Let a circle passes through origin and the points $\mathrm { A } ( - \sqrt { 2 } \alpha , 0 ) , \mathrm { B } ( 0, \sqrt { 2 } \beta )$, where $\alpha$ and $\beta$ are non zero real parameters, such that its radius is 4 . Then the radius of locus of centroid of triangle $O A B$ is
(A) $\frac { 2 } { 3 }$
(B) $\frac { 4 } { 3 }$
(C) $\frac { 11 } { 3 }$

We have a triangle ABC such that
$$\mathrm { AB } = 9 , \quad \mathrm { BC } = 12 , \quad \angle \mathrm { ABC } = 90 ^ { \circ } .$$
There are also two circles $\mathrm { O } _ { 1 }$ and $\mathrm { O } _ { 2 }$ with radii of length $2r$ and $r$, respectively. The two circles $\mathrm { O } _ { 1 }$ and $\mathrm { O } _ { 2 }$ are tangential to each other. Further, $\mathrm { O } _ { 1 }$ is tangent to the two sides AB and AC, and $\mathrm { O } _ { 2 }$ is tangent to the two sides CA and CB. We are to find the value of $r$.
First, let D and E denote the points at which the segment AC is tangent to the circles $\mathrm { O } _ { 1 }$ and $\mathrm { O } _ { 2 }$ respectively, and set $\alpha = \angle \mathrm { O } _ { 1 } \mathrm { AC }$. Then, since $\tan 2 \alpha = \frac { \square \mathbf { A } } { \square }$, we have $\tan \alpha = \frac { \mathbf { C } } { \mathbf { D } }$ using the double-angle formula. Thus, we obtain $\mathrm { AD } = \mathbf { E }$.
Next, set $\beta = \angle \mathrm { O } _ { 2 } \mathrm { CA }$. Since $\alpha + \beta = \mathbf { F G } ^ { \circ }$, we have $\tan \beta = \frac { \mathbf { H } } { \square \mathbf{I} }$ using the addition theorem. Thus, we obtain $\mathrm { CE } = \square r$.
Moreover, it follows that $\mathrm { AC } = \mathbf { K L }$ and $\mathrm { DE } = \mathbf { M } \sqrt { \mathbf { N } } r$.
Finally we obtain
$$r = \frac { \mathbf { O P } ( \mathbf { Q } - \mathbf { R } \sqrt { \mathbf { S } } ) } { 41 }$$

Let $a$, $b$ and $c$ be positive real numbers. Consider a triangle ABC whose vertices are the three points $\mathrm{A}(a, 0)$, $\mathrm{B}(3, b)$ and $\mathrm{C}(0, c)$ on a plane with the coordinate system. Assume that the circumscribed circle of the triangle ABC passes through the origin $\mathrm{O}(0,0)$ and that $\angle\mathrm{BAC} = 60^\circ$.
(1) Since $\angle\mathrm{AOB} = \mathbf{AB}^\circ$, we obtain $b = \sqrt{\mathbf{C}}$.
(2) The equation of the circumscribed circle is
$$\left(x - \frac{a}{\mathbf{D}}\right)^2 + \left(y - \frac{c}{\mathbf{E}}\right)^2 = \frac{a^2 + c^2}{\mathbf{F}},$$
and $c$ can be expressed in terms of $a$ as $c = \sqrt{\mathbf{G}}\,(\mathbf{H} - a)$.
(3) Let D denote the point of intersection of the segment OB and the segment AC.
Set $\alpha = \angle\mathrm{OAC}$ and $\beta = \angle\mathrm{ADB}$. When $a = 2\sqrt{3}$, it follows that
$$\tan\alpha = \mathbf{I} - \sqrt{\mathbf{J}}, \quad \tan\beta = \mathbf{K}.$$

