A rectangle is formed by lines $x = 0 , y = 0 , x = 3 , y = 4$. A line perpendicular to $3 x + 4 y + 6 = 0$ divides the rectangle into two equal parts, then the distance of the line from ( $- 1 , \frac { 3 } { 2 }$ ) is\ (A) 2\ (B) $\frac { 8 } { 5 }$\ (C) $\frac { 6 } { 5 }$\ (D) $\frac { 17 } { 10 }$

The image of the parabola $\mathrm { x } ^ { 2 } = 4 \mathrm { y }$ in the line $\mathrm { x } - \mathrm { y } = 1$ is
(A) $( y - 1 ) ^ { 2 } = 4 ( x + 1 )$
(B) $( y + 1 ) ^ { 2 } = 4 ( x + 1 )$
(C) $( y + 1 ) ^ { 2 } = 4 ( x - 1 )$
(D) $( y - 1 ) ^ { 2 } = 4 ( x - 1 )$

Let side AB of an equilateral triangle ABC is given by $x + 2 \sqrt { 2 } y - 4 = 0$, where $A$ is on $x$-axis and $B$ is an $y$-axis. If origin $( 0,0 )$ is the orthocentre of the triangle ABC and vertex C is $( \alpha , \beta )$, then the value of $| \alpha - \sqrt { 2 \beta } |$ is (A) 0 (B) 2 (C) 4 (D) 6

Let $m$ be a real number. On a plane with the coordinate system, in which the origin is denoted by O, consider the parabola $y = x^2$ and the two points on it,
$$\mathrm{A}(a,\, ma+1), \quad \mathrm{B}(b,\, mb+1) \quad (a < 0 < b)$$
(1) The $x$-coordinates $a$ and $b$ of the two points A and B can be expressed in terms of $m$ as
$$a = \frac{m - \sqrt{D}}{\mathbf{A}}, \quad b = \frac{m + \sqrt{D}}{\mathbf{B}},$$
where the expression $D$ is
$$D = m^2 + \mathbf{C}.$$
(2) Let the coordinates of the point of intersection of the segment AB and the $y$-axis be denoted by $(0, c)$. Then $c = \mathbf{D}$.
(3) Further, when the area $S$ of the triangle OAB with the three vertices O, A and B is expressed in terms of $a$ and $b$, we have
$$S = \frac{1}{2}\mathbf{E},$$
where $E$ is the appropriate choice from among (0) $\sim$ (5). (0) $a + b$
(1) $a - b$
(2) $b - a$
(3) $a^2 + b^2$
(4) $a^2 - b^2$
(5) $b^2 - a^2$
Also, when $S$ is represented in terms of $m$, we have
$$S = \frac{\mathbf{F}}{\mathbf{G}} \sqrt{m^2 + \mathbf{H}}.$$
Hence the value of $S$ is minimalized when $m = \mathbf{I}$, and its minimum value is $S = \mathbf{J}$.

Consider two squares as in the figure to the right. Let the coordinates of their vertexes be
$$\begin{array} { l l } \mathrm { A } ( 2 t , 0 ) , \quad \mathrm { B } ( 0,2 t ) , & \mathrm { C } ( - 2 t , 0 ) , \quad \mathrm { D } ( 0 , - 2 t ) , \\ \mathrm { P } \left( 4 - t ^ { 2 } , 4 - t ^ { 2 } \right) , & \mathrm { Q } \left( - 4 + t ^ { 2 } , 4 - t ^ { 2 } \right) , \\ \mathrm { R } \left( - 4 + t ^ { 2 } , - 4 + t ^ { 2 } \right) , & \mathrm { S } \left( 4 - t ^ { 2 } , - 4 + t ^ { 2 } \right) , \end{array}$$
where $0 < t < 2$. Denote the areas of the two squares ABCD and PQRS by $S _ { 1 }$ and $S _ { 2 }$, respectively.
Then we have
$$S _ { 1 } = \mathbf { M } t ^ { 2 } \text { and } S _ { 2 } = \mathbf { N } \left( t ^ { 2 } - \mathbf { O } \right) ^ { 2 } .$$
(1) $S _ { 1 } + S _ { 2 }$ is minimized at $t = \sqrt { \mathbf { P } }$, and the minimum value is $\mathbf { Q } \mathbf { R }$.
(2) For $\mathbf { W }$ and $\mathbf { X }$ below, choose the correct answer from among (0) $\sim$ (9), and for the other $\square$, enter the correct numbers.
We are to find the range of $t$ such that $S _ { 1 } < S _ { 2 }$. If $S _ { 1 } < S _ { 2 }$, then $t$ satisfies the inequality
$$t ^ { 4 } - \mathbf { ST } t ^ { 2 } + \mathbf { UV } > 0 .$$
From the above inequality, a condition on $t ^ { 2 }$ is $\mathbf { W }$. Hence, $S _ { 1 } < S _ { 2 }$ if and only if $t$ satisfies $\mathbf { X }$.
(0) $t ^ { 2 } < 4$ or $6 < t ^ { 2 }$ (1) $4 < t ^ { 2 } < 6$ (2) $t ^ { 2 } < 2$ or $8 < t ^ { 2 }$ (3) $2 < t ^ { 2 } < 8$ (4) $t ^ { 2 } \neq 4$ (5) $0 < t < 2$ (6) $0 < t < \sqrt { 2 }$ (7) $\sqrt { 2 } < t < 2$ (8) $2 < t < \sqrt { 6 }$ (9) $t \neq 2$

