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) $2 \sqrt { 6 }$

The half-life $T$ of a radioactive substance is defined as ``every time period $T$ passes, the mass of the substance decays to half of its original amount''. A lead container contains two radioactive substances $A$ and $B$ with half-lives $T _ { A }$ and $T _ { B }$ respectively. At the start of recording, the masses of these two substances are equal. After 112 days, measurement shows that the mass of substance $B$ is one-quarter of the mass of substance $A$. Based on the above, which of the following is the relationship between $T _ { A }$ and $T _ { B }$?
(1) $- 2 + \frac { 112 } { T _ { A } } = \frac { 112 } { T _ { B } }$
(2) $2 + \frac { 112 } { T _ { A } } = \frac { 112 } { T _ { B } }$
(3) $- 2 + \log _ { 2 } \frac { 112 } { T _ { A } } = \log _ { 2 } \frac { 112 } { T _ { B } }$
(4) $2 + \log _ { 2 } \frac { 112 } { T _ { A } } = \log _ { 2 } \frac { 112 } { T _ { B } }$
(5) $2 \log _ { 2 } \frac { 112 } { T _ { A } } = \log _ { 2 } \frac { 112 } { T _ { B } }$

What is the limit
$$\lim _ { n \rightarrow \infty } \frac { 3 } { n ^ { 2 } } \left( \sqrt { 4 n ^ { 2 } + 9 \times 1 ^ { 2 } } + \sqrt { 4 n ^ { 2 } + 9 \times 2 ^ { 2 } } + \cdots + \sqrt { 4 n ^ { 2 } + 9 \times ( n - 1 ) ^ { 2 } } \right)$$
which of the following definite integrals can represent?
(1) $\int _ { 0 } ^ { 3 } \sqrt { 1 + x ^ { 2 } } d x$
(2) $\int _ { 0 } ^ { 3 } \sqrt { 1 + 9 x ^ { 2 } } d x$
(3) $\int _ { 0 } ^ { 3 } \sqrt { 4 + x ^ { 2 } } d x$
(4) $\int _ { 0 } ^ { 3 } \sqrt { 4 + 9 x ^ { 2 } } d x$
(5) $\int _ { 0 } ^ { 3 } \sqrt { 4 x ^ { 2 } + 9 } d x$

Let $a , b$ be real numbers. It is known that the four numbers $- 3 , - 1, 4, 7$ all satisfy the inequality $| x - a | \leq b$ in $x$. Select the correct options.
(1) $\sqrt { 10 }$ also satisfies the inequality $| x - a | \leq b$ in $x$
(2) $3, 1 , - 4 , - 7$ satisfy the inequality $| x + a | \leq b$ in $x$
(3) $- \frac { 3 } { 2 } , - \frac { 1 } { 2 } , 2 , \frac { 7 } { 2 }$ satisfy the inequality $| x - a | \leq \frac { b } { 2 }$ in $x$
(4) $b$ could equal 4
(5) $a$ and $b$ could be equal

Consider the real coefficient polynomial $f ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 2 x ^ { 2 } + a x + b$. It is known that the equation $f ( x ) = 0$ has a complex root $1 + 2 i$ (where $i = \sqrt { - 1 }$). Select the correct options.
(1) $1 - 2i$ is also a root of $f ( x ) = 0$
(2) Both $a$ and $b$ are positive numbers
(3) $f ^ { \prime } ( 2.1 ) < 0$
(4) The function $y = f ( x )$ has a local minimum at $x = 1$
(5) The $x$-coordinates of all inflection points of the graph $y = f ( x )$ are greater than 0

Let $a , b , c , d , r , s , t$ all be real numbers. It is known that three non-zero vectors $\vec { u } = ( a , b , 0 )$, $\vec { v } = ( c , d , 0 )$, and $\vec { w } = ( r , s , t )$ in coordinate space satisfy the dot products $\vec { w } \cdot \vec { u } = \vec { w } \cdot \vec { v } = 0$. Consider the $3 \times 3$ matrix $A = \left[ \begin{array} { l l l } a & b & 0 \\ c & d & 0 \\ r & s & t \end{array} \right]$. Select the correct options.
(1) If $\vec { u } \cdot \vec { v } = 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(2) If $t \neq 0$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(3) If there exists a vector $\overrightarrow { w ^ { \prime } }$ satisfying $\overrightarrow { w ^ { \prime } } \cdot \vec { u } = \overrightarrow { w ^ { \prime } } \cdot \vec { v } = 0$ and cross product $\overrightarrow { w ^ { \prime } } \times \vec { w } \neq \overrightarrow { 0 }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(4) If for any three real numbers $e , f , g$, the vector $( e , f , g )$ can be expressed as a linear combination of $\vec { u } , \vec { v } , \vec { w }$, then the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$
(5) If the determinant $\left| \begin{array} { l l } a & b \\ c & d \end{array} \right| \neq 0$, then the determinant of $A$ is not equal to 0

There is a circular clock with numbers $1, 2 , \ldots , 12$ marked in clockwise order (as shown in the figure). Initially, a game piece is placed at the ``12'' o'clock position on this clock. Then, each time a fair coin is tossed, the game piece is moved according to the following rules:

• If heads appears, move the game piece 5 clock positions clockwise from its current position.
• If tails appears, move the game piece 5 clock positions counterclockwise from its current position.

