Let $a > 1$. We divide the region defined by the two inequalities
$$0 \leqq x \leqq \frac { \pi } { 6 } , \quad 0 \leqq y \leqq a \cos 3 x$$
into two sections by the straight line $y = 1$. Let us denote the area of the section where $y \geq 1$ by $S$ and the area of the section where $y \leq 1$ by $T$. We are to find the value of $a$ such that $T - S$ is maximized, and also find the maximum value of $T - S$.
Let $t$ denote the value of $x \left( 0 \leqq x \leqq \frac { \pi } { 6 } \right)$ satisfying the equation $a \cos 3 x = 1$. Then we have
$$\begin{aligned} S & = \frac { \sin 3 t } { \mathbf { A } \cos 3 t } - t \\ S + T & = \frac { 1 } { \mathbf { B } \cos 3 t } . \end{aligned}$$
When we set $f ( t ) = T - S$, we see that
$$f ^ { \prime } ( t ) = \frac { ( \mathbf { C } - \mathbf { D } \sin 3 t ) \sin 3 t } { \cos ^ { \mathbf { E } } 3 t } .$$
Hence $T - S$ is maximized at $t = \frac { \pi } { \mathbf { F G } }$. Thus, $T - S$ is maximized at $a = \frac { \mathbf { H } \sqrt { \mathbf { I } } } { \mathbf{J} }$, and the maximum value is $\frac { \pi } { \mathbf { K } }$.

Consider the function $f ( x ) = x \sin ^ { 2 } x$ on the interval $0 \leqq x \leqq \pi$. Let $\ell$ be the tangent to the curve $y = f ( x )$ that passes through the origin, where $\ell$ is not the $x$-axis. We are to find the area $S$ of the region bounded by the curve $y = f ( x )$ and the tangent $\ell$.
(1) For each of $\mathbf{A}$ $\sim$ $\mathbf{D}$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
When we denote the point of tangency of the curve $y = f ( x )$ and the tangent $\ell$ by $( t , f ( t ) )$, we have the equality $\mathbf{A}$, since $\ell$ passes through the origin. Further, since
$$f ^ { \prime } ( t ) = \mathbf { B } + 2 t \, \mathbf { C }$$
the $x$-coordinate of the point of tangency is $t = \mathbf { D }$.
(0) $f ( t ) = t f ^ { \prime } ( t )$ (1) $f ^ { \prime } ( t ) = t f ( t )$ (2) $\sin t$ (3) $\sin ^ { 2 } t$ (4) $\cos ^ { 2 } t$ (5) $\sin t \cos t$ (6) $\frac { \pi } { 2 }$ (7) $\frac { \pi } { 3 }$ (8) $\frac { \pi } { 4 }$ (9) $\frac { \pi } { 6 }$
(2) For each of $\mathbf { E }$ $\sim$ $\mathbf { G }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) below.
The antiderivative of the function $f ( x )$ is
$$\int f ( x ) d x = \mathbf { E } \left( 2 x ^ { 2 } - 2 x \mathbf { F } - \mathbf { G } \right) + C ,$$
where $C$ is the integral constant.
(0) $\frac { 1 } { 8 }$ (1) $\frac { 1 } { 4 }$ (2) $\frac { 1 } { 2 }$ (3) 2 (4) 4 (5) 8 (6) $\sin x$ (7) $\cos x$ (8) $\sin 2 x$ (9) $\cos 2 x$
(3) Thus, the area $S$ of the region bounded by the curve $y = f ( x )$ and the tangent $\ell$ is
$$S = \frac { \mathbf { H } } { \mathbf { I J } } \pi ^ { \mathbf { K } } - \frac { \mathbf { L } } { \mathbf{M} } .$$