Q2 As shown in the figure, on an $xy$-plane whose origin is O, let us consider an isosceles triangle ABC satisfying $\mathrm { AB } = \mathrm { AC }$. Furthermore, suppose that side AB passes through $\mathrm { P } ( - 1,5 )$ and side AC passes through $\mathrm{Q}(3, 3)$.
Let us consider the radius of the inscribed circle of the triangle ABC.
Denote the straight line passing through the two points A and B by $\ell _ { 1 }$ and the straight line passing through the two points A and C by $\ell _ { 2 }$. When we denote the slope of $\ell _ { 1 }$ by $a$, the equations of $\ell _ { 1 }$ and $\ell _ { 2 }$ are
$$\begin{aligned} & \ell _ { 1 } : y = a x + a + \mathbf { M } , \\ & \ell _ { 2 } : y = - a x + \mathbf { N } a + \mathbf { O } . \end{aligned}$$
Denote the center and the radius of the inscribed circle by I and $r$, respectively. Then the coordinates of I are $\left( \mathbf { P } - \frac { \mathbf { Q } } { a } , r \right)$.
Hence $r$ can be expressed in terms of $a$ as
$$r = \frac { \mathbf { R } } { \mathbf { T } + \sqrt { \mathbf { S } } }$$
In particular, when $r = \frac { 5 } { 2 }$, the coordinates of vertex A are $\left( \frac { \mathbf { V } } { \mathbf{U} } , \frac { \mathbf { X Y } } { \mathbf { W } } \right)$.

Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \hline & & & & & & & & & & & & & & \hline \end{tabular}

1. For ALL APPLICANTS.

For each part of the question on pages $3 - 7$ you will be given four possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), or (d) you think is correct with a tick ( ✓ ) in the corresponding column in the table below. Please show any rough working in the space provided between the parts.

