Q67. Let the line $2 x + 3 y - \mathrm { k } = 0 , \mathrm { k } > 0$, intersect the $x$-axis and $y$-axis at the points A and B , respectively. If the equation of the circle having the line segment AB as a diameter is $x ^ { 2 } + y ^ { 2 } - 3 x - 2 y = 0$ and the length of the latus rectum of the ellipse $x ^ { 2 } + 9 y ^ { 2 } = k ^ { 2 }$ is $\frac { m } { n }$, where $m$ and $n$ are coprime, then $2 \mathrm {~m} + \mathrm { n }$ is equal to
(1) 11
(2) 10
(3) 12
(4) 13

Q66. Let $A B C D$ and $A E F G$ be squares of side 4 and 2 units, respectively. The point $E$ is on the line segment AB and the point F is on the diagonal AC . Then the radius r of the circle passing through the point F and touching the line segments BC and CD satisfies:
(1) $r = 0$
(2) $2 r ^ { 2 } - 4 r + 1 = 0$
(3) $2 r ^ { 2 } - 8 r + 7 = 0$
(4) $r ^ { 2 } - 8 r + 8 = 0$

Q67. Let $C$ be the circle of minimum area touching the parabola $y = 6 - x ^ { 2 }$ and the lines $y = \sqrt { 3 } | x |$. Then, which one of the following points lies on the circle $C$ ?
(1) $( 1,2 )$
(2) $( 1,1 )$
(3) $( 2,2 )$
(4) $( 2,4 )$

Q66. If $\mathrm { P } ( 6,1 )$ be the orthocentre of the triangle whose vertices are $\mathrm { A } ( 5 , - 2 ) , \mathrm { B } ( 8,3 )$ and $\mathrm { C } ( \mathrm { h } , \mathrm { k } )$, then the point $C$ lies on the circle:
(1) $x ^ { 2 } + y ^ { 2 } - 61 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 52 = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 65 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 74 = 0$

Q66. If the image of the point $( - 4,5 )$ in the line $x + 2 y = 2$ lies on the circle $( x + 4 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = r ^ { 2 }$, then r is equal to:
(1) 2
(2) 3
(3) 1
(4) 4

Q66. Let a circle passing through $( 2,0 )$ have its centre at the point $( h , k )$. Let $\left( x _ { c } , y _ { c } \right)$ be the point of intersection of the lines $3 x + 5 y = 1$ and $( 2 + c ) x + 5 c ^ { 2 } y = 1$. If $\mathrm { h } = \lim _ { \mathrm { c } \rightarrow 1 } x _ { \mathrm { c } }$ and $\mathrm { k } = \lim _ { \mathrm { c } \rightarrow 1 } y _ { \mathrm { c } }$, then the equation of the circle is :
(1) $25 x ^ { 2 } + 25 y ^ { 2 } - 2 x + 2 y - 60 = 0$
(2) $5 x ^ { 2 } + 5 y ^ { 2 } - 4 x + 2 y - 12 = 0$
(3) $5 x ^ { 2 } + 5 y ^ { 2 } - 4 x - 2 y - 12 = 0$
(4) $25 x ^ { 2 } + 25 y ^ { 2 } - 20 x + 2 y - 60 = 0$

Q83. Let the centre of a circle, passing through the points $( 0,0 ) , ( 1,0 )$ and touching the circle $x ^ { 2 } + y ^ { 2 } = 9$, be $( h , k )$ - Then for all possible values of the coordinates of the centre $( h , k ) , 4 \left( h ^ { 2 } + k ^ { 2 } \right)$ is equal to

Question 1 is a multiple choice question with ten parts. Marks are given solely for correct answers but any rough working should be shown in the space between parts. Answer Question 1 on the grid on Page 2. Each part is worth 4 marks.
Answers to questions 2-7 should be written in the space provided, continuing on to the blank pages at the end of this booklet if necessary. Each of Questions 2-7 is worth 15 marks.
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Signature of Invigilator: $\_\_\_\_$ FOR OFFICE USE ONLY:

