In the coordinate plane, let $O$ be the origin, and let $A$ and $B$ be two distinct points different from $O$. Let $C_{1}$, $C_{2}$, $C_{3}$ be three points in the plane satisfying $\overrightarrow{OC}_{n} = \overrightarrow{OA} + n\overrightarrow{OB}$, $n = 1, 2, 3$. Select the correct options.
(1) $\overrightarrow{OC}_{1} \neq \overrightarrow{0}$
(2) $\overline{OC_{1}} < \overline{OC_{2}} < \overline{OC_{3}}$
(3) $\overrightarrow{OC}_{1} \cdot \overrightarrow{OA} < \overrightarrow{OC}_{2} \cdot \overrightarrow{OA} < \overrightarrow{OC}_{3} \cdot \overrightarrow{OA}$
(4) $\overrightarrow{OC_{1}} \cdot \overrightarrow{OB} < \overrightarrow{OC_{2}} \cdot \overrightarrow{OB} < \overrightarrow{OC_{3}} \cdot \overrightarrow{OB}$
(5) $C_{1}$, $C_{2}$, $C_{3}$ are collinear

A biased coin has probability $\frac { 1 } { 3 }$ of showing heads and probability $\frac { 2 } { 3 }$ of showing tails. On a coordinate plane, a game piece moves to the next position based on the result of flipping this coin, according to the following rules: (I) If heads appears, the piece moves from its current position in the direction and distance of vector $( - 1,2 )$ to the next position; (II) If tails appears, the piece moves from its current position in the direction and distance of vector $( 1,0 )$ to the next position. For example: If the game piece is currently at coordinates $( 2,4 )$ and tails appears, the piece moves to coordinates $( 3,4 )$. Suppose the game piece starts at the origin $( 0,0 )$ and, according to the above rules, flips the coin 6 times consecutively, with each flip being independent. After 6 moves, the game piece is most likely to stop at coordinates (12), (13)).

Consider a trapezoid $ABCD$ where $\overline { AB }$ is parallel to $\overline { DC }$ . It is known that points $E$ and $F$ lie on diagonals $\overline { AC }$ and $\overline { BD }$ respectively, and $\overline { AB } = \frac { 2 } { 5 } \overline { DC }$ , $\overline { AE } = \frac { 3 } { 2 } \overline { EC }$ , $\overline { BF } = \frac { 2 } { 3 } \overline { FD }$ . If vector $\overrightarrow { FE }$ is expressed as $\alpha \overrightarrow { AC } + \beta \overrightarrow { AD }$ , then the real numbers $\alpha = \frac { \text{(16)} } { \text{(17)(18)} } , \beta = \frac { \text{(19)(20)} } { \text{(21)(22)} }$ . (Express as fractions in lowest terms)

As shown in the figure, a robot starts from a point $P$ on the ground and moves according to the following rules: First, move forward 1 meter in a certain direction, then rotate counterclockwise $45 ^ { \circ }$ in the direction of movement; move forward 1 meter in the new direction, then rotate clockwise $90 ^ { \circ }$ in the direction of movement; move forward 1 meter in the new direction, then rotate counterclockwise $45 ^ { \circ }$ in the direction of movement; move forward 1 meter in the new direction, then rotate clockwise $90 ^ { \circ }$ in the direction of movement, and so on. The path traced by the robot forms a closed region. The area of this closed region is (28) + (29) $\sqrt { (30) }$ square meters. (Express as a fraction in simplest radical form)

Let $\vec { a }$ and $\vec { b }$ be non-zero vectors in the plane. If the area of the triangle formed by $2 \vec { a } + \vec { b }$ and $\vec { a } + 2 \vec { b }$ is 6, what is the area of the triangle formed by $3 \vec { a } + \vec { b }$ and $\vec { a } + 3 \vec { b }$?
(1) 8
(2) 9
(3) 12
(4) 13.5
(5) 16

On a coordinate plane, it is known that vector $\overrightarrow{PQ} = \left(\log \frac{1}{5}, -10^{-5}\right)$, where point $P$ has coordinates $\left(\log \frac{1}{2}, 2^{-5}\right)$. Select the correct option.
(1) Point $Q$ is in the first quadrant
(2) Point $Q$ is in the second quadrant
(3) Point $Q$ is in the third quadrant
(4) Point $Q$ is in the fourth quadrant
(5) Point $Q$ is on a coordinate axis

Let $P$ be a point inside $\triangle A B C$, and $\overrightarrow { A P } = a \overrightarrow { A B } + b \overrightarrow { A C }$ , where $a , b$ are distinct real numbers. Let $Q , R$ be on the same plane, with $\overrightarrow { A Q } = b \overrightarrow { A B } + a \overrightarrow { A C } , ~ \overrightarrow { A R } = a \overrightarrow { A B } + ( b - 0.05 ) \overrightarrow { A C }$ . Select the correct options.
(1) $Q , R$ are also both inside $\triangle A B C$
(2) $| \overrightarrow { A P } | = | \overrightarrow { A Q } |$
(3) Area of $\triangle A B P$ = Area of $\triangle A C Q$
(4) Area of $\triangle B C P$ = Area of $\triangle B C Q$
(5) Area of $\triangle A B P$ > Area of $\triangle A B R$

Consider points $O(0,0), A, B, C, D, E, F, G$ on a coordinate plane, where points $B$, $C$ and $D$, $E$ and $F$, $G$ and $A$ are located in the first, second, third, and fourth quadrants respectively. If $\vec{v}$ is a vector on the coordinate plane satisfying $\vec{v} \cdot \overrightarrow{OA} > 0$ and $\vec{v} \cdot \overrightarrow{OB} > 0$, then the dot product of $\vec{v}$ with which of the following vectors must be negative?
(1) $\overrightarrow{OC}$
(2) $\overrightarrow{OD}$
(3) $\overrightarrow{OE}$
(4) $\overrightarrow{OF}$
(5) $\overrightarrow{OG}$

On a coordinate plane, there is a circle with radius 7 and center $O$. It is known that points $A, B$ are on the circle and $\overline{AB} = 8$. Then the dot product $\overrightarrow{OA} \cdot \overrightarrow{OB} =$ (14--1) (14--2).

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)

On a coordinate plane, a particle starts from point $( - 3 , - 2 )$ and moves 5 units in the direction of vector $( a , 1 )$ and arrives exactly at the $x$-axis, where $a$ is a positive real number. What is the value of $a$?
(1) $\frac { \sqrt { 13 } } { 2 }$
(2) 2
(3) $\sqrt { 5 }$
(4) $\frac { \sqrt { 21 } } { 2 }$
(5) $2 \sqrt { 6 }$

In coordinate space, consider a unit cube with edge length 1, with one vertex $O$ fixed. From the seven vertices other than $O$, two distinct points are randomly selected, denoted as $P$ and $Q$. What is the expected value of the dot product $\overrightarrow{OP} \cdot \overrightarrow{OQ}$ among the following options?
(1) $\frac{4}{7}$
(2) $\frac{5}{7}$
(3) $\frac{6}{7}$
(4) 1
(5) $\frac{8}{7}$

It is known that a right triangle $\triangle A B C$ has side lengths $\overline { A B } = \sqrt { 7 }$, $\overline { A C } = \sqrt { 3 }$, $\overline { B C } = 2$. If isosceles triangles $\triangle M A B$ and $\triangle N A C$ with vertex angles equal to $120 ^ { \circ }$ are constructed outside $\triangle A B C$ using $\overline { A B }$ and $\overline { A C }$ as bases respectively, then $\overline { M N } ^ { 2 } =$ . (Express as a fraction in lowest terms)