	(a)	(b)	(c)	(d)
A
B
C
D
E
F
G
H
I
J

A. The point lying between $P ( 2,3 )$ and $Q ( 8 , - 3 )$ which divides the line $P Q$ in the ratio $1 : 2$ has co-ordinates
(a) $( 4 , - 1 )$
(b) $( 6 , - 2 )$
(c) $\left( \frac { 14 } { 3 } , 2 \right)$
(d) $( 4,1 )$
B. The diagram below shows the graph of the function $y = f ( x )$. [Figure]
The graph of the function $y = - f ( x + 1 )$ is drawn in which of the following diagrams? [Figure]
(a) [Figure]
(c) [Figure]
(b) [Figure]
(d)
C. Which of the following numbers is largest in value? (All angles are given in radians.)
(a) $\tan \left( \frac { 5 \pi } { 4 } \right)$
(b) $\sin ^ { 2 } \left( \frac { 5 \pi } { 4 } \right)$
(c) $\log _ { 10 } \left( \frac { 5 \pi } { 4 } \right)$
(d) $\log _ { 2 } \left( \frac { 5 \pi } { 4 } \right)$
D. The numbers $x$ and $y$ satisfy the following inequalities
$$\begin{aligned} 2 x + 3 y & \leqslant 23 \\ x + 2 & \leqslant 3 y \\ 3 y + 1 & \leqslant 4 x \end{aligned}$$
The largest possible value of $x$ is
(a) 6
(b) 7
(c) 8
(d) .9
E. In the range $0 \leqslant x < 2 \pi$ the equation
$$\cos ( \sin x ) = \frac { 1 } { 2 }$$
has
(a) no solutions;
(b) one solution;
(c) two solutions;
(d) three solutions. F. The turning point of the parabola
$$y = x ^ { 2 } - 2 a x + 1$$
is closest to the origin when
(a) $a = 0$
(b) $a = \pm 1$
(c) $a = \pm \frac { 1 } { \sqrt { 2 } }$ or $a = 0$
(d) $a = \pm \frac { 1 } { \sqrt { 2 } }$. G. The four digit number 2652 is such that any two consecutive digits from it make a multiple of 13 . Another number $N$ has this same property, is 100 digits long, and begins in a 9 . What is the last digit of $N$ ?
(a) 2
(b) 3
(c) 6
(d) 9 H. The equation
$$\left( x ^ { 2 } + 1 \right) ^ { 10 } = 2 x - x ^ { 2 } - 2$$
(a) has $x = 2$ as a solution;
(b) has no real solutions;
(c) has an odd number of real solutions;
(d) has twenty real solutions. I. .Observe that $2 ^ { 3 } = 8,2 ^ { 5 } = 32,3 ^ { 2 } = 9$ and $3 ^ { 3 } = 27$. From these facts, we can deduce that $\log _ { 2 } 3$, the logarithm of 3 to base 2 , is
(a) between $1 \frac { 1 } { 3 }$ and $1 \frac { 1 } { 2 }$;
(b) between $1 \frac { 1 } { 2 }$ and $1 \frac { 2 } { 3 }$;
(c) between $1 \frac { 2 } { 3 }$ and 2;
(d) between 2 and 3. J. Into how many regions is the plane divided when the following three parabolas are drawn?
$$\begin{aligned} & y = x ^ { 2 } \\ & y = x ^ { 2 } - 2 x \\ & y = x ^ { 2 } + 2 x + 2 \end{aligned}$$
(a) 4
(b) 5
(c) 6
(d) 7

Q1 & Q2 & Q3 & Q4 & Q5 & Q6 & Q7 & Total \hline & & & & & & & & & & & & & & \hline \end{tabular}

1. For ALL APPLICANTS.

For each part of the question on pages $3 - 7$ you will be given four possible answers, just one of which is correct. Indicate for each part $\mathbf { A } - \mathbf { J }$ which answer (a), (b), (c), or (d) you think is correct with a tick ( ✓ ) in the corresponding column in the table below. Please show any rough working in the space provided between the parts.

