Let $a_1, a_2, a_3, a_4$ be a geometric sequence with first term 10 and common ratio 10. Let $b = \sum_{n=1}^{3} \log_{a_n} a_{n+1}$. Select the correct option.
(1) $2 < b \leq 3$
(2) $3 < b \leq 4$
(3) $4 < b \leq 5$
(4) $5 < b \leq 6$
(5) $6 < b \leq 7$

Let $c$ be a real number such that the system of linear equations $\left\{ \begin{array}{c} x - y + z = 0 \\ 2x + cy + 3z = 1 \\ 3x - 3y + cz = 0 \end{array} \right.$ has no solution. Select the value of $c$.
(1) $-3$
(2) $-2$
(3) $0$
(4) $2$
(5) $3$

In coordinate space, $O$ is the origin, and point $P$ is in the first octant with $\overline{OP} = 1$. The line $OP$ makes an angle of $45^\circ$ with the $x$-axis, and the distance from point $P$ to the $y$-axis is $\frac{\sqrt{6}}{3}$. Select the $z$-coordinate of point $P$.
(1) $\frac{1}{2}$
(2) $\frac{\sqrt{2}}{4}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{6}}{6}$
(5) $\frac{\sqrt{3}}{6}$

Let the polynomials $f(x) = x^3 + 2x^2 - 2x + k$ and $g(x) = x^2 + ax + 1$, where $k, a$ are real numbers. Given that $g(x)$ divides $f(x)$ and the equation $g(x) = 0$ has complex roots, select the option that is a root of the equation $f(x) = 0$.
(1) $-3$
(2) $0$
(3) $1$
(4) $\frac{1 + \sqrt{-3}}{2}$
(5) $\frac{3 + \sqrt{-5}}{2}$

On the coordinate plane, there is a figure $\Gamma$ with equation $(x-1)^2 + (y-1)^2 = 101$. Select the correct options.
(1) $\Gamma$ intersects the negative $x$-axis and negative $y$-axis at $(-9, 0)$ and $(0, -9)$ respectively
(2) The point on $\Gamma$ with the maximum $x$-coordinate is $(11, 0)$
(3) The maximum distance from a point on $\Gamma$ to the origin is $\sqrt{2} + \sqrt{101}$
(4) Points on $\Gamma$ in the third quadrant can be expressed in polar coordinates as $[9, \theta]$, where $\pi < \theta < \frac{3}{2}\pi$
(5) After a rotational linear transformation, the figure can still be expressed by a quadratic equation in two variables without an $xy$ term

Suppose a $2 \times 2$ matrix $\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$ representing a linear transformation maps three points $O(0,0), A(1,0), B(0,1)$ on the coordinate plane to $O(0,0), A'(3, \sqrt{3}), B'(-\sqrt{3}, 3)$ respectively, and maps a point $C(x, y)$ at distance 1 from the origin to point $C'(x', y')$. Select the correct options.
(1) The determinant $\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| = 6$
(2) $\overline{OC'} = 2\sqrt{3}$
(3) The angle between $\overrightarrow{OC}$ and $\overrightarrow{OC'}$ is $60^\circ$
(4) It is possible that $y = y'$
(5) If $x < y$ then $x' < y'$

Suppose $A, B$ are two points on a parabola $\Gamma$ and their connecting line segment passes through the focus $F$ of $\Gamma$. Let the projections of $A, F, B$ onto the directrix of $\Gamma$ be $A', F', B'$ respectively. Select the option equal to $\frac{\overline{AF}}{\overline{A'F'}}$. (Note: This schematic diagram only illustrates the relative positions of the points; the distance relationships between points are not accurate)
(1) $\tan \angle 1$, where $\angle 1 = \angle A'F'A$
(2) $\sin \angle 2$, where $\angle 2 = \angle AF'F$
(3) $\sin \angle 3$, where $\angle 3 = \angle A'AF$
(4) $\cos \angle 4$, where $\angle 4 = \angle F'FB$
(5) $\tan \angle 5$, where $\angle 5 = \angle FF'B$

