4. For fill-in-the-blank questions, write the answer directly. For solution questions, write out explanations, proofs, or calculation steps.
Section I (50 points total)
I. Multiple Choice Questions: This section has 10 questions, each worth 5 points, totaling 50 points. For each question, only one of the four options is correct.
(1) Given sets $A = \{ x \mid 2 < x < 4 \}$ and $B = \{ x \mid (x - 1)(x - 3) < 0 \}$, then $A \cap B =$
(A) $(1,3)$
(B) $(1,4)$
(C) $(2,3)$
(D) $(2,4)$
(2) If the complex number $z$ satisfies $\frac{\bar{z}}{1-i} = i$, where $i$ is the imaginary unit, then $z =$
(A) $1 - i$
(B) $1 + i$
(C) $-1$
(D) $-1 + i$
(3) Let $a = 0.6^{0.6}$, $b = 0.6^{1.5}$, $c = 1.5^{0.6}$. The size relationship among $a$, $b$, $c$ is
(A) $a < b < c$
(B) $a < c < b$
(C) $b < a < c$
(D) $b < c < a$
(4) To obtain the graph of the function $y = \sin(4x - \frac{\pi}{3})$, we only need to shift the graph of the function $y = \sin 4x$ ()
(A) to the left by $\frac{\pi}{12}$ units
(B) to the right by $\frac{\pi}{12}$ units
(C) to the left by $\frac{\pi}{3}$ units
(D) to the right by $\frac{\pi}{3}$ units
(5) If $m \in \mathbb{N}$ and the proposition is ``if $m > 0$, then the equation $x^2 + x - m = 0$ has real roots'', then its contrapositive is
(A) If the equation $x^2 + x - m = 0$ has real roots, then $m > 0$
(B) If the equation $x^2 + x - m = 0$ has real roots, then $m \leq 0$
(C) If the equation $x^2 + x - m = 0$ has no real roots, then $m > 0$
(D) If the equation $x^2 + x - m = 0$ has no real roots, then $m \leq 0$ (6) To compare the 2 PM temperature situation in two locations A and B in a certain month, 5 days are randomly selected from that month. The temperature data at 2 PM on these 5 days (in ${}^{\circ}C$) are presented in the stem-and-leaf plot shown below. Consider the following conclusions:
| \multicolumn{2}{c|}{Location A} | | \multicolumn{2}{|c}{Location B} | | |
| 9 | 9 | 6 | 2 | 8 | 9 | |
| 1 | 1 | 3 | 0 | 1 | 2 |
(1) The average temperature at 2 PM in Location A is lower than that in Location B;
(2) The average temperature at 2 PM in Location A is higher than that in Location B;
(3) The standard deviation of the temperature at 2 PM in Location A is smaller than that in Location B;
(4) The standard deviation of the temperature at 2 PM in Location A is larger than that in Location B. The numbers of the statistical conclusions that can be obtained from the stem-and-leaf plot are
(A) (1)(3)
(B) (1)(4)
(C) (2)(3)
(D) (2)(4) (7) A number $x$ is randomly selected from the interval $[0,2]$. The probability that the event ``$-1 \leq \log_{\frac{1}{2}}(x + \frac{1}{2}) \leq 1$'' occurs is
(A) $\frac{3}{4}$
(B) $\frac{2}{3}$
(C) $\frac{1}{3}$
(D) $\frac{1}{4}$ (8) If the function $f(x) = \frac{2^x + 1}{2^x - a}$ is an odd function, then the range of $x$ for which $f(x) > 3$ holds is
(A) $(-\infty, -1)$
(B) $(-1, 0)$
(C) $(0, 1)$
(D) $(1, +\infty)$ (9) An isosceles right triangle has legs of length 2. When the triangle is rotated one full revolution around the line containing its hypotenuse, the volume of the solid formed by the resulting surface is
(A) $\frac{2\sqrt{2}\pi}{3}$
(B) $\frac{4\sqrt{2}\pi}{3}$
(C) $2\sqrt{2}\pi$
(D) $4\sqrt{2}\pi$ (10) Let the function $f(x) = \begin{cases} 3x - b, & x < 1 \\ 2^x, & x \geq 1 \end{cases}$. If $f(f(\frac{5}{6})) = 4$, then $b =$
(A) $1$
(B) $\frac{7}{8}$
(C) $\frac{3}{4}$
(D) $\frac{1}{2}$
Section II (100 points total)
II. Fill-in-the-Blank Questions: This section has 5 questions, each worth 5 points, totaling 25 points
(11) Execute the flowchart on the right. If the input value of $x$ is 1, then the output value of $y$ is
[Figure] (12) If $x, y$ satisfy the constraints $\begin{cases} y - x < 1 \\ x + y \leq 3 \\ y > 1 \end{cases}$, then the maximum value of $z = x + 3y$ is \_\_\_\_. (13) Two tangent lines are drawn from point $P(1, \sqrt{3})$ to the circle $x^2 + y^2 = 1$, with tangent points $A$ and $B$ respectively. Then $\overrightarrow{PA} \cdot \overrightarrow{PB} =$ (14) Define the operation ``$\otimes$'': $x \otimes y = \frac{x^2 - y^2}{xy}$ $(x, y \in \mathbb{R}, xy \neq 0)$. When $x > 0, y > 0$, the minimum value of $x \otimes y + 2y \otimes x$ is \_\_\_\_. (15) A line through the right focus of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$ parallel to one of its asymptotes intersects $C$ at point $P$. If the x-coordinate of point $P$ is $2a$, then the eccentricity of $C$ is \_\_\_\_.
