In an open space, there are three utility poles perpendicular to the ground with equal heights and equally spaced bases on a straight line. A person uses one-point perspective to draw these three utility poles on a canvas. A coordinate system is set up on the canvas so that the utility poles are parallel to the $y$-axis. The three base points are $A_{1}(0,0)$, $A_{2}$, $A_{3}$, all on the line $L: x + 3y = 0$; the three top points are $B_{1}(0,3)$, $B_{2}$, $B_{3}$, all on the line $M: 2x - 3y + 9 = 0$, as shown in the figure. It is known that $\overline{A_{3}B_{3}} = 2\overline{A_{1}B_{1}}$, and by one-point perspective, the intersection of lines $A_{1}B_{3}$ and $A_{3}B_{1}$ lies on line $A_{2}B_{2}$. Let $P$ be the intersection of $L$ and $M$ (this point is also called the "vanishing point").
Suppose a bee stops on the middle utility pole at a position where the ratio of distances from the base to the top is $1:2$. The person wants to draw this bee on the line segment $A_{2}B_{2}$ on the canvas. Assuming the bee's position on the canvas is point $Q$, that is, the ratio of the distance from point $Q$ to the base $A_{2}$ of line segment $A_{2}B_{2}$ to the distance to the top $B_{2}$ is $1:2$, find the coordinates of point $Q$.

On the coordinate plane, there is a square and a regular hexagon, with the square to the right of the hexagon. Both regular polygons have one side on the $x$-axis, and the center $A$ of the square and the center $B$ of the hexagon are both above the $x$-axis. The two polygons have exactly one intersection point $P$. The side length of the square is 6, and the distance from point $P$ to the $x$-axis is $2\sqrt{3}$. Select the correct options.
(1) The distance from point $A$ to the $x$-axis is greater than the distance from point $B$ to the $x$-axis
(2) The side length of the regular hexagon is 6
(3) $\overrightarrow{BA} = (7, 3 - 2\sqrt{3})$
(4) $\overline{AP} > \sqrt{10}$
(5) The slope of line $AP$ is greater than $-\frac{1}{\sqrt{3}}$

On the coordinate plane, $P ( a , 0 )$ is a point on the $x$-axis, where $a > 0$. Let $L _ { 1 }$ and $L _ { 2 }$ be lines passing through point $P$ with slopes $- \frac { 4 } { 3 }$ and $- \frac { 3 } { 2 }$ respectively. Given that the difference in areas of the two right triangles formed by $L _ { 1 }$ and $L _ { 2 }$ with the two coordinate axes is 3, what is the value of $a$?
(1) $3 \sqrt { 2 }$
(2) 6
(3) $6 \sqrt { 2 }$
(4) 9
(5) $8 \sqrt { 2 }$

A person uses single-point perspective with a point on the horizon as the vanishing point to draw six vertical pillars $A , B , C , D , E , F$ on a coordinate plane. The coordinates of the top and base of each pillar are shown in the table below, with point $V ( 4,9 )$ representing the vanishing point, as shown in the figure. Since the base line and top line of pillars $A$ and $F$ in the figure are both parallel to the horizon, the actual heights of pillars $A$ and $F$ are equal. Based on the above, select the pillar with the maximum actual height.

Pillar	$A$	$B$	$C$	$D$	$E$	$F$
Top coordinate	$( 0,8 )$	$( 2,3 )$	$( 4,6 )$	$( 6,8 )$	$( 8,5 )$	$( 10,8 )$
Base coordinate	$( 0,6 )$	$( 2,0 )$	$( 4,3 )$	$( 6,5 )$	$( 8,1 )$	$( 10,6 )$

(1) $A$
(2) $B$
(3) $C$
(4) $D$
(5) $E$

On the coordinate plane, given three points $A ( 0,2 )$ , $B ( - 1,0 )$ , $C ( 4,0 )$ . If the line $y = m x$ divides triangle $A B C$ into two equal areas, then $m = \frac { \text{(14--1)} } { \text{(14--2)} }$ . (Reduce to lowest terms)

3. The perpendicular bisector of the line segment joining the points $( 2 , - 6 )$ and $( 5,4 )$ cuts the $x$-axis at the point with $x$-coordinate
A $\frac { 1 } { 20 }$
B $\frac { 1 } { 6 }$
C $\frac { 1 } { 3 }$
D $\frac { 19 } { 5 }$
E $\frac { 41 } { 6 }$

$PQRS$ is a rectangle.
The coordinates of $P$ and $Q$ are $( 0,6 )$ and $( 1,8 )$ respectively.
The perpendicular to $PQ$ at $Q$ meets the $x$-axis at $R$.
What is the area of $PQRS$ ?
A $\frac { 5 } { 2 }$
B $4 \sqrt { 10 }$
C 20
D $8 \sqrt { 10 }$
E 40