	(a)	(b)	(c)	(d)
A
B
C
D
E
F
G
H
I
J

A. The point lying between $P ( 2,3 )$ and $Q ( 8 , - 3 )$ which divides the line $P Q$ in the ratio $1 : 2$ has co-ordinates
(a) $( 4 , - 1 )$
(b) $( 6 , - 2 )$
(c) $\left( \frac { 14 } { 3 } , 2 \right)$
(d) $( 4,1 )$
B. The diagram below shows the graph of the function $y = f ( x )$. [Figure]
The graph of the function $y = - f ( x + 1 )$ is drawn in which of the following diagrams? [Figure]
(a) [Figure]
(c) [Figure]
(b) [Figure]
(d)
C. Which of the following numbers is largest in value? (All angles are given in radians.)
(a) $\tan \left( \frac { 5 \pi } { 4 } \right)$
(b) $\sin ^ { 2 } \left( \frac { 5 \pi } { 4 } \right)$
(c) $\log _ { 10 } \left( \frac { 5 \pi } { 4 } \right)$
(d) $\log _ { 2 } \left( \frac { 5 \pi } { 4 } \right)$
D. The numbers $x$ and $y$ satisfy the following inequalities
$$\begin{aligned} 2 x + 3 y & \leqslant 23 \\ x + 2 & \leqslant 3 y \\ 3 y + 1 & \leqslant 4 x \end{aligned}$$
The largest possible value of $x$ is
(a) 6
(b) 7
(c) 8
(d) .9
E. In the range $0 \leqslant x < 2 \pi$ the equation
$$\cos ( \sin x ) = \frac { 1 } { 2 }$$
has
(a) no solutions;
(b) one solution;
(c) two solutions;
(d) three solutions. F. The turning point of the parabola
$$y = x ^ { 2 } - 2 a x + 1$$
is closest to the origin when
(a) $a = 0$
(b) $a = \pm 1$
(c) $a = \pm \frac { 1 } { \sqrt { 2 } }$ or $a = 0$
(d) $a = \pm \frac { 1 } { \sqrt { 2 } }$. G. The four digit number 2652 is such that any two consecutive digits from it make a multiple of 13 . Another number $N$ has this same property, is 100 digits long, and begins in a 9 . What is the last digit of $N$ ?
(a) 2
(b) 3
(c) 6
(d) 9 H. The equation
$$\left( x ^ { 2 } + 1 \right) ^ { 10 } = 2 x - x ^ { 2 } - 2$$
(a) has $x = 2$ as a solution;
(b) has no real solutions;
(c) has an odd number of real solutions;
(d) has twenty real solutions. I. .Observe that $2 ^ { 3 } = 8,2 ^ { 5 } = 32,3 ^ { 2 } = 9$ and $3 ^ { 3 } = 27$. From these facts, we can deduce that $\log _ { 2 } 3$, the logarithm of 3 to base 2 , is
(a) between $1 \frac { 1 } { 3 }$ and $1 \frac { 1 } { 2 }$;
(b) between $1 \frac { 1 } { 2 }$ and $1 \frac { 2 } { 3 }$;
(c) between $1 \frac { 2 } { 3 }$ and 2;
(d) between 2 and 3. J. Into how many regions is the plane divided when the following three parabolas are drawn?
$$\begin{aligned} & y = x ^ { 2 } \\ & y = x ^ { 2 } - 2 x \\ & y = x ^ { 2 } + 2 x + 2 \end{aligned}$$
(a) 4
(b) 5
(c) 6
(d) 7

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Mathematics \& Computer Science and Computer Science applicants should turn to page 14.
Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$.
(i) The mirror image of the point $P$ in the $x$-axis is the point $R$ with co-ordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided opposite.
(ii) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such parth, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.)
(iii) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$.
(iv) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the shortest such path, giving clear reasoning for your answer. [Figure]

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Mathematics \& Computer Science and Computer Science applicants should turn to page 14.
Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$.
(i) The mirror image of the point $P$ in the $x$-axis is the point $R$ with co-ordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided opposite.
(ii) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such parth, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.)
(iii) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$.
(iv) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the shortest such path, giving clear reasoning for your answer. [Figure]

3. (a) Write down the equation of the straight line through the point $( 1,2 )$ with slope - 1 .
(b) Let $l$ be a line with equation
$$y = ( 2 - a ) + a x$$
where $a$ is a constant. Show that, for any $a$, the line passes through the point $( 1,2 )$. Find the equation of the line perpendicular to this line which also passes through the point $( 1,2 )$.
(c) Find the equations of the lines which pass through the point $( 1,2 )$ and have perpendicular distance 1 from the origin.