Q1	Q2	Q3	Q4	Q5	Q6	Q7

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 $( \checkmark )$ 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. Which of the following lines is a tangent to the circle with equation
$$x ^ { 2 } + y ^ { 2 } = 4 ?$$
(a) $x + y = 2$;
(b) $y = x - 2 \sqrt { 2 }$;
(c) $x = \sqrt { 2 }$;
(d) $y = \sqrt { 2 } - x$.
B. Let $N = 2 ^ { k } \times 4 ^ { m } \times 8 ^ { n }$ where $k , m , n$ are positive whole numbers. Then $N$ will definitely be a square number whenever
(a) $k$ is even;
(b) $k + n$ is odd;
(c) $k$ is odd but $m + n$ is even;
(d) $k + n$ is even.
C. Which is the smallest of the following numbers?
(a) $( \sqrt { 3 } ) ^ { 3 }$,
(b) $\quad \log _ { 3 } \left( 9 ^ { 2 } \right)$,
(c) $\quad \left( 3 \sin \frac { \pi } { 3 } \right) ^ { 2 }$,
(d) $\quad \log _ { 2 } \left( \log _ { 2 } \left( 8 ^ { 5 } \right) \right)$.
D. Shown below is a diagram of the square with vertices $( 0,0 ) , ( 0,1 ) , ( 1,1 ) , ( 1,0 )$ and the line $y = x + c$. The shaded region is the region of the square which lies below the line; this shaded region has area $A ( c )$. [Figure]
Which of the following graphs shows $A ( c )$ as $c$ varies? [Figure]
(a) [Figure]
(b) [Figure]
(c) [Figure]
(d)
E. Which one of the following equations could possibly have the graph given below?
(a) $y = ( 3 - x ) ^ { 2 } ( 3 + x ) ^ { 2 } ( 1 - x )$;
(b) $y = - x ^ { 2 } ( x - 9 ) \left( x ^ { 2 } - 3 \right)$;
(c) $y = ( x - 6 ) ( x - 2 ) ^ { 2 } ( x + 2 ) ^ { 2 }$;
(d) $y = \left( x ^ { 2 } - 1 \right) ^ { 2 } ( 3 - x )$. [Figure] F. Let
$$T = \left( \int _ { - \pi / 2 } ^ { \pi / 2 } \cos x \mathrm {~d} x \right) \times \left( \int _ { \pi } ^ { 2 \pi } \sin x \mathrm {~d} x \right) \times \left( \int _ { 0 } ^ { \pi / 8 } \frac { \mathrm {~d} x } { \cos 3 x } \right)$$
Which of the following is true?
(a) $\quad T = 0$;
(b) $T < 0$;
(c) $T > 0$;
(d) $T$ is not defined. G. There are positive real numbers $x$ and $y$ which solve the equations
$$2 x + k y = 4 , \quad x + y = k$$
for
(a) all values of $k$;
(b) no values of $k$;
(c) $k = 2$ only;
(d) only $k > - 2$. H. In the region $0 < x \leqslant 2 \pi$, the equation
$$\int _ { 0 } ^ { x } \sin ( \sin t ) d t = 0$$
has
(a) no solution;
(b) one solution;
(c) two solutions;
(d) three solutions. I. The vertices of an equilateral triangle are labelled $X , Y$ and $Z$. The points $X , Y$ and $Z$ lie on a circle of circumference 10 units. Let $P$ and $A$ be the numerical values of the triangle's perimeter and area, respectively. Which of the following is true?
(a) $\frac { A } { P } = \frac { 5 } { 4 \pi } ;$
(b) $P < A$;
(c) $\frac { P } { A } = \frac { 10 } { 3 \pi }$;
(d) $P ^ { 2 }$ is rational. J. If two chords $Q P$ and $R P$ on a circle of radius 1 meet in an angle $\theta$ at $P$, for example as drawn in the diagram below, [Figure] then the largest possible area of the shaded region $R P Q$ is
(a) $\theta \left( 1 + \cos \left( \frac { \theta } { 2 } \right) \right) ;$
(b) $\theta + \sin \theta$;
(c) $\frac { \pi } { 2 } ( 1 - \cos \theta )$;
(d) $\quad \theta$.