Suppose two sequences $\langle a_n \rangle$ and $\langle b_n \rangle$ satisfy $b_n + \frac{4n-1}{n} < a_n < 3b_n$ for all positive integers $n$. Given that $\lim_{n \to \infty} a_n = 6$, select the correct options.
(1) $b_n < 6 - \frac{4n-1}{n}$
(2) $b_n > \frac{4n-1}{2n}$
(3) The sequence $\langle b_n \rangle$ may diverge
(4) $a_{10000} < 6.1$
(5) $a_{10000} > 5.9$

A department store is preparing many red envelopes for customers to draw during the Lunar New Year period, claiming that the activity will continue until all red envelopes are distributed. The drawing box contains 5 sticks, of which only 1 stick is marked ``Great Fortune'', and each stick has an equal chance of being drawn. Each customer draws one stick from the box, records it, puts it back, and draws again for the next round, drawing at most 3 times. When two consecutive draws result in ``Great Fortune'', the customer stops drawing and receives a red envelope. We can view whether each customer receives a red envelope as a Bernoulli trial. Let $X$ be the position of the first customer to receive a red envelope in the entire activity, and let $E(X)$ denote the expected value of the random variable $X$. Then $E(X) = $ \hspace{2cm}. (Round to the nearest integer)

A teacher requires the class monitor to distribute review sheets for Chinese, English, Mathematics, Social Studies, and Science—5 subjects in total—over Monday, Tuesday, Wednesday, and Thursday of next week. At least one subject's sheet must be distributed each day for students to take home for practice and submit the next day. Since there are Chinese and English classes on Tuesday, the Chinese teacher requires that the Chinese sheet must be distributed on Monday for review; and the English teacher, having assigned other work that day, requires that the English sheet not be distributed on Tuesday. According to these requirements, the class monitor has \hspace{2cm} ways to arrange the distribution.

In the complex plane, a complex number $z$ is in the first quadrant and satisfies $|z| = 1$ and $\left|\frac{-3+4i}{5} - z^3\right| = \left|\frac{-3+4i}{5} - z\right|$, where $i = \sqrt{-1}$. If the real part of $z$ is $a$ and the imaginary part is $b$, then $a = \dfrac{\sqrt{\phantom{0}}}{\sqrt{\phantom{0}}}$ and $b = \dfrac{\sqrt{\phantom{0}}}{\sqrt{\phantom{0}}}$. (Express in simplest radical form)

There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. The intersection of a horizontal plane at distance $x$ from $A$ (where $0 < x < 15$) with the wooden block is a rectangle of area $20x + \frac{4}{9}x^2$. Divide the line segment $\overline{AP}$ into $n$ equal parts, and denote the division points along the direction of vector $\overrightarrow{AP}$ as $A = P_0, P_1, \ldots, P_{n-1}, P_n = P$. For each segment $\overline{P_{k-1}P_k}$, consider the rectangular prism formed by taking the rectangle formed by the intersection of the horizontal plane passing through $P_k$ with this wooden block as the base and $\overline{P_{k-1}P_k}$ as the height. Please use this slicing method to write down the Riemann sum estimating the volume of this wooden block (no need to simplify), express the volume of this wooden block as a definite integral, and find its value. (Non-multiple choice question, 6 points)

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $\frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$. Let this expression be $f(x)$, with domain $\{x \mid 1 < x < 8\}$. Find $f(x)$ and its derivative. (Non-multiple choice question, 4 points)

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $f(x) = \frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$, with domain $\{x \mid 1 < x < 8\}$. Explain where $f(x)$ is increasing and decreasing in its domain. Determine the value of $x$ for which the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is maximum. (Non-multiple choice question, 4 points)

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $f(x) = \frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$, with domain $\{x \mid 1 < x < 8\}$. Using the linear approximation (first-order approximation) of $f(x)$, find the approximate value of $\cos\theta$ when $x = 4.96$. (Non-multiple choice question, 4 points)