$A ( 0,2 )$ and $C ( 4,0 )$ are opposite vertices of the square $A B C D$. What is the equation of the straight line through $B$ and $D$ ?
A $y = - 2 x + 5$
B $y = - \frac { 1 } { 2 } x - 3$
C $y = - \frac { 1 } { 2 } x + 2$
D $y = x$
E $y = 2 x - 3$ F $y = 2 x + 2$

The curve with equation
$$x = y ^ { 2 } - 6 y + 11$$
is rotated $90 ^ { \circ }$ clockwise about the point $P$ to give the curve $C$. $P$ has $x$-coordinate - 2 and $y$-coordinate 3 . What is the equation of $C$ ?
A $y = - x ^ { 2 } - 4 x - 3$ B $y = - x ^ { 2 } - 4 x - 5$ C $y = - x ^ { 2 } - 6 x - 7$ D $y = - x ^ { 2 } - 6 x - 11$ E $y = x ^ { 2 } - 4 x + 5$ F $y = x ^ { 2 } + 4 x + 3$ G $y = x ^ { 2 } - 6 x + 11$ H $y = x ^ { 2 } + 6 x + 7$

The graph of the line $a x + b y = c$ is drawn, where $a , b$ and $c$ are real non-zero constants. Which one of the following is a necessary but not sufficient condition for the line to have a positive gradient and a positive $y$-intercept?
A $\frac { c } { b } > 0$ and $\frac { a } { b } < 0$ B $\frac { c } { b } < 0$ and $\frac { a } { b } > 0$ C $a > b > c$ D $a < b < c$ E $\quad a$ and $c$ have opposite signs F $\quad a$ and $c$ have the same sign

Three lines are given by the equations:
$$\begin{aligned} & a x + b y + c = 0 \\ & b x + c y + a = 0 \\ & c x + a y + b = 0 \end{aligned}$$
where $a$, $b$ and $c$ are non-zero real numbers. Which one of the following is correct? A If two of the lines are parallel, then all three are parallel. B If two of the lines are parallel, then the third is perpendicular to the other two. C If two of the lines are parallel, then the third is parallel to $y = x$. D If two of the lines are parallel, then the third is perpendicular to $y = x$. E If two of the lines are perpendicular, then all three meet at a point. F If two of the lines are perpendicular, then the third is parallel to $y = x$. G If two of the lines are perpendicular, then the third is perpendicular to $y = x$.

A right-angled triangle has vertices at $( 2,3 ) , ( 9 , - 1 )$ and $( 5 , k )$.
Find the sum of all the possible values of $k$.

In the coordinate plane, let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two points. Suppose two points $\mathrm{P}(p,0)$ and $\mathrm{Q}(q,0)$ on the $x$-axis satisfy both of the following conditions (i) and (ii).

• [(i)] $0 < p < 1$ and $p < q$
• [(ii)] Let $\mathrm{M}$ be the midpoint of segment $\mathrm{AP}$; then $\angle \mathrm{OAP} = \angle \mathrm{PMQ}$

(1) Express $q$ in terms of $p$.
(2) Find the value of $p$ such that $q = \dfrac{1}{3}$.
(3) Let $S$ be the area of $\triangle \mathrm{OAP}$ and $T$ be the area of $\triangle \mathrm{PMQ}$. Find the range of $p$ such that $S > T$.

A line on a two-dimensional plane can be expressed as $\alpha x + \beta y + \gamma = 0$, where $( x , y )$ is a point on the line in the Cartesian coordinate system. We call the column vector $( \alpha , \beta , \gamma ) ^ { \mathrm { T } }$ a coefficient vector of the line. Answer the following questions. Note that the coefficient vector in your answer must satisfy $\alpha ^ { 2 } + \beta ^ { 2 } = 1$.
(1) Find a coefficient vector of a line that passes through a point $\vec { a }$ and is perpendicular to a unit vector $\vec { v }$ on a two-dimensional plane.
(2) Let a line B pass through a point $\vec { b }$ and be perpendicular to a unit vector $\vec { n }$. Given a line $A$, let the line $A ^ { \prime }$ be the mirror transformation of the line $A$ over the line $B$. Using $\vec { b }$ and $\vec { n }$, write a three-dimensional square matrix that transforms a coefficient vector of the line A to a coefficient vector of the line $\mathrm { A } ^ { \prime }$.
(3) Find the determinant of the matrix derived in Question (2).
(4) Consider the movement of the line $\mathrm { D } _ { t }$ whose coefficient vector changes with the real variable $t$ as $\left( 4 t , 4 t ^ { 2 } - 1 , t \right) ^ { \mathrm { T } }$. This line passes through a point regardless of $t$. Find the coordinate of that point.
(5) Suppose that, with the mirror transformation over a line $M _ { t }$, which also changes with $t$, the line $\mathrm { D } _ { t }$ in Question (4) is transformed to the line with a coefficient vector $( 0,1 , - t ) ^ { \mathrm { T } }$. Find the coefficient vector $\left( \alpha _ { t } , \beta _ { t } , \gamma _ { t } \right) ^ { \mathrm { T } }$ of the line $\mathrm { M } _ { t }$, where $\alpha _ { t } > 0$ and $\beta _ { t } > 0$ for $t > 0$.
(6) When $t$ changes from 0 to $+ \infty$, consider the region where the line $\mathrm { M } _ { t }$ in Question (5) can exist. Describe the region using a simple mathematical expression and draw a diagram of the region.