On the coordinate plane, take point $\mathrm { A } ( 2,0 )$, and on the circle centered at origin O with radius 2, take points $\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ such that points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }$ are the vertices of a regular hexagon in order. Here, B is in the first quadrant.
(1) The coordinates of point B are (ア ア (ア $)$, and the coordinates of point D are $( -$ ウ, $0 )$.
(2) Let M be the midpoint of segment BD, and let N be the intersection of line AM and line CD. We want to find $\overrightarrow { \mathrm { ON } }$.
$\overrightarrow { \mathrm { ON } }$ can be expressed in two ways using real numbers $r, s$: $\overrightarrow { \mathrm { ON } } = \overrightarrow { \mathrm { OA } } + r \overrightarrow { \mathrm { AM } } , \overrightarrow { \mathrm { ON } } = \overright

Let $C$ be a circle with a radius of 4, centered at the point $( 5,0 )$ on the $x$-axis.
(1) If $\mathrm { P } ( p , q )$ is a point on circle $C$, then
$$p ^ { 2 } - \mathbf { PQ } p + q ^ { 2 } + \mathbf { R } = 0 .$$
Also, the equation of the tangent to circle $C$ at point $\mathrm { P } ( p , q )$ is
$$( p - \mathbf { S } ) x + q y = \mathbf { T } p - \mathbf { U } .$$
(2) Let us draw a line tangent to circle $C$ from point $\mathrm { A } ( 0 , a )$ on the $y$-axis, where $a \geqq 0$, and let $\mathrm { P } ( p , q )$ be the tangent point.
The length of the segment AP is minimized at $a = \mathbf { V }$, and the length in this case is $\mathbf { W }$.
Furthermore, the two tangents to circle $C$ from point A are orthogonal when the length of AP is $\mathbf { X }$. In this case, the value of $a$ is $a = \sqrt { \mathbf { Y } }$.

(Course 2) On a coordinate plane, consider a circle $C$ with the radius of 1 centered at the origin O. We denote by P and Q the points of intersection of $C$ and the radii which are rotated at angles of $\theta$ and $3\theta$ respectively from the positive section of the $x$ axis, where $0 \leqq \theta \leqq \pi$.
Also, we denote by A the point at which the straight line which is perpendicular to the $x$ axis and passes through point P intersects the $x$ axis, and we denote by B the point at which the straight line which is perpendicular to the $x$ axis and passes through point Q intersects the $x$ axis. Furthermore, we denote the length of line segment AB by $\ell$.
(1) When $\theta = \frac { \pi } { 3 }$, we see that $\ell = \frac { \mathbf { A } } { \mathbf { B } }$.
(2) We are to find the maximum value of $\ell$. When we set $\cos \theta = t$ and express $\ell$ in terms of $t$, we have
$$\ell = \left| \mathbf { C } t ^ { \mathbf { D } } - \mathbf { E } t \right| .$$
Next, when we set $g ( t ) = \mathrm { C } t ^ { \mathrm { D } } - \mathrm { E } t$, we have
$$g ^ { \prime } ( t ) = \mathbf { F } \left( \mathbf { G } t ^ { \mathbf { H } } - 1 \right) .$$
Hence, when
$$\cos \theta = \pm \frac { \sqrt { \mathbf { J } } } { \mathbf { J } }$$
$\ell$ is maximized and its value is $\frac { \mathbf { K } \sqrt { \mathbf { L } } } { \mathbf { M } }$.
(3) For $\mathbf { N } \sim \mathbf{S}$ in the following sentence, choose the correct answer from among choices (0) $\sim$ (9) below.
There are two pairs of points P and Q at which $\ell$ is maximized, and their coordinates are
$$\mathrm { P } \left( \frac { \sqrt { \mathbf{I} } } { \mathbf{J} } , \mathbf{N} \right) \text{ and } \mathrm { Q } \left( \mathbf{O} , \mathbf{P} \right)$$
and
$$\mathrm { P } \left( - \frac { \sqrt { \mathbf{I} } } { \mathbf{J} } , \mathbf{Q} \right) \text{ and } \mathrm { Q } \left( \mathbf{R} , \mathbf{S} \right)$$
(0) $\frac { \sqrt { 6 } } { 3 }$
(1) $\frac { \sqrt { 6 } } { 2 }$
(2) $\frac { 4 \sqrt { 3 } } { 9 }$
(3) $- \frac { 4 \sqrt { 3 } } { 9 }$
(4) $\frac { 5 \sqrt { 3 } } { 9 }$
(5) $- \frac { 5 \sqrt { 3 } } { 9 }$ (6) $\frac { \sqrt { 6 } } { 9 }$ (7) $- \frac { \sqrt { 6 } } { 9 }$ (8) $\frac { 2 \sqrt { 6 } } { 9 }$ (9) $- \frac { 2 \sqrt { 6 } } { 9 }$