(a) The straight line in the $( x , y )$ plane through the points $( - 1,3 )$ and $( 2,1 )$ is defined by the equation\ (i) $3 x + 2 y = 3$,\ (ii) $- x + 3 y = 1$,\ (iii) $2 x + 3 y = 7$,\ (iv) $x + 2 y = 5$.\ (b) There is a solution to the equation $x ^ { 3 } + x ^ { 2 } + 3 = 0$ between\ (i) - 2 and - 1 ,\ (ii) - 1 and 0 ,\ (iii) 0 and 1 ,\ (iv) 1 and 2 .\ (c) Anne, Bert, Clare, Derek and Emily are planning to play a game for which they need to divide themselves into three teams. Each team must have at least one member. The number of different ways they can do this is\ (i) 10 ,\ (ii) 15 ,\ (iii) 25 ,\ (iv) 30 .\ (d) For the following statements
$$P : \frac { x ( x - 2 ) } { 1 - x } > 0 , \quad Q : 1 < x < 2$$
about a real number $x$,\ (i) $P$ implies $Q$, but $Q$ does not imply $P$,\ (ii) $Q$ implies $P$, but $P$ does not imply $Q$,\ (iii) $P$ implies $Q$, and $Q$ implies $P$,\ (iv) $P$ and $Q$ contradict each other.\ (e) The least and greatest values of $\cos ( \cos x )$ in the range $0 \leq x \leq \pi$ are\ (i) 0 and 1 ,\ (ii) - $\cos 1$ and 1 ,\ (iii) - 1 and 1 , (iv) $\cos 1$ and 1 .\ (f) As the integer $n$ becomes very large and positive,
$$\frac { \sqrt { n } + ( - 1 ) ^ { n } } { \sqrt { n } }$$
(i) approaches (that is, converges to) 0 ,\ (ii) approaches (that is, converges to) 1 ,\ (iii) approaches infinity,\ (iv) oscillates, but does not converge.\ (g) The power of $x$ which has the greatest coefficient in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }$ is\ (i) $x ^ { 2 }$,\ (ii) $x ^ { 3 }$,\ (iii) $x ^ { 5 }$,\ (iv) $x ^ { 10 }$.\ (h) The (shaded) area under the graph of $y = f ( x )$ between $x = 1$ and $x = 2$ is given to be 1 .
The area under the graph of $y = 2 f ( 3 - x )$ between $x = 1$ and $x = 2$ is therefore\ (i) 1 ,\ (ii) 2 ,\ (iii) 3 ,\ (iv) 6 .\ (j) In a plane there are given $n$ straight lines, no two of them parallel and no three of them meeting at a point. The number of parts they divide the plane into is\ (i) $n + 1$,\ (ii) $n ^ { 2 } - n + 2$,\ (iii) $\frac { 1 } { 2 } n ( n + 1 ) + 1$,\ (iv) $2 ^ { n }$.\ (k) The simultaneous equations
$$\begin{aligned} & x - 2 y + 3 z = 1 \\ & 2 x + 3 y - z = 4 \\ & 4 x - y + 5 z = 6 \end{aligned}$$
have\ (i) no solutions,\ (ii) exactly one solution,\ (iii) exactly three solutions,\ (iv) infinitely many solutions.

3. Let $P$ and $Q$ be the points with co-ordinates $( 7,1 )$ and $( 11,2 )$.
(a) The mirror image of the point $P$ in the $x$-axis is the point $R$ with coordinates $( 7 , - 1 )$. Mark the points $P , Q$ and $R$ on the grid provided.
(b) Consider paths from $P$ to $Q$ each of which consists of two straight line segments $P X$ and $X Q$ where $X$ is a point on the $x$-axis. Find the length of the shortest such path, giving clear reasoning for your answer. (You may refer to the diagram to help your explanation, if you wish.)
(c) Sketch in the line $\ell$ with equation $y = x$. Find the co-ordinates of $S$, the mirror image in the line $\ell$ of the point $Q$, and mark in the point $S$.
(d) Consider paths from $P$ to $Q$ each of which consists of three straight line segments $P Y , Y Z$ and $Z Q$, where $Y$ is on the $x$-axis and $Z$ is on the line $\ell$. Find the length of the shortest such path, giving clear reasoning for your answer. [Figure]

(a) Show that the line $y = m x + c$ passes through the point $( 1,1 )$ if $c = 1 - m$.
(b) Let $L$ be a line with gradient $m > 0$, which passes through ( 1,1 ). Find the equation of the line $L ^ { \prime }$ which is perpendicular to $L$, and which passes through the point $( 1 , a )$, given $a \neq 1$.
(c) Find the area of the triangle which has ( 1,1 ) and ( $1 , a$ ) as two of its vertices and the intersection of $L$ and $L ^ { \prime }$ as the third vertex.
(d) For what value of $m$ is the triangle isosceles (two sides of equal length)?

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

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

6. How many lines on the coordinate plane are there such that the distance from point $O(0,0)$ to the line is 1, and the distance from point $A(3,0)$ to the line is 2?
(1) 1 line
(2) 2 lines
(3) 3 lines
(4) 4 lines
(5) Infinitely many lines

II. Multiple-Choice Questions (25 points)

Instructions: For questions 7 through 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 deductions are made for incorrect answers. Full credit (5 points) is given for all five options correct; 2.5 points are given for exactly one incorrect option; no credit is given for two or more incorrect options.

