Let S be the focus of the hyperbola $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 5 } = 1$, on the positive $x$-axis. Let C be the circle with its centre at $A ( \sqrt { 6 } , \sqrt { 5 } )$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to $\_\_\_\_$

Let $A , B$ and $C$ be three points on the parabola $y ^ { 2 } = 6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ and $B$ on $L$. Then $\left( \frac { A M \cdot B N } { C D } \right) ^ { 2 }$ is equal to $\_\_\_\_$

Let the line $x + y = 1$ meet the circle $x^2 + y^2 = 4$ at the points A and B. If the line perpendicular to $AB$ and passing through the mid point of the chord $AB$ intersects the circle at $C$ and $D$, then the area of the quadrilateral ADBC is equal to:
(1) $\sqrt{14}$
(2) $3\sqrt{7}$
(3) $2\sqrt{14}$
(4) $5\sqrt{7}$

Q67. A square is inscribed in the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 6 y + 30 = 0$. One side of this square is parallel to $y = x + 3$. If $\left( x _ { i } , y _ { i } \right)$ are the vertices of the square, then $\boldsymbol { \Sigma } \left( x _ { i } ^ { 2 } + y _ { i } ^ { 2 } \right)$ is equal to:
(1) 148
(2) 152
(3) 160
(4) 156

Q84. Let S be the focus of the hyperbola $\frac { x ^ { 2 } } { 3 } - \frac { y ^ { 2 } } { 5 } = 1$, on the positive $x$-axis. Let C be the circle with its centre at $A ( \sqrt { 6 } , \sqrt { 5 } )$ and passing through the point $S$. If $O$ is the origin and $S A B$ is a diameter of $C$, then the square of the area of the triangle OSB is equal to $\_\_\_\_$

If $4 x ^ { 2 } + y ^ { 2 } \leq 52 , x , y \in I$ then number of ordered pairs ( $x , y$ ) is (A) 67 (B) 77 (C) 87 (D) 38

The line $y = x + 1$ intersects the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 1 } = 1$ at $A$ and $B$. Find the angle sub-stained by segment AB and centre of ellipse is\ (A) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$\ (B) $\frac { \pi } { 2 } + 2 \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$\ (C) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$\ (D) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$

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]
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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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 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 $Q$ ?

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 14.
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 $x$.

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
What are the coordinates of point $B$? (Single-choice question, 3 points)
(1) $( 0,2 )$
(2) $( 1 , \sqrt { 3 } )$
(3) $( \sqrt { 2 } , \sqrt { 2 } )$
(4) $( \sqrt { 3 } , 1 )$
(5) $( 2,0 )$

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
(Continuing from Question 19) Let $\Omega$ denote the region swept by the scanning rod in the first quadrant. Find the areas of $\Omega$ and $R$ respectively. (Non-multiple choice question, 6 points)

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, there is a parallelogram $\Gamma$, where two sides lie on lines parallel to $5x - y = 0$, and the other two sides lie on lines perpendicular to $3x - 2y = 0$. Let $Q$ be the intersection point of the two diagonals of $\Gamma$. It is known that $\Gamma$ has a vertex $P$ satisfying $\overrightarrow{PQ} = (10, -1)$. The area of $\Gamma$ is (11--1)(11--2)(11--3).

A circle has centre $O$ and radius 6 .
$P , Q$ and $R$ are points on the circumference with angle $P O Q \geq \frac { \pi } { 2 }$
The area of the triangle $P O Q$ is $9 \sqrt { 3 }$
What is the greatest possible area of triangle $P R Q$ ?

The figure below shows the construction used to obtain a square with an area equal to that of a given rectangle.
ABCD is a rectangle, HDFG is a square, semicircle with center O
$$A ( ABCD ) = A ( HDFG )$$
The F vertex of the square HDFG in the figure lies on the semicircle with center O.
Given that the perimeter of rectangle ABCD is 36 cm, what is the diameter of the circle in cm?
A) 12
B) 15
C) 18
D) 21
E) 24

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

ABCD is a square $\mathrm { AF } \perp \mathrm { FB }$ $\mathrm { DE } \perp \mathrm { AF }$ $| E F | = 4 \mathrm {~cm}$
Given that the area of triangle AFB in the figure is $30 \mathrm {~cm} ^ { 2 }$, what is the area of square ABCD in $\mathrm { cm } ^ { 2 }$?
A) 81
B) 100
C) 120
D) 136
E) 144

Teacher Aslı created the number 3 on a piece of paper by painting identical equilateral triangles inside an equilateral triangle ABC as shown in the figure.
If the area of equilateral triangle ABC is 96 square units, what is the painted area in square units?
A) 22 B) 27 C) 33 D) 36 E) 44

Given two squares as shown; the area of square ABCD is equal to 2 times the area of square CEFG.
Accordingly, what is the ratio $\frac { | \mathrm { AF } | } { | \mathrm { AG } | }$?
A) $\frac { \sqrt { 5 } } { 2 }$ B) $\frac { 2 \sqrt { 2 } } { 3 }$ C) $\frac { \sqrt { 10 } } { 3 }$ D) $\frac { 2 \sqrt { 2 } } { 5 }$ E) $\frac { \sqrt { 10 } } { 5 }$

A rectangle ABCD with short side 12 units and long side 18 units is folded along AL and KC such that $| \mathrm { KB } | = | \mathrm { LD } | = 4$ units. Then, with M and N being the midpoints of the sides they are on, this resulting shape is folded again along the line MN as shown to form a trapezoid.
Accordingly, what is the area of this trapezoid in square units?
A) 108 B) 105 C) 102 D) 99 E) 96

$$6 | \mathrm { AB } | = 3 | \mathrm { BC } | = 2 | \mathrm { CD } |$$
Above, three semicircles with diameters $[ \mathrm { AB } ] , [ \mathrm { BC } ]$ and $[ \mathrm { CD } ]$ with collinear centers are drawn inside a semicircle with diameter [AD], and the region between them is painted as shown in the figure.
If the perimeter of the painted region is $\mathbf { 24 \pi }$ units, what is its area in square units?
A) $44 \pi$ B) $48 \pi$ C) $52 \pi$ D) $56 \pi$ E) $60 \pi$

A square frame made by assembling four wires of equal length and fixed to the wall with nails at its corners as shown in Figure 1 covers an area of 100 square units on the wall.
As a result of the nails on corners A and B coming loose, one side slides down to form a rhombus shape as shown in Figure 2. In this frame, the height of corners A and B from the ground has decreased by 6 units each, while the position of the other two corners has not changed.
Accordingly, by how many square units has the area covered by the frame on the wall decreased?
A) 18 B) 20 C) 26 D) 30 E) 32

Identical boards in the shape of an isosceles trapezoid are joined together as shown in the figure to form a rectangular frame with a short side of 16 cm and a long side of 26 cm on the outside.
A picture is placed inside the frame of this frame such that the entire picture is visible and completely covers the inside of the frame. Accordingly, what is the area of this picture placed in $\mathbf { c m } ^ { \mathbf { 2 } }$?
A) 336
B) 312
C) 288
D) 264
E) 240