3. Let $O , P , P _ { 1 } , P _ { 2 }$ be the points in the $( x , y )$-plane with coordinates $( 0,0 ) , ( s , 1 / s )$, $\left( s _ { 1 } , 1 / s _ { 1 } \right) , \left( s _ { 2 } , 1 / s _ { 2 } \right)$ respectively.
(i) Using the axes below, sketch the curve traced out by $P$ as $s$ varies over non-zero real values, and find an equation for the curve in the form $y = f ( x )$.
(ii) Write down the equation of the straight line $P P _ { 1 }$ joining $P$ to $P _ { 1 }$, giving your answer in the form $y = m _ { 1 } x + c _ { 1 }$.
(iii) Show that the line $P P _ { 1 }$ is perpendicular to $P P _ { 2 }$ if, and only if, $s _ { 1 } s _ { 2 } = - 1 / s ^ { 2 }$.
(iv) Let $m _ { 1 } , m _ { 2 } , n _ { 1 } , n _ { 2 }$ be the gradients of the lines $P P _ { 1 } , P P _ { 2 } , O P _ { 1 } , O P _ { 2 }$ respectively. Show that
$$\left( \frac { m _ { 1 } } { m _ { 2 } } \right) ^ { 2 } = \frac { n _ { 1 } } { n _ { 2 } }$$
[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.
In the diagram below is sketched the circle with centre ( 1,1 ) and radius 1 and a line $L$. The line $L$ is tangential to the circle at $Q$; further $L$ meets the $y$-axis at $R$ and the $x$-axis at $P$ in such a way that the angle $O P Q$ equals $\theta$ where $0 < \theta < \pi / 2$. [Figure]
(i) Show that the co-ordinates of $Q$ are
$$( 1 + \sin \theta , 1 + \cos \theta ) ,$$
and that the gradient of $P Q R$ is $- \tan \theta$. Write down the equation of the line $P Q R$ and so find the co-ordinates of $P$.
(ii) The region bounded by the circle, the $x$-axis and $P Q$ has area $A ( \theta )$; the region bounded by the circle, the $y$-axis and $Q R$ has area $B ( \theta )$. (See diagram.)
Explain why
$$A ( \theta ) = B ( \pi / 2 - \theta )$$
for any $\theta$. Calculate $A ( \pi / 4 )$.
(iii) Show that
$$A \left( \frac { \pi } { 3 } \right) = \sqrt { 3 } - \frac { \pi } { 3 } .$$

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. [Figure]
Let $p$ and $q$ be positive real numbers. Let $P$ denote the point ( $p , 0$ ) and $Q$ denote the point $( 0 , q )$.
(i) Show that the equation of the circle $C$ which passes through $P , Q$ and the origin $O$ is
$$x ^ { 2 } - p x + y ^ { 2 } - q y = 0 .$$
Find the centre and area of $C$.
(ii) Show that
$$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } \geqslant \pi \text {. }$$
(iii) Find the angles $O P Q$ and $O Q P$ if
$$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } = 2 \pi$$