Consider the following two curves
$$x ^ { 2 } + y ^ { 2 } = 1 , \tag{1}$$ $$4 x y = 1 , \tag{2}$$
where $x > 0 , y > 0$. We are to find the area $S$ of the region bounded by curve (1) and curve (2).
(1) First, let P and Q be the intersection points of curves (1) and (2), and let us denote the $x$-coordinates of P and Q by $p$ and $q$ $(p < q)$, respectively.
From (1), the coordinates $( x , y )$ of the intersection points of curves (1) and (2) can be expressed as $x = \cos \theta , y = \sin \theta \left( 0 < \theta < \frac { \pi } { 2 } \right)$. Then from (2) we have
$$\sin \mathbf { A } \theta = \frac { \mathbf { B } } { \mathbf{C} } .$$
From this we know that
$$\theta = \frac { \mathbf { D } } { \mathbf { E F } } \pi \quad \text { or } \quad \frac { \mathbf { G } } { \mathbf { H I } } \pi$$
(Write the answers in the order such that $\frac { \mathbf{D} } { \mathbf{EF} } < \frac { \mathbf{G} } { \mathbf{HI} }$.) Hence we have
$$p = \cos \frac { \mathbf { J } } { \mathbf { KL } } \pi , \quad q = \cos \frac { \mathbf { M } } { \mathbf { N } } \pi .$$
(2) Now we can find the value of $S$. Since
$$S = \int _ { p } ^ { q } \left( \sqrt { 1 - x ^ { 2 } } - \frac { 1 } { 4 x } \right) d x$$
we have to find the values of
$$I = \int _ { p } ^ { q } \sqrt { 1 - x ^ { 2 } } \, d x , \quad J = \int _ { p } ^ { q } \frac { 1 } { x } \, d x$$
For $I$, when we set $x = \cos \theta$ and calculate by substituting it for $x$ in the integral, we have
$$I = \frac { \mathbf { P } } { \mathbf { Q } }$$
For $J$, we have
$$J = \log \left( \mathbf { R } ^ { \mathbf{S} } + \sqrt { \mathbf { S } } \right) ,$$
where $\log$ is the natural logarithm. From these, we obtain
$$S = \frac { \mathbf{P} } { \mathbf{Q} } \pi - \frac { \mathbf { T } } { \mathbf{U} } \log ( \mathbf { R } + \sqrt { \mathbf { S } } ) .$$

Let $k$ be a positive real number. Consider the two curves
$$C _ { 1 } : y = \sin ^ { 2 } x , \quad C _ { 2 } : y = k \cos 2 x \quad \left( 0 \leqq x \leqq \frac { \pi } { 2 } \right)$$
Let $S _ { 1 }$ be the area of the region bounded by the two curves $C _ { 1 } , C _ { 2 }$ and the $y$-axis, and let $S _ { 2 }$ be the area of the region bounded by the two curves $C _ { 1 } , C _ { 2 }$ and the straight line $x = \frac { \pi } { 2 }$. We are to show that the value of $S _ { 2 } - S _ { 1 }$ is a constant independent of the value of $k$.
When we denote the $x$ satisfying the equation $\sin ^ { 2 } x = k \cos 2 x$ by $\alpha$, we have
$$\sin \alpha = \sqrt { \frac { k } { \mathbf { A } k + \mathbf { B } } } , \quad \cos \alpha = \sqrt { \frac { k + \mathbf { C } } { \mathbf { D } k + \mathbf { E } } } .$$
Then we have
$$\begin{aligned} & S _ { 1 } = \frac { \mathbf { F } } { \mathbf { F G } } \int _ { 0 } ^ { \alpha } \{ ( \mathbf { H } k + \mathbf { I } ) \cos \mathbf { J } x - 1 \} d x \\ & = \frac { \mathbf { K } } { \mathbf { L } } \{ \sqrt { k ( k + \mathbf { M } ) } - \alpha \} , \\ & S _ { 2 } = \frac { \mathbf { N } } { \mathbf { O } } \{ \sqrt { k ( k + \mathbf{P} ) } - \alpha \} + \frac { \pi } { \mathbf { Q } } . \end{aligned}$$
Hence, we obtain
$$S _ { 2 } - S _ { 1 } = \frac { \pi } { \mathbf { R } } ,$$
which shows that the value of $S _ { 2 } - S _ { 1 }$ is a constant independent of the value of $k$.