In the Cartesian coordinate plane, the perpendicular drawn from point $A ( 1,0 )$ to the line $\mathbf { y } + \mathbf { 2 x } - \mathbf { 1 } = \mathbf { 0 }$ intersects the Y-axis at which point?
A) $\frac { - 1 } { 2 }$
B) $\frac { - 1 } { 3 }$
C) $\frac { - 1 } { 4 }$
D) $\frac { - 1 } { 5 }$
E) $\frac { - 1 } { 6 }$

When 130 liters of milk in a dairy is used to make cheese, the graph of the linear relationship between the remaining milk and the amount of cheese produced is given.
According to this, when 10 kg of cheese is produced in this dairy, how many liters of milk remain?
A) 50
B) 60
C) 65
D) 75
E) 80

OABC is a parallelogram $\mathrm { A } = ( 5,0 )$ $\mathrm { C } = ( 3,4 )$
According to the given information above, what is the sum of the diagonal lengths of parallelogram OABC in units?
A) $5 \sqrt { 5 }$
B) $6 \sqrt { 5 }$
C) $7 \sqrt { 5 }$
D) $7 \sqrt { 3 }$
E) $8 \sqrt { 3 }$

In the right coordinate plane shown in the figure, lines d and e are perpendicular to each other.
Accordingly, what is the abscissa of the point where line d intersects the x-axis?
A) $\frac { 9 } { 2 }$
B) $\frac { 11 } { 2 }$
C) $\frac { 13 } { 3 }$
D) $\frac { 14 } { 3 }$
E) $\frac { 25 } { 6 }$

The following information is known about points A, B, C, D, and E in the plane.
$$\begin{aligned} & { [ A B ] \perp [ B C ] } \\ & { [ A B ] \cap [ C D ] = E } \\ & | A E | = | B C | = 4 \text { units } \\ & | A B | = | C D | = 7 \text { units } \end{aligned}$$
Given this, what is the length |DE| in units?
A) $\sqrt { 3 }$
B) $\sqrt { 5 }$
C) $\sqrt { 7 }$
D) 2
E) 3

What is the perimeter of rectangle ABCD given in the coordinate plane in the figure?
A) 18
B) 21
C) 24
D) 27
E) 30

In the rectangular coordinate plane, the sides of rectangle $A B C D$ are parallel to the axes.
If the coordinates of vertices A and $C$ are $(1, -1)$ and $(3, 5)$ respectively, what is the area of rectangle ABCD in square units?
A) 8 B) 10 C) 12 D) 15 E) 16

According to the given information above, what is the sum of the coordinates of point C?
A) 3 B) 4 C) 5 D) 6 E) 7

In the rectangular coordinate plane, the line $y = \frac { x } { 7 }$ intersects the lines $x = 2$ and $x = 9$ at points $P$ and $R$ respectively.
Accordingly, what is the length $| \mathrm { PR } |$ in units?
A) $5 \sqrt { 2 }$
B) $6 \sqrt { 2 }$
C) $4 \sqrt { 10 }$
D) 8
E) 9

In the rectangular coordinate plane, a parallelogram whose vertices are the intersection points of the lines $y = 2$ and $y = 6$ with the line $y = 2x$ has diagonals intersecting at the point $(0,4)$.
What is the area of this parallelogram in square units?
A) 16 B) 18 C) 20 D) 22 E) 24

The square ABCD given above is divided into four rectangles of equal area.
Accordingly, what is the ratio $\frac { | AE | } { | AD | }$?\ A) $\frac { 2 } { 3 }$\ B) $\frac { 3 } { 4 }$\ C) $\frac { 3 } { 5 }$\ D) $\frac { 5 } { 8 }$\ E) $\frac { 9 } { 16 }$