III. Solution Questions: This section has 6 questions, totaling 75 points
(16) (This question is worth 12 points)
A school surveyed all 45 students in a certain class regarding their participation in calligraphy club and speech club. The data is shown in the table below (in persons):
| Participate in Calligraphy Club | Do Not Participate in Calligraphy Club |
| Participate in Speech Club | 8 | 5 |
| Do Not Participate in Speech Club | 30 | |
(1) If one student is randomly selected from the class, find the probability that the student participates in at least one of the two clubs;
(2) Among the 8 students who participate in both calligraphy club and speech club, there are 5 male students $A_1, A_2, A_3, A_4, A_5$ and 3 female students $B_1, B_2, B_3$. One person is randomly selected from the 5 male students and one person from the 3 female students. Find the probability that $A_1$ is selected and $B_1$ is not selected. (17) (This question is worth 12 points) In $\triangle ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively. Given that $\cos B = \frac{\sqrt{3}}{3}$, $\sin(A + B) = \frac{\sqrt{6}}{9}$, and $ac = 2\sqrt{3}$, find $\sin A$ and the value of $c$. (18) As shown in the figure, in the triangular frustum $DEF - ABC$, $AB = 2DE$, and $G, H$ are the midpoints of $AC$ and $BC$ respectively. (I) Prove that $BD \parallel$ plane $FGH$; (II) If $CF \perp BC$ and $AB \perp BC$, prove that plane $BCD \perp$ plane $FGH$.
[Figure] (19) (This question is worth 12 points) The sequence $\{a_n\}$ is an arithmetic sequence with a positive first term. The sum of the first $n$ terms of the sequence $\{\frac{1}{a_n \cdot a_{n+1}}\}$ is $\frac{n}{2n+1}$.
(1) Find the general term formula of the sequence $\{a_n\}$; (II) Let $b_n = (a_n + 1) \cdot 2^{a_n}$. Find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$. (20) (This question is worth 13 points) Let $f(x) = (x + a)\ln x$ and $g(x) = \frac{x^2}{e^x}$. The tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ is parallel to the line $2x - y = 0$. (I) Find the value of $a$; (II) Does there exist a natural number $k$ such that the equation $f(x) = g(x)$ has a unique root in the interval $(k, k+1)$? If such a $k$ exists, find it; if not, explain why; (III) Let the function $m(x) = \min\{f(x), g(x)\}$ (where $\min\{p, q\}$ denotes the smaller of $p$ and $q$). Find the maximum value of $m(x)$.
(21) (This question is worth 14 points)
In the rectangular coordinate system $xOy$, the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ has eccentricity $\frac{\sqrt{3}}{2}$, and the point $(\sqrt{3}, \frac{1}{2})$ lies on the ellipse $C$. (I) Find the equation of ellipse $C$; (II) Let the ellipse $E: \frac{x^2}{4a^2} + \frac{y^2}{4b^2} = 1$. Let $P$ be an arbitrary point on ellipse $C$. A line $y = kx + m$ through point $P$ intersects ellipse $E$ at points $A$ and $B$. The ray $PO$ intersects ellipse $E$ at point $Q$.
(i) Find the value of $\frac{|OQ|}{|OP|}$;
(ii) Find the maximum area of $\triangle ABQ$.