For example: If the coin is tossed three times and all are heads, the game piece moves to the ``5'' o'clock position on the first move, the ``10'' o'clock position on the second move, and the ``3'' o'clock position on the third move.
For any positive integer $n$, let the random variable $X _ { n }$ represent the clock position of the game piece after $n$ moves according to the above rules, $P \left( X _ { n } = k \right)$ represents the probability that $X _ { n } = k$ (where $k = 1, 2 , \ldots , 12$), and let $E \left( X _ { n } \right)$ represent the expected value of $X _ { n }$. Select the correct options.
(1) $E \left( X _ { 1 } \right) = 6$
(2) $P \left( X _ { 2 } = 12 \right) = \frac { 1 } { 4 }$
(3) $P \left( X _ { 8 } = 5 \right) \geq \frac { 1 } { 2 ^ { 8 } }$
(4) $P \left( X _ { 8 } = 4 \right) = P \left( X _ { 8 } = 8 \right)$
(5) $E \left( X _ { 8 } \right) \leq 7$

On the complex plane, let $\bar { z }$ denote the complex conjugate of complex number $z$, and $i = \sqrt { - 1 }$. Select the correct options.
(1) If $z = 2 i$, then $z ^ { 3 } = 4 \bar { i } \bar { z }$
(2) If non-zero complex number $\alpha$ satisfies $\alpha ^ { 3 } = 4 i \bar { \alpha }$, then $| \alpha | = 2$
(3) If non-zero complex number $\alpha$ satisfies $\alpha ^ { 3 } = 4 i \bar { \alpha }$ and let $\beta = i \alpha$, then $\beta ^ { 3 } = 4 i \bar { \beta }$
(4) Among all non-zero complex numbers $z$ satisfying $z ^ { 3 } = 4 i \bar { z }$, the minimum possible value of the principal argument is $\frac { \pi } { 6 }$
(5) There are exactly 3 distinct non-zero complex numbers $z$ satisfying $z ^ { 3 } = 4 i \bar { z }$

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 $\overline { M N } ^ { 2 } =$ \hspace{2cm}. (Express as a fraction in lowest terms)

In coordinate space, there is a line $L$ with direction vector $( 1 , - 2, 2 )$, plane $E _ { 1 } : 2 x + 3 y + 6 z = 10$, and plane $E _ { 2 } : 2 x + 3 y + 6 z = - 4$. The length of the line segment of $L$ cut off by $E _ { 1 }$ and $E _ { 2 }$ is \hspace{2cm}. (Express as a fraction in lowest terms)

A department store holds a Father's Day card drawing promotion with the following rules: The organizer prepares ten cards numbered $1, 2, \ldots, 9$, of which there are two cards numbered 8, and only one card for each other number. Four cards are randomly drawn from these ten cards without replacement and arranged from left to right in order to form a four-digit number. If the four-digit number satisfies any one of the following conditions, a prize is won:
(1) The four-digit number is greater than 6400
(2) The four-digit number contains two digits 8 For example: If the four cards drawn are numbered $5, 8, 2, 8$ in order, then the four-digit number is 5828, and a prize is won. According to the above rules, there are (11-1)(11-2)(11-3)(11-4) four-digit numbers that can win prizes.

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Find the cosine of the angle between vectors $\overrightarrow { C O }$ and $\overrightarrow { C P }$.

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Prove that the graph of $y = f ( x )$ and $\Omega$ have a common tangent line at point $P$.

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Find the area of the region bounded by the graph of $y = f ( x )$ above and the lower semicircular arc of $\Omega$.

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
The length of the major axis of ellipse $\Gamma ^ { \prime }$ is (15-1) $\sqrt{\underline{(15-2)}}$. (Express as a simplified radical)

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
Find the equation of the line containing the minor axis of $\Gamma ^ { \prime }$ and the length of the minor axis.

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
It is known that a point $P$ on $\Gamma$ is transformed by this rotation to a point $P ^ { \prime }$ that falls on the $x$-axis, and the $x$-coordinate of $P ^ { \prime }$ is positive. Find the coordinates of point $P$.