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.
[Figure]
Diagram when $h > 2 / \sqrt { 5 }$
[Figure]
Diagram when $h < \sqrt { 3 } / 2$
The three corners of a triangle $T$ are $( 0,0 ) , ( 3,0 ) , ( 1,2 h )$ where $h > 0$. The circle $C$ has equation $x ^ { 2 } + y ^ { 2 } = 4$. The angle of the triangle at the origin is denoted as $\theta$. The circle and triangle are drawn in the diagrams above for different values of $h$.
(i) Express $\tan \theta$ in terms of $h$.
(ii) Show that the point $( 1,2 h )$ lies inside $C$ when $h < \sqrt { 3 } / 2$.
(iii) Find the equation of the line connecting $( 3,0 )$ and $( 1,2 h )$. Show that this line is tangential to the circle $C$ when $h = 2 / \sqrt { 5 }$.
(iv) Suppose now that $h > 2 / \sqrt { 5 }$. Find the area of the region inside both $C$ and $T$ in terms of $\theta$.
(v) Now let $h = 6 / 7$. Show that the point ( $8 / 5,6 / 5$ ) lies on both the line (from part (iii)) and the circle $C$.
Hence show that the area of the region inside both $C$ and $T$ equals
$$\frac { 27 } { 35 } + 2 \alpha$$
where $\alpha$ is an angle whose tangent, $\tan \alpha$, you should determine. [0pt] [You may use the fact that the area of a triangle with corners $( 0,0 ) , ( a , b ) , ( c , d )$ equals $\frac { 1 } { 2 } | a d - b c |$.]
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Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
Let $Q$ denote the quarter-disc of points $( x , y )$ such that $x \geqslant 0 , y \geqslant 0$ and $x ^ { 2 } + y ^ { 2 } \leqslant 1$ as drawn in Figures A and B below.
[Figure]
Figure A
[Figure]
Figure B
(i) On the axes in Figure A, sketch the graphs of
$$x + y = \frac { 1 } { 2 } , \quad x + y = 1 , \quad x + y = \frac { 3 } { 2 } .$$
What is the largest value of $x + y$ achieved at points $( x , y )$ in $Q$ ? Justify your answer.
(ii) On the axes in Figure B, sketch the graphs of
$$x y = \frac { 1 } { 4 } , \quad x y = 1 , \quad x y = 2 .$$
What is the largest value of $x ^ { 2 } + y ^ { 2 } + 4 x y$ achieved at points $( x , y )$ in $Q$ ? What is the largest value of $x ^ { 2 } + y ^ { 2 } - 6 x y$ achieved at points $( x , y )$ in $Q$ ?
(iii) Describe the curve
$$x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = k$$
where $k > - 5$. What is the smallest value of $x ^ { 2 } + y ^ { 2 } - 4 x - 2 y$ achieved at points ( $x , y$ ) in $Q$ ?

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.
For Test Supervisors Use Only: [0pt] [ ] Tick here if special arrangements were made for the test. Please either include details below or securely attach to the test script a letter with the details.
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) $\quad \theta$.

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Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
The diagram below shows the parabola $y = x ^ { 2 }$ and a circle with centre $( 0,2 )$ just 'resting' on the parabola. By 'resting' we mean that the circle and parabola are tangential to each other at the points $A$ and $B$. [Figure]
(i) Let ( $x , y$ ) be a point on the parabola such that $x \neq 0$. Show that the gradient of the line joining this point to the centre of the circle is given by
$$\frac { x ^ { 2 } - 2 } { x } .$$
(ii) With the help of the result from part (i), or otherwise, show that the coordinates of $B$ are given by
$$\left( \sqrt { \frac { 3 } { 2 } } , \frac { 3 } { 2 } \right) .$$
(iii) Show that the area of the sector of the circle enclosed by the radius to $A$, the minor $\operatorname { arc } A B$ and the radius to $B$ is equal to
$$\frac { 7 } { 4 } \cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 7 } } \right)$$
(iv) Suppose now that a circle with centre ( $0 , a$ ) is resting on the parabola, where $a > 0$. Find the range of values of $a$ for which the circle and parabola touch at two distinct points.
(v) Let $r$ be the radius of a circle with centre ( $0 , a$ ) that is resting on the parabola. Express $a$ as a function of $r$, distinguishing between the cases in which the circle is, and is not, in contact with the vertex of the parabola.

