Let $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is
(1) 60
(2) 53
(3) 62
(4) 30

Let the parabola $y = x ^ { 2 } + \mathrm { p } x - 3$, meet the coordinate axes at the points $\mathrm { P } , \mathrm { Q }$ and R. If the circle C with centre at $( - 1 , - 1 )$ passes through the points $P , Q$ and $R$, then the area of $\triangle P Q R$ is:
(1) 7
(2) 4
(3) 6
(4) 5

Q66. A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n ^ { 2 }$ is equal to
(1) 408
(2) 414
(3) 396
(4) 312

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
$$\tan\alpha = \mathbf{I} - \sqrt{\mathbf{J}}, \quad \tan\beta = \mathbf{K}.$$

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 $\left( \frac { \mathbf { V } } { \mathbf{U} } , \frac { \mathbf { X Y } } { \mathbf { W } } \right)$.

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
$$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } = 2 \pi$$

A circle has equation $x ^ { 2 } + y ^ { 2 } - 18 x - 22 y + 178 = 0$
A regular hexagon is drawn inside this circle so that the vertices of the hexagon touch the circle.
What is the area of the hexagon?
A 6
B $6 \sqrt { 3 }$
C 18
D $18 \sqrt { 3 }$
E 36
F $36 \sqrt { 3 }$
G 48
H $48 \sqrt { 3 }$

The diagram shows a kite $P Q R S$ whose diagonals meet at $O$.
$$\begin{aligned} & O P = x \\ & O Q = y \\ & O R = x \\ & O S = z \end{aligned}$$
Which of the following is necessary and sufficient for angle $S P Q$ to be a right angle?
A $x = y = z$
B $2 x = y + z$
C $\quad x ^ { 2 } = y z$
D $y = z$
E $y ^ { 2 } = x ^ { 2 } + z ^ { 2 }$