1. Let $f(x) = ax^6 - bx^4 + 3x - \sqrt{2}$, where $a, b$ are non-zero real numbers. Then the value of $f(5) - f(-5)$ is (1) $-30$ (2) $0$ (3) $2\sqrt{2}$ (4) $30$ (5) Cannot be determined (depends on $a, b$)
2. How many positive integers $n$ are there such that the line passing through points $A(-n, 0)$ and $B(0, 2)$ on the coordinate plane also passes through point $P(7, k)$, where $k$ is a positive integer? (1) 2 (2) 4 (3) 6 (4) 8 (5) Infinitely many
3. The temperature function for a certain desert region during a certain period is $f(t) = -t^2 + 10t + 11$, where $1 \leq t \leq 10$. The maximum temperature difference in this region during this period is (1) 9 (2) 16 (3) 20 (4) 25 (5) 36
4. On the coordinate plane, how many intersection points do the graphs of the equation $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and $\frac{(x+1)^2}{16} - \frac{y^2}{9} = 1$ have? (1) 1 (2) 2 (3) 3 (4) 4 (5) 0
5. Regarding the number of intersection points between the graph of the function $y = \sin x$ and the graph of $y = \frac{x}{10\pi}$ on the coordinate plane, which of the following options is correct? (1) The number of intersection points is infinite (2) The number of intersection points is odd and greater than 20 (3) The number of intersection points is odd and less than 20 (4) The number of intersection points is even and greater than or equal to 20 (5) The number of intersection points is even and less than 20
II. Multiple-Choice Questions (30 points)
Instructions: For questions 6 to 11, each of the five options is independent, and at least one option is correct. Select the correct options and mark them on the "Answer Sheet". No deduction for wrong answers. Five points are awarded for all five options correct, 2.5 points for only one wrong option, and no points for two or more wrong options.
6. If $\Gamma = \{z \mid z \text{ is a complex number and } |z - 1| = 1\}$, which of the following points lie on the graph $\Omega = \{w \mid w = iz, z \in \Gamma\}$? (1) $2i$ (2) $-2i$ (3) $1 + i$ (4) $1 - i$ (5) $-1 + i$
7. On the coordinate plane, there are two distinct points $P$ and $Q$, where point $P$ has coordinates $(s, t)$. The perpendicular bisector $L$ of segment $\overline{PQ}$ has equation $3x - 4y = 0$. Which of the following options are correct? (1) Vector $\overrightarrow{PQ}$ is parallel to vector $(3, -4)$ (2) The length of segment $\overline{PQ}$ equals $\frac{|6s - 8t|}{5}$ (3) Point $Q$ has coordinates $(t, s)$ (4) The line passing through $Q$ and parallel to line $L$ must pass through point $(-s, -t)$ (5) If $O$ denotes the origin, then the dot product of vector $\overrightarrow{OP} + \overrightarrow{OQ}$ and vector $\overrightarrow{PQ}$ must be 0
10. Let $a$ be a real number greater than 1. Consider the functions $f(x) = a^x$ and $g(x) = \log_a x$. Which of the following options are correct? (1) If $f(3) = 6$, then $g(36) = 6$ (2) $\frac{f(238)}{f(219)} = \frac{f(38)}{f(19)}$ (3) $g(238) - g(219) = g(38) - g(19)$ (4) If $P, Q$ are two distinct points on the graph of $y = g(x)$, then the slope of line $PQ$ must be positive (5) If the line $y = 5x$ and the graph of $y = f(x)$ have two intersection points, then the line $y = \frac{1}{5}x$ and the graph of $y = g(x)$ also have two intersection points
11. Let $f(x)$ be a real cubic polynomial with leading coefficient 1. Given that $f(1) = 1, f(2) = 2, f(5) = 5$, in which of the following intervals must $f(x) = 0$ have a real root? (1) $(-\infty, 0)$ (2) $(0, 1)$ (3) $(1, 2)$ (4) $(2, 5)$ (5) $(5, \infty)$
Part Two: Fill-in Questions (45 points)
Instructions: 1. For questions A through I, mark your answers on the "Answer Sheet" at the row numbers indicated (12–41). 2. Each completely correct answer receives 5 points. Wrong answers do not result in deduction. Incomplete answers receive no points. A. Let real number $x$ satisfy $0 < x < 1$ and $\log_x 4 - \log_2 x = 1$. Then $x = $ (12). (Express as a fraction in lowest terms) B. In $\triangle ABC$ on the coordinate plane, $P$ is the midpoint of side $\overline{BC}$, and $Q$ is on side $\overline{AC}$ such that $\overline{AQ} = 2\overline{QC}$. Given that $\overrightarrow{PA} = (4, 3)$ and $\overrightarrow{PQ} = (1, 5)$, then $\overrightarrow{BC} = ($ (14) (15), (16) (17) $)$. C. In a certain talent competition, to avoid excessive subjective influence from individual judges on contestants' scores, the