8. On the coordinate plane, four lines $L_{1}, L_{2}, L_{3}, L_{4}$ have the relative positions with respect to the $x$-axis, $y$-axis, and the line $y = x$ as shown in the figure. $L_{1}$ is perpendicular to $L_{3}$, and $L_{3}$ is parallel to $L_{4}$. The equations of $L_{1}, L_{2}, L_{3}, L_{4}$ are $y = m_{1}x$, $y = m_{2}x$, $y = m_{3}x$, and $y = m_{4}x + c$ respectively. Which of the following options are correct?
(1) $m_{3} > m_{2} > m_{1}$
(2) $m_{1} \cdot m_{4} = -1$
(3) $m_{1} < -1$
(4) $m_{2} \cdot m_{3} < -1$
(5) $c > 0$ [Figure]

4. On a coordinate plane, two points $A(1, 0)$ and $B(0, 1)$ are given. Consider three additional points $P(\pi, 1)$, $Q(-\sqrt{3}, 6)$, and $R\left(2, \log_{4} 32\right)$. Let the area of $\triangle PAB$ be $p$, the area of $\triangle QAB$ be $q$, and the area of $\triangle RAB$ be $r$. Which of the following options is correct?
(1) $p < q < r$
(2) $p < r < q$
(3) $q < p < r$
(4) $q < r < p$
(5) $r < q < p$

On the coordinate plane, two parallel lines $L _ { 1 } , L _ { 2 }$ both have slope 2 and are at distance 5 apart. Point $A ( 2 , - 1 )$ is a point on $L _ { 1 }$ in the fourth quadrant. Point $B$ is a point on $L _ { 2 }$ in the second quadrant with $\overline { A B } = 5$ . Line $L _ { 3 }$ has slope 3, passes through point $A$, and intersects $L _ { 2 }$ at point $C$. Answer the following questions:
(1) Find the slope of line $AB$ . (2 points)
(2) Find the vector $\overrightarrow { A B }$ . (4 points)
(3) Find the dot product $\overrightarrow { A B } \cdot \overrightarrow { A C }$ . (3 points)
(4) Find the vector $\overrightarrow { A C }$ . (4 points)

On a coordinate plane, there are two points $A ( - 3,4 ) , B ( 3,2 )$ and a line $L$. Points $A$ and $B$ are on opposite sides of line $L$, and $\vec { n } = ( 4 , - 3 )$ is a normal vector to line $L$. The distance from point $A$ to line $L$ is 5 times the distance from point $B$ to line $L$. Based on the above, answer the following questions.
(1) Find the dot product of vector $\overrightarrow { A B }$ and vector $\vec { n }$. (4 points)
(2) Find the equation of line $L$. (4 points)
(3) Point $P$ is on line $L$ and $\overline { P A } = \overline { P B }$. Find the coordinates of point $P$. (4 points)

Two lines $L _ { 1 } , L _ { 2 }$ on the coordinate plane both have positive slopes, and the angle bisector of one of the angles formed by $L _ { 1 } , L _ { 2 }$ has slope $\frac { 11 } { 9 }$ . Another line $L$ passes through the point $( 2 , \frac { 1 } { 3 } )$ and forms a bounded region with $L _ { 1 } , L _ { 2 }$ that is an equilateral triangle. Which of the following options is the equation of $L$?
(1) $11 x - 9 y = 19$
(2) $9 x + 11 y = 25$
(3) $11 x + 9 y = 25$
(4) $27 x - 33 y = 43$
(5) $27 x + 33 y = 65$

Consider the line $L: 5y + (2k-4)x - 10k = 0$ on the coordinate plane (where $k$ is a real number), and the rectangle $OABC$ with vertices at $O(0,0)$, $A(10,0)$, $B(10,6)$, $C(0,6)$. Let $L$ intersect the line $OC$ and the line $AB$ at points $D$ and $E$ respectively. Select the correct options.
(1) When $k = 4$, the line $L$ passes through point $A$
(2) If the line $L$ passes through point $C$, then the slope of $L$ is $-\frac{5}{2}$
(3) If point $D$ is on the line segment $\overline{OC}$, then $0 \leq k \leq 3$
(4) If $k = \frac{1}{2}$, then the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary)
(5) If the line segment $\overline{DE}$ is inside the rectangle $OABC$ (including the boundary), then the slope of $L$ could be $\frac{3}{10}$

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").
If $\overrightarrow{PA_{1}} = k\overrightarrow{PA_{3}}$, then the value of $k$ is $\square$. (Express as a fraction in lowest terms)

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").
Find the coordinates of points $P$ and $B_{3}$.