct.

1. Let $A = \left\{ x \mid x ^ { 2 } - 5 x + 6 > 0 \right\} , B = \{ x \mid x - 1 < 0 \}$, then $A \cap B =$
A. $( - \infty , 1 )$
B. $( - 2,1 )$
C. $( - 3 , - 1 )$
D. $( 3 , + \infty )$
2. Let $z = - 3 + 2 \mathrm { i }$. In the complex plane, the point corresponding to $\bar { z }$ is located in
A. the first quadrant
B. the second quadrant
C. the third quadrant
D. the fourth quadrant
3. Given $\overrightarrow { A B } = ( 2,3 ) , \overrightarrow { A C } = ( 3 , t ) , | \overrightarrow { B C } | = 1$, then $\overrightarrow { A B } \cdot \overrightarrow { B C } =$
A. $- 3$
B. $- 2$
C. $2$
D. $3$
4. On January 3, 2019, the Chang'e-4 probe successfully achieved humanity's first soft landing on the far side of the moon, marking another major achievement in China's space program. A key technical challenge in achieving soft landing on the far side of the moon is maintaining communication between the ground and the probe. To solve this problem, the Queqiao relay satellite was launched, which orbits around the Earth-Moon Lagrange point $L _ { 2 }$. The $L _ { 2 }$ point is an equilibrium point located on the extension of the Earth-Moon line. Let the Earth's mass be $M _ { 1 }$, the Moon's mass be $M _ { 2 }$, the Earth-Moon distance be $R$, and the distance from the $L _ { 2 }$ point to the Moon be $r$. According to Newton's laws of motion and the law of universal gravitation, $r$ satisfies the equation: $\frac { M _ { 1 } } { ( R + r ) ^ { 2 } } + \frac { M _ { 2 } } { r ^ { 2 } } = ( R + r ) \frac { M _ { 1 } } { R ^ { 3 } }$. Let $\alpha = \frac { r } { R }$. Since $\alpha$ is very small, in approximate calculations $\frac { 3 \alpha ^ { 3 } + 3 \alpha ^ { 4 } + \alpha ^ { 5 } } { ( 1 + \alpha ) ^ { 2 } } \approx 3 \alpha ^ { 3 }$. Then the approximate value of $r$ is
A. $\sqrt { \frac { M _ { 2 } } { M _ { 1 } } } R$
B. $\sqrt { \frac { M _ { 2 } } { 2 M _ { 1 } } } R$
C. $\sqrt [ 3 ] { \frac { 3 M _ { 2 } } { M _ { 1 } } } R$
D. $\sqrt [ 3 ] { \frac { M _ { 2 } } { 3 M _ { 1 } } } R$
5. In a speech competition, 9 judges each give an original score to a contestant. When determining the contestant's final score, 1 highest score and 1 lowest score are removed from the 9 original scores, leaving 7 valid scores. Compared with the 9 original scores, the numerical characteristic that remains unchanged for the 7 valid scores is
A. median
B. mean
C. variance
D. range

Given set $A = \left\{ x \mid x ^ { 2 } - 3 x - 4 < 0 \right\} , B = \{ - 4,1,3,5 \}$ , then $A \cap B =$
A. $\{ - 4,1 \}$
B. $\{ 1,5 \}$
C. $\{ 3,5 \}$
D. $\{ 1,3 \}$

Let sets $A = \left\{ x \mid x ^ { 2 } - 4 \leqslant 0 \right\}$, $B = \{ x \mid 2 x + a \leqslant 0 \}$, and $A \cap B = \{ x \mid - 2 \leqslant x \leqslant 1 \}$. Then $a =$
A. $- 4$
B. $- 2$
C. 2
D. 4

Given sets $U = \{ - 2 , - 1,0,1,2,3 \} , A = \{ - 1,0,1 \} , B = \{ 1,2 \}$ , then $\complement _ { U } ( A \cup B ) =$
A． $\{ - 2,3 \}$
B． $\{ - 2,2,3 \}$
C． $\{ - 2 , - 1,0,3 \}$
D． $\{ - 2 , - 1,0,2,3 \}$

Given sets $A = \{ 1,2,3,5,7,11 \} , B = \{ x \mid 3 < x < 15 \}$, the number of elements in $A \cap B$ is
A. 2
B. 3
C. 4
D. 5