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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 $\tan ( \theta ) = x$.
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The line $l$ passes through the origin at angle $2 \alpha$ above the $x$-axis, where $2 \alpha < \frac { \pi } { 2 }$. [Figure]
Circles $C _ { 1 }$ of radius 1 and $C _ { 2 }$ of radius 3 are drawn between $l$ and the $x$-axis, just touching both lines.
(i) What is the centre of circle $C _ { 1 }$ ?
(ii) What is the equation of circle $C _ { 1 }$ ?
(iii) For what value of $\alpha$ do circles $C _ { 1 }$ and $C _ { 2 }$ touch?
(iv) For this value of $\alpha$ (for which the circles $C _ { 1 }$ and $C _ { 2 }$ touch) a third circle, $C _ { 3 }$, larger than $C _ { 2 }$, is to be drawn between $l$ and the $x$-axis. $C _ { 3 }$ just touches both lines and also touches $C _ { 2 }$. What is the radius of this circle $C _ { 3 }$ ?
(v) For the same value of $\alpha$, what is the area of the region bounded by the $x$-axis and the circles $C _ { 1 }$ and $C _ { 2 }$ ?
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Consider two circles $S _ { 1 }$ and $S _ { 2 }$ centred at $A$ and $B$ and with radii $\sqrt { 6 }$ and $\sqrt { 3 } - 1$, respectively. Suppose that the two circles intersect at two distinct points $C$ and $D$. Suppose further that the two centres $A$ and $B$ are of distance 2 apart. The sketch below is not to scale. [Figure]
(i) Find the angle $\angle C B A$, and deduce that $A$ and $B$ lie on the same side of the line $C D$.
(ii) Show that $C D$ has length $3 - \sqrt { 3 }$ and hence calculate the angle $\angle C A D$.
(iii) Show that the area of the region lying inside the circle $S _ { 2 }$ and outside of the circle $S _ { 1 }$ (that is the shaded region in the picture) is equal to
$$\frac { \pi } { 6 } ( 5 - 4 \sqrt { 3 } ) + 3 - \sqrt { 3 } .$$
(iv) Suppose that a line through $C$ is drawn such that the total area covered by $S _ { 1 }$ and $S _ { 2 }$ is split into two equal areas. Let $E$ be the intersection of this line with $S _ { 1 }$ and $x$ denote the angle $\angle C A E$. You may assume that $E$ lies on the larger $\operatorname { arc } C D$ of $S _ { 1 }$. Write down an equation which $x$ satisfies and explain why there is a unique solution $x$.

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In this question we will consider subsets $S$ of the $x y$-plane and points $( a , b )$ which may or may not be in $S$. We will be interested in those points of $S$ which are nearest to the point $( a , b )$. There may be many such points, a unique such point, or no such point.
(i) Let $S$ be the disc $x ^ { 2 } + y ^ { 2 } \leqslant 1$. For a given point ( $a , b$ ), find the unique point of $S$ which is closest to $( a , b )$. [0pt] [You will need to consider separately the cases when $a ^ { 2 } + b ^ { 2 } > 1$ and when $a ^ { 2 } + b ^ { 2 } \leqslant 1$.]
(ii) Describe (without further justification) an example of a subset $S$ and a point $( a , b )$ such that there is no point of $S$ nearest to $( a , b )$.
(iii) Describe (without further justification) an example of a subset $S$ and a point $( a , b )$ such that there is more that one point of $S$ nearest to $( a , b )$.
(iv) Let $S$ denote the line with equation $y = m x + c$. Obtain an expression for the distance of $( a , b )$ from a general point $( x , m x + c )$ of $S$.
Show that there is a unique point of $S$ nearest to $( a , b )$.
(v) For some subset $S$, and for any point ( $a , b$ ), the nearest point of $S$ to ( $a , b$ ) is
$$\left( \frac { a + 2 b - 2 } { 5 } , \frac { 2 a + 4 b + 1 } { 5 } \right) .$$
Describe the subset $S$.
(vi) Say now that $S$ has the property that for any two points $P$ and $Q$ in $S$ the line segment $P Q$ is also in $S$.
Show that, for a given point $( a , b )$, there cannot be two distinct points of $S$ which are nearest to $( a , b )$.
If you require additional space please use the pages at the end of the booklet