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

Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 16.
A circle $A$ passes through the points $( - 1,0 )$ and $( 1,0 )$. Circle $A$ has centre $( m , h )$, and radius $r$.
(i) Determine $m$ and write $r$ in terms of $h$.
(ii) Given a third point $\left( x _ { 0 } , y _ { 0 } \right)$ and $y _ { 0 } \neq 0$ show that there is a unique circle passing through the three points $( - 1,0 ) , ( 1,0 ) , \left( x _ { 0 } , y _ { 0 } \right)$.
For the remainder of the question we consider three circles $A , B$, and $C$, each passing through the points $( - 1,0 ) , ( 1,0 )$. Each circle is cut into regions by the other two circles. For a group of three such circles, we will say the lopsidedness of a circle is the fraction of the full area of that circle taken by its largest region.
(iii) Let circle $A$ additionally pass through the point ( 1,2 ), circle $B$ pass through ( 0,1 ), and let circle $C$ pass through the point $( 0 , - 4 )$. What is the lopsidedness of circle $A$ ?
(iv) Let $p > 0$. Now let $A$ pass through ( $1,2 p$ ), $B$ pass through ( 0,1 ), and $C$ pass through $( - 1 , - 2 p )$. Show that the value of $p$ minimising the lopsidedness of circle $B$ satisfies the equation
$$\left( p ^ { 2 } + 1 \right) \tan ^ { - 1 } \left( \frac { 1 } { p } \right) - p = \frac { \pi } { 6 }$$
Note that $\tan ^ { - 1 } ( x )$ is sometimes written as $\arctan ( x )$ and is the value of $\theta$ in the range $\frac { - \pi } { 2 } < \theta < \frac { \pi } { 2 }$ such that $\tan ( \theta ) = x$.
If you require additional space please use the pages at the end of the booklet

Consider three distinct points $A$, $B$, $C$ in the coordinate plane, where point $A$ is $(1, 1)$. Circles are drawn with line segments $\overline{AB}$ and $\overline{AC}$ as diameters. These two circles intersect at point $A$ and point $P(4, 2)$. Given that $\overline{PB} = 3\sqrt{10}$ and point $B$ is in the fourth quadrant, the coordinates of point $B$ are (（12），（13）（14）).

The line segment joining the points $( 3,3 )$ and ( 7,5 ) is a diameter of a circle. This circle is translated by 3 units in the negative $x$-direction, then reflected in the $x$-axis, and then enlarged by a scale factor of 4 about the centre of the resulting circle.
The equation of the final circle is
A $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 320$ B $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 320$ C $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 80$ D $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 80$ E $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 20$ F $( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20$

Each interior angle of a regular polygon with $n$ sides is $\frac { 3 } { 4 }$ of each interior angle of a second regular polygon with $m$ sides.
How many pairs of positive integers $n$ and $m$ are there for which this statement is true?

Find the length of the curve with equation
$$2 \log _ { 10 } ( x - y ) = \log _ { 10 } ( 2 - 2 x ) + \log _ { 10 } ( y + 5 )$$
A 5 B 10 C 15 D $3 \pi$ E $9 \pi$ F $12 \pi$

4

Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t,\, f(t))$ on the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.

1. [(1)] Let the center of circle $C_t$ be $(c(t),\, 0)$ and the radius be $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
   
2. [(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that circle $C_t$ passes through the point $(3,\, a)$?

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Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t, f(t))$ in the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.

1. [(1)] Let the center of circle $C_t$ have coordinates $(c(t), 0)$ and radius $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
   
2. [(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that the circle $C_t$ passes through the point $(3, a)$?

OAEF is a rectangle, ABCD is a square $| \mathrm { FE } | = 7$ units $| \mathrm { AB } | = 2$ units $| \mathrm { DE } | = x$
In the figure, points E and C are on a quarter circle with center O.
Accordingly, what is $x$ in units?
A) $\frac { 7 } { 2 }$ B) $\frac { 9 } { 2 }$ C) $\frac { 13 } { 4 }$ D) 3 E) 4

In the rectangular coordinate plane, a circle divided into two equal parts by the line $x + y = 4$ intersects the x-axis at a single point and the y-axis at two different points. Given that the distance between the points where the circle intersects the y-axis is 4 units, what is the circumference of the circle in units?
A) $4 \pi$
B) $5 \pi$
C) $6 \pi$
D) $7 \pi$
E) $8 \pi$

In a rectangular coordinate plane, point $A(11, 9)$ is located in the interior of a circle that is tangent to the line $y = x$ at point $B(7, 7)$.
Accordingly, what is the smallest integer value that the radius of this circle can take in units?
A) 6 B) 8 C) 10 D) 12 E) 14

Two identical blue ropes have one end each tied to two nails on a wall at equal heights from the ground and 48 units apart. Then a circular plate is hung on these ropes such that the other ends of the ropes are attached to two points on the circumference of the plate and the ropes are perpendicular to the ground, as shown in Figure 1. Later, one of these ropes broke and the plate hung on the remaining rope, and when the rope is perpendicular to the ground, the view in Figure 2 is obtained, and the height of the plate from the ground decreased by 16 units compared to the initial situation.
Accordingly, what is the radius of this plate in units?
A) 25 B) 26 C) 29 D) 30 E) 32