Given the sets $A = \left\{ ( x , y ) \mid x , y \in \mathbf { N } ^ { * } , y \geqslant x \right\} , B = \{ ( x , y ) \mid x + y = 8 \}$ , the number of elements in $A \cap B$ is
A. 2
B. 3
C. 4
D. 6

Given sets $A = \{ 1,2,4 \} , B = \{ 2,3,4 \}$, find $A \cap B =$ $\_\_\_\_$

1. If $M = \{ x \mid \sqrt { x } < 4 \}$ and $N = \{ x \mid 3 x \geqslant 1 \}$, then $M \cap N =$
A. $\{ x \mid 0 \leqslant x < 2 \}$
B. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leqslant x < 2 \right. \right\}$
C. $\{ x \mid 3 \leqslant x < 16 \}$
D. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leqslant x < 16 \right. \right\}$

Let $A = \{(x,y) : x^4 + y^2 \leq 1\}$ and $B = \{(x,y) : x^6 + y^4 \leq 1\}$. Which of the following is true?
(A) $A = B$
(B) $A \subset B$ (A is a proper subset of B)
(C) $B \subset A$ (B is a proper subset of A)
(D) Neither $A \subset B$ nor $B \subset A$

If $A = \{ x \in R : | x | < 2 \}$ and $B = \{ x \in R : | x - 2 | \geq 3 \}$; then
(1) $A \cap B = ( - 2 , - 1 )$
(2) $B - A = R - ( - 2,5 )$
(3) $A \cup B = R - ( 2,5 )$
(4) $A - B = [ - 1,2 )$

Let $A = \{x \in R : |x + 1| < 2\}$ and $B = \{x \in R : |x - 1| \geq 2\}$. Then which one of the following statements is NOT true?
(1) $A - B = (-1,1)$
(2) $B - A = R - (-3,1)$
(3) $A \cap B = (-3,-1]$
(4) $A \cup B = R - [1,3)$

Let $S = \left\{ x \in [ - 6,3 ] - \{ - 2,2 \} : \frac { | x + 3 | - 1 } { | x | - 2 } \geq 0 \right\}$ and $T = \left\{ x \in Z : x ^ { 2 } - 7 | x | + 9 \leq 0 \right\}$. Then the number of elements in $S \cap T$ is
(1) 7
(2) 5
(3) 4
(4) 3

Let $\mathrm { A } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x + y | \geqslant 3 \}$ and $\mathrm { B } = \{ ( x , y ) \in \mathbf { R } \times \mathbf { R } : | x | + | y | \leq 3 \}$. If $\mathrm { C } = \{ ( x , y ) \in \mathrm { A } \cap \mathrm { B } : x = 0$ or $y = 0 \}$, then $\sum _ { ( x , y ) \in \mathrm { C } } | x + y |$ is :
(1) 15
(2) 24
(3) 18
(4) 12

Let $a , b , c$ and $d$ be real numbers satisfying $a < b < c < d$. Suppose that the two subsets of real numbers
$$A = \{ x \mid a \leqq x \leqq c \} , \quad B = \{ x \mid b \leqq x \leqq d \}$$
satisfy
$$A \cap B = \left\{ x \mid x ^ { 2 } - 4 x + 3 \leqq 0 \right\} .$$
Then, answer the questions for cases (1) and (2).
(1) Let the union of $A$ and $B$ be
$$A \cup B = \left\{ x \mid x ^ { 2 } - 5 x - 24 \leqq 0 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { \text { NO } } , \quad b = \mathbf { P } , \quad c = \mathbf { Q } , \quad d = \mathbf { Q } .$$
(2) Let the intersection of $A$ and the complement $\bar { B }$ of $B$ be
$$A \cap \bar { B } = \left\{ x \mid x ^ { 2 } + 5 x - 6 \leqq 0 \text { and } x \neq 1 \right\} ,$$
and let the intersection of the complement $\bar { A }$ of $A$ and $B$ be
$$\bar { A } \cap B = \left\{ x \mid x ^ { 2 } - 9 x + 18 \leqq 0 \text { and } x \neq 3 \right\} .$$
Then the values of $a , b , c$ and $d$ are
$$a = \mathbf { S T } , \quad b = \mathbf { U } , \quad c = \mathbf { V } , \quad d = \mathbf { W } .$$