If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x - 2 y - 2 \leq 0 , \\ x - y + 1 \geq 0 , \\ y \leq 0 , \end{array} \right.$ then the maximum value of $z = 3 x + 2 y$ is \_\_\_\_

Given the constraints $\left\{ \begin{array} { l } x + 2 y - 5 \geq 0 , \\ x - 2 y + 3 \geq 0 , \\ x - 5 \leq 0 , \end{array} \right.$ the minimum value of $z = x + y$ is \_\_\_\_.

If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + 2 y - 5 \geqslant 0 , \\ x - 2 y + 3 \geqslant 0 , \\ x - 5 \leqslant 0 , \end{array} \right.$ then the maximum value of $z = x + y$ is $\_\_\_\_$.

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. $\{ 0,1,2 \}$

If $x , y$ satisfy $| x | \leqslant 1 - y$ and $y \geqslant - 1$, then the maximum value of $3 x + y$ is (A) $- 7$ (B) 1 (C) 5 (D) 7

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

11. If the range of the function $f ( x ) = a \cdot \left( \frac { 1 } { 3 } \right) ^ { x } \left( \frac { 1 } { 2 } \leq x \leq 1 \right)$ is a subset of the range of the function $g ( x ) = \frac { x ^ { 2 } - 1 } { x ^ { 2 } + x + 1 } ( x \in \mathbb{R} )$, then the range of positive number $a$ is
A. $(0,2]$
B. $(0,1]$
C. $( 0,2 \sqrt { 3 } ]$
D. $( 0 , \sqrt { 3 } ]$

11. Let the plane region represented by the system of inequalities $\left\{ \begin{array} { l } x + y \geq 6 , \\ 2 x - y \geq 0 \end{array} \right.$ be $D$ . Proposition $p : \exists ( x , y ) \in D , 2 x + y \geq 9$ ; Proposition $q : \forall ( x , y ) \in D , 2 x + y \leq 12$ . Four propositions are given below:
(1) $p \vee q$
(2) $\neg p \vee q$
(3) $p \wedge \neg q$
(4) $\neg p \wedge \neg q$
The numbers of all true propositions among these four are
A. (1)(3)
B. (1)(2)
C. (2)(3)
D. (3)(4)

13. If variables $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { l } 2 x + 3 y - 6 \geq 0 , \\ x + y - 3 \leq 0 , \\ y - 2 \leq 0 , \end{array} \right.$ then the maximum value of $z = 3 x - y$ is $\_\_\_\_$ .

Let $x, y, z \in \mathbf{R}$ and $x + y + z = 1$.
(1) Find the minimum value of $(x-1)^2 + (y+1)^2 + (z+1)^2$;
(2) If $(x-2)^2 + (y-1)^2 + (z-a)^2 \geqslant \frac{1}{3}$ holds for all $x, y, z$ satisfying $x + y + z = 1$, prove that $a \leqslant -3$ or $a \geqslant -1$.

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. $\{ 1,3 \}$

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 $A \cap B =$ $\_\_\_\_$

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 $\left\{ \begin{array} { l } x + y \geq 2 \\ y \geq 0 \\ x + 2 y - 3 \leq 0 \end{array} \right.$, find the maximum value of $z = y - 2 x$ as $\_\_\_\_$

If $2 ^ { x } - 2 ^ { y } < 3 ^ { - x } - 3 ^ { - y }$ , then
A． $\ln ( y - x + 1 ) > 0$
B． $\ln ( y - x + 1 ) < 0$
C． $\ln | x - y | > 0$
D． $\ln | x - y | < 0$

If $x , y$ satisfy the constraint conditions $\left\{ \begin{array} { l } 2 x + y - 2 \leqslant 0 , \\ x - y - 1 \geqslant 0 , \\ y + 1 \geqslant 0 , \end{array} \right.$ then the maximum value of $z = x + 7 y$ is $\_\_\_\_$

If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + y \geqslant 0 , \\ 2 x - y \geqslant 0 , \\ x \leqslant 1 , \end{array} \right.$ then the maximum value of $z = 3 x + 2 y$ is $\_\_\_\_$ .

If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + y \geqslant 0 , \\ 2 x - y \geqslant 0 , \\ x \leqslant 1 , \end{array} \right.$ then the maximum value of $z = 3 x + 2 y$ is $\_\_\_\_$ .

Which of the following inequalities always holds? ( )
A. $a ^ { 2 } + b ^ { 2 } \leq 2 a b$
B. $a ^ { 2 } + b ^ { 2 } \geq - 2 a b$
C. $a + b \geq - 2 \sqrt { | a b | }$
D. $a + b \leq 2 \sqrt { | a b | }$

Given the function $f ( x ) = \left| x - a ^ { 2 } \right| + | x - 2 a + 1 |$ .
(1) When $a = 2$, find the solution set of the inequality $f ( x ) \geqslant 4$;
(2) If $f ( x ) \geqslant 4$ for all $x$, find the range of values of $a$.

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. $\left\{ x \left\lvert \, \frac { 1 } { 3 } \leqslant x < 16 \right. \right\}$