QUESTION 179
The midpoint of the segment with endpoints $A(2, 4)$ and $B(6, 8)$ is
(A) $(3, 5)$
(B) $(4, 6)$
(C) $(5, 7)$
(D) $(6, 8)$
(E) $(8, 12)$

Determine the coordinates of $D$ and give the coordinates of the midpoint $M$ of the line segment $[ A C ]$.

Consider a triangle $ABC$. The sides $AB$ and $AC$ are extended to points $D$ and $E$, respectively, such that $AD = 3AB$ and $AE = 3AC$. Then one diagonal of $BDEC$ divides the other diagonal in the ratio
(A) $1 : 3$
(B) $1 : \sqrt{3}$
(C) $1 : 2$
(D) $1 : \sqrt{2}$.

14. A straight line through the origin $O$ meets the parallel lines $4 x + 2 y = 9$ and $2 x + y + 6 = 0$ at points $P$ and $Q$ respectively. Then the point $O$ divides the segment $P Q$ in the ratio
(A) $1 : 2$
(B) $\quad 3 : 4$
(C) $\quad 2 : 1$
(D) $4 : 3$

The lines $L_{1}: y-x=0$ and $L_{2}: 2x+y=0$ intersect the line $L_{3}: y+2=0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$. This question has Statement-1 and Statement-2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement-1: The ratio $PR:RQ$ equals $2\sqrt{2}:\sqrt{5}$. Statement-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.
(1) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
(2) Statement-1 is true, Statement-2 is false.
(3) Statement-1 is false, Statement-2 is true.
(4) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

A straight line through origin $O$ meets the lines $3y = 10 - 4x$ and $8x + 6y + 5 = 0$ at points $A$ and $B$ respectively. Then $O$ divides the segment $AB$ in the ratio: (1) $2:3$ (2) $1:2$ (3) $4:1$ (4) $3:4$

A straight line through origin $O$ meets the lines $3 y = 10 - 4 x$ and $8 x + 6 y + 5 = 0$ at points $A$ and $B$ respectively. Then, $O$ divides the segment $A B$ in the ratio
(1) $2 : 3$
(2) $1 : 2$
(3) $4 : 1$
(4) $3 : 4$

Q65. The equations of two sides AB and AC of a triangle ABC are $4 x + y = 14$ and $3 x - 2 y = 5$, respectively. The point $\left( 2 , - \frac { 4 } { 3 } \right)$ divides the third side BC internally in the ratio $2 : 1$. the equation of the side BC is
(1) $x + 3 y + 2 = 0$
(2) $x - 6 y - 10 = 0$
(3) $x - 3 y - 6 = 0$
(4) $x + 6 y + 6 = 0$

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

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").
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 $Q$.

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

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) $\frac { 9 } { 16 }$