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

Computer Science and Computer Science \& Philosophy applicants should turn to page 20.
Below is a sketch of the curve $S$ with equation $y ^ { 2 } - y = x ^ { 3 } - x$. The curve crosses the $x$-axis at the origin and at $( a , 0 )$ and at $( b , 0 )$ for some real numbers $a < 0$ and $b > 0$. The curve only exists for $\alpha \leqslant x \leqslant \beta$ and for $x \geqslant \gamma$. The three points with coordinates $( \alpha , \delta ) , ( \beta , \delta )$, and $( \gamma , \delta )$ are all on the curve. [Figure]
(i) What are the values of $a$ and $b$ ?
(ii) By completing the square, or otherwise, find the value of $\delta$.
(iii) Explain why the curve is symmetric about the line $y = \delta$.
(iv) Find a cubic equation in $x$ which has roots $\alpha , \beta , \gamma$. (Your expression for the cubic should not involve $\alpha , \beta$, or $\gamma$ ). Justify your answer.
(v) By considering the factorization of this cubic, find the value of $\alpha + \beta + \gamma$.
(vi) Let $C$ denote the circle which has the points $( \alpha , \delta )$ and $( \beta , \delta )$ as ends of a diameter. Write down the equation of $C$. Show that $C$ intersects $S$ at two other points and find their common $x$-co-ordinate in terms of $\gamma$.
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Let the planes $\pi _ { 1 } \equiv x + y = 1$ and $\pi _ { 2 } \equiv x + z = 1$.\ a) ( 1.5 points) Find the planes parallel to plane $\pi _ { 1 }$ such that their distance to the origin of coordinates is 2.\ b) ( 0.5 points) Find the line that passes through the point ( $0,2,0$ ) and is perpendicular to plane $\pi _ { 2 }$.\ c) ( 0.5 points) Find the distance between the points of intersection of plane $\pi _ { 1 }$ with the $x$ and $y$ axes.

12. Let $\Gamma : x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0$ be a circle in the coordinate plane. Which of the following statements are correct?
(1) The center of $\Gamma$ is at $( 5,0 )$
(2) The maximum distance from a point on $\Gamma$ to the line $L : 3 x + 4 y - 15 = 0$ equals 4
(3) The line $L _ { 1 } : 3 x + 4 y + 15 = 0$ is tangent to $\Gamma$
(4) There

6. In coordinate space, $O$ is the origin and point $A$ has coordinates $(1, 2, 1)$. Let $S$ be the sphere with center $O$ and radius 4. What is the figure formed by all points $P$ on $S$ that satisfy the dot product $\overrightarrow{OA} \cdot \overrightarrow{OP} = 6$?
(1) Empty set
(2) A single point
(3) Two points
(4) A circle
(5) Two circles

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）).

On the coordinate plane, a circle with radius 12 intersects the line $x + y = 0$ at two points, and the distance between these two points is 8. If this circle intersects the line $x + y = 24$ at points $P$ and $Q$, then the length of segment $\overline { P Q }$ is $\_\_\_\_$ (14)$\sqrt { (15) }$. (Express as a simplified radical)

On the coordinate plane, there is a regular hexagon $A B C D E F$ with side length 3, where $A ( 3,0 ) , D ( - 3,0 )$. How many intersection points does the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$ have with the regular hexagon $A B C D E F$?
(1) 0
(2) 2
(3) 4
(4) 6
(5) 8

As shown in the figure, $L$ is a line passing through the origin $O$ on the coordinate plane, $\Gamma$ is a circle centered at $O$, and $L$ and $\Gamma$ have one intersection point $A ( 3,4 )$. It is known that $B , C$ are two distinct points on $\Gamma$ satisfying $\overrightarrow { B C } = \overrightarrow { O A }$. Select the correct options.
(1) The other intersection point of $L$ and $\Gamma$ is $( - 4 , - 3 )$
(2) The slope of line $B C$ is $\frac { 3 } { 4 }$
(3) $\angle A O C = 60 ^ { \circ }$
(4) The area of $\triangle A B C$ is $\frac { 25 \sqrt { 3 } } { 2 }$
(5) $B$ and $C$ are in the same quadrant