8. At two points $A$ and $B$ that are 2 kilometers apart, target $C$ is measured. If $\angle CAB = 75°, \angle CBA = 60°$, then the distance between points $A$ and $C$ is $\_\_\_\_$ kilometers.
10. A research group conducted a survey on urban air quality and divided 24 cities into three groups (A, B, C) by region, with corresponding numbers of cities being $4, 12, 8$. If stratified sampling is used to select 6 cities, then the number of cities to be selected from group C is $\_\_\_\_$
11. The determinant $\left| \begin{array}{ll} a & b \\ c & d \end{array} \right|$ where $a, b, c, d \in \{-1, 1, 2\}$. Among all possible values, the maximum is $\_\_\_\_$
13. Among 9 randomly selected students, the probability that at least 2 students are born in the same month is $\_\_\_\_$ (assuming each month has the same number of days; round the result to 0.001)
14. Let $g(x)$ be a function defined on $\mathbb{R}$ with period 1. If the function $f(x) = x + g(x)$ has range $[-2, 5]$ on the interval $[0, 1]$, then the range of $f(x)$ on the interval $[0, 3]$ is $\_\_\_\_$ II. Multiple Choice Questions (Total: 20 points) This section contains 4 questions. Each question has exactly one correct answer. Candidates should shade the corresponding box on the answer sheet. Each correct answer is worth 5 points; otherwise, zero points are awarded.
15. Among the following functions, which one is both an even function and monotonically decreasing on the interval $(0, +\infty)$? (A) $y = x^{-2}$ (B) $y = x^{-1}$ (C) $y = x^2$ (D) $y = x^{\frac{1}{3}}$
16. If $a, b \in \mathbb{R}$ and $ab > 0$, then which of the following inequalities always holds? (A) $a^2 + b^2 > 2ab$ (B) $a + b \geq 2\sqrt{ab}$ (C) $\frac{1}{a} + \frac{1}{b} > \frac{2}{\sqrt{ab}}$ (D) $\frac{b}{a} + \frac{a}{b} \geq 2$
17. If the solution sets of the trigonometric equations $\sin x = 0$ and $\sin 2x = 0$ are $E$ and $F$ respectively, then ( ) (A) $E \subset F$ (B) $E \supset F$ (C) $E = F$ (D) $E \cap F = \varnothing$
18. Let $A_1, A_2, A_3, A_4$ be 4 distinct points in the plane. Then the number of points $M$ such that $\overrightarrow{MA_1} + \overrightarrow{MA_2} + \overrightarrow{MA_3} + \overrightarrow{MA_4} = \overrightarrow{0}$ is ( ) (A) 0 (B) 1 (C) 2 (D) 4 III. Solution Questions (Total: 74 points) This section contains 5 questions. When solving each question, necessary steps must be shown in the designated area on the answer sheet.
19. (Total: 12 points) Given that the complex number $z_1$ satisfies $(z_1 - 2)(1 + i) = 1 - i$ (where $i$ is the imaginary unit), and the imaginary part of complex number $z_2$ is 2, and $z_1 \cdot z_2$ is a real number, find $z_2$.
20. (Total: 14 points; Part 1: 7 points; Part 2: 7 points) $ABCD - A_1B_1C_1D_1$ is a right square prism with base edge length 1 and height $AA_1 = 2$. Find: (1) The angle between skew lines $BD$ and $AB_1$ (express the result using inverse trigonometric functions); (2) The volume of tetrahedron $AB_1D_1C$ [Figure]
21. (Total: 14 points; Part 1: 6 points; Part 2: 8 points) Given the function $f(x) = a \cdot 2^x + b \cdot 3^x$, where constants $a, b$ satisfy $a \cdot b \neq 0$ (1) If $a \cdot b > 0$, determine the monotonicity of function $f(x)$; (2) If $a \cdot b < 0$, find the range of $x$ when $f(x+1) > f(x)$.
22. (Total: 16 points; Part 1: 4 points; Part 2: 6 points; Part 3: 6 points) Given the ellipse $C: \frac{x^2}{m^2} + y^2 = 1$ (constant $m > 1$), $P$ is a moving point on curve $C$, $M$ is the right vertex of curve $C$, and the fixed point $A$ has coordinates $(2, 0)$ (1) If $M$ coincides with $A$, find the coordinates of the foci of curve $C$; (2) If $m = 3$, find the maximum and minimum values of $|PA|$; (3) If the minimum value of $|PA|$ is $|MA|$, find the range of the real number $m$.
23. (Total: 18 points; Part 1: 4 points; Part 2: 6 points; Part 3: 8 points) Given that the general terms of sequences $\{a_n\}$ and $\{b_n\}$ are $a_n = 3n + 6, b_n = 2n + 7$ $(n \in \mathbb{N}^*)$ respectively. The elements in the set $\{x \mid x = a_n, n \in \mathbb{N}^*\} \cup \{x \mid x = b_n, n \in \mathbb{N}^*\}$ are arranged in increasing order to form a sequence $c_1, c_2, c_3, \cdots, c_n, \cdots$ (1) Find the three smallest numbers that are both terms of sequence $\{a_n\}$ and terms of sequence $\{b_n\}$; (2) Among the terms $c_1, c_2, c_3, \cdots, c_{40}$, how many are not terms of sequence $\{b_n\}$? Please explain your reasoning; (3) Find the sum of the first $4n$ terms $S_{4n}$ $(n \in \mathbb{N}^*)$ of sequence $\{c_n\}$. 2011 Shanghai College Entrance Examination Mathematics (Liberal Arts) Answers 1. $\{x \mid x < 1\}$; 2. $-2$; 3. $-\frac{3}{2}$; 4. $\sqrt{5}$; 5. $x + 2y - 11 = 0$; 6. $x < 0$ or $x > 1$; 7. $3\pi$; 8. $\sqrt{6}$; 9. $\frac{5}{2}$; 10. 2; 11. 6; 12. $\frac{15}{2}$; 13. 0.985; 14. $[-2, 7]$. 15. A; 16. D; 17. A; 18. B. 19. Solution: $(z_1 - 2)(1 + i) = 1 - i \Rightarrow z_1 = 2 - i$ Let $z_2 = a + 2i, a \in \mathbb{R}$, then $z_1 z_2 = (2 - i)(a + 2i) = (2a + 2) + (4 - a)i$. Since $z_1 z_2 \in \mathbb{R}$, we have $4 - a = 0$, so $z_2 = 4 + 2i$. 20. Solution: (1) Connect $BD, AB_1, B_1D_1, AD_1$. Since $BD \parallel B_1D_1$ and $AB_1 = AD_1$, the angle between skew lines $BD$ and $AB_1$ is $\angle AB_1D_1$. Let $\angle AB_1D_1 = \theta$. $\cos \theta = \frac{AB_1^2 + B_1D_1^2 - AD_1^2}{2AB_1 \times B_1D_1} = \frac{\sqrt{10}}{10}$. Therefore, the angle between skew lines $BD$ and $AB_1$ is $\arccos \frac{\sqrt{10}}{10}$. (2) Connect $AC, CB_1, CD_1$. The volume of the tetrahedron is [Figure] $V = V_{ABCD - A_1B_1C_1D_1} - 4 \times V_{C - B_1C_1D_1} = 2 - 4 \times \frac{1}{3} = \frac{2}{3}$. 21. Solution: (1) When $a > 0, b > 0$, for any $x_1, x_2 \in \mathbb{R}, x_1 < x_2$, we have $f(x_1) - f(x_2) = a(2^{x_1} - 2^{x_2}) + b(3^{x_1} - 3^{x_2})$. Since $2^{x_1} < 2^{x_2}, a > 0 \Rightarrow a(2^{x_1} - 2^{x_2}) < 0$, and $3^{x_1} < 3^{x_2}, b > 0 \Rightarrow b(3^{x_1} - 3^{x_2}) < 0$, we have $f(x_1) - f(x_2) < 0$, so function $f(x)$ is increasing on $\mathbb{R}$. When $a < 0, b < 0$, by similar reasoning, function $f(x)$ is decreasing on $\mathbb{R}$. (2) $f(x+1) - f(x) = a \cdot 2^x + 2b \cdot 3^x > 0$. When $a < 0, b > 0$, we have $\left(\frac{3}{2}\right)^x > -\frac{a}{2b}$, so $x > \log_{1.5}\left(-\frac{a}{2b}\right)$. When $a > 0, b < 0$, we have $\left(\frac{3}{2}\right)^x < -\frac{a}{2b}$, so $x < \log_{1.5}\left(-\frac{a}{2b}\right)$. 22. Solution: (1) $m = 2$, the ellipse equation is $\frac{x^2}{4} + y^2 = 1$, and $