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 area of the region bounded by the curves $y = x ^ { 2 }$ and $y = x + 2$ equals
(a) $\frac { 7 } { 3 }$
(b) $\frac { 7 } { 2 }$
(c) $\frac { 9 } { 2 }$
(d) $\frac { 11 } { 2 }$
B. The smallest value of the function
$$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x + 3$$
in the range $0 \leq x \leq 2$ is
(a) 1
(b) 3
(c) 5
(d) 7
C. What is the reflection of the point $( 3,4 )$ in the line $3 x + 4 y = 50$ ?
(a) $( 9,12 )$
(b) $( 6,8 )$
(c) $( 12,16 )$
(d) $( 16,12 )$
D. The equation $x ^ { 3 } - 30 x ^ { 2 } + 108 x - 104 = 0$
(a) no real roots;
(b) exactly one real root;
(c) three distinct real roots;
(d) a repeated root.
E. The fact that
$$6 \times 7 = 42$$
is a counter-example to which of the following statements?
(a) the product of any two odd integers is odd;
(b) if the product of two integers is not a multiple of 4 then the integers are not consecutive;
(c) if the product of two integers is a multiple of 4 then the integers are not consecutive;
(d) any even integer can be written as the product of two even integers. F. How many values of $x$ satisfy the equation
$$2 \cos ^ { 2 } x + 5 \sin x = 4$$
in the range $0 \leqslant x < 2 \pi$ ?
(a) 2
(b) 4
(c) 6
(d) 8 G. The inequalities $x ^ { 2 } + 3 x + 2 > 0$ and $x ^ { 2 } + x < 2$, are met by all $x$ in the region:
(a) $x < - 2$;
(b) $- 1 < x < 1$;
(c) $x > - 1$;
(d) $x > - 2$. H. Given that
$$\log _ { 10 } 2 = 0.3010 \text { to } 4 \text { d.p. and that } 10 ^ { 0.2 } < 2$$
it is possible to deduce that
(a) $2 ^ { 100 }$ begins in a 1 and is 30 digits long;
(b) $2 ^ { 100 }$ begins in a 2 and is 30 digits long;
(c) $2 ^ { 100 }$ begins in a 1 and is 31 digits long;
(d) $2 ^ { 100 }$ begins in a 2 and is 31 digits long. I. The power of $x$ which has the greatest coefficient in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }$ is
(a) $x ^ { 2 }$
(b) $x ^ { 3 }$
(c) $x ^ { 5 }$
(d) $x ^ { 10 }$ J. A sketch of the curve with equation $x ^ { 2 } y ^ { 2 } ( x + y ) = 1$ is drawn in which of the following diagrams? [Figure]
(a) [Figure]
(c) [Figure]
(b) [Figure]
(d)

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 area of the region bounded by the curves $y = x ^ { 2 }$ and $y = x + 2$ equals
(a) $\frac { 7 } { 3 }$
(b) $\frac { 7 } { 2 }$
(c) $\frac { 9 } { 2 }$
(d) $\frac { 11 } { 2 }$
B. The smallest value of the function
$$f ( x ) = 2 x ^ { 3 } - 9 x ^ { 2 } + 12 x + 3$$
in the range $0 \leq x \leq 2$ is
(a) 1
(b) 3
(c) 5
(d) 7
C. What is the reflection of the point $( 3,4 )$ in the line $3 x + 4 y = 50$ ?
(a) $( 9,12 )$
(b) $( 6,8 )$
(c) $( 12,16 )$
(d) $( 16,12 )$
D. The equation $x ^ { 3 } - 30 x ^ { 2 } + 108 x - 104 = 0$
(a) no real roots;
(b) exactly one real root;
(c) three distinct real roots;
(d) a repeated root.
E. The fact that
$$6 \times 7 = 42$$
is a counter-example to which of the following statements?
(a) the product of any two odd integers is odd;
(b) if the product of two integers is not a multiple of 4 then the integers are not consecutive;
(c) if the product of two integers is a multiple of 4 then the integers are not consecutive;
(d) any even integer can be written as the product of two even integers. F. How many values of $x$ satisfy the equation
$$2 \cos ^ { 2 } x + 5 \sin x = 4$$
in the range $0 \leqslant x < 2 \pi$ ?
(a) 2
(b) 4
(c) 6
(d) 8 G. The inequalities $x ^ { 2 } + 3 x + 2 > 0$ and $x ^ { 2 } + x < 2$, are met by all $x$ in the region:
(a) $x < - 2$;
(b) $- 1 < x < 1$;
(c) $x > - 1$;
(d) $x > - 2$. H. Given that
$$\log _ { 10 } 2 = 0.3010 \text { to } 4 \text { d.p. and that } 10 ^ { 0.2 } < 2$$
it is possible to deduce that
(a) $2 ^ { 100 }$ begins in a 1 and is 30 digits long;
(b) $2 ^ { 100 }$ begins in a 2 and is 30 digits long;
(c) $2 ^ { 100 }$ begins in a 1 and is 31 digits long;
(d) $2 ^ { 100 }$ begins in a 2 and is 31 digits long. I. The power of $x$ which has the greatest coefficient in the expansion of $\left( 1 + \frac { 1 } { 2 } x \right) ^ { 10 }$ is
(a) $x ^ { 2 }$
(b) $x ^ { 3 }$
(c) $x ^ { 5 }$
(d) $x ^ { 10 }$ J. A sketch of the curve with equation $x ^ { 2 } y ^ { 2 } ( x + y ) = 1$ is drawn in which of the following diagrams? [Figure]
(a) [Figure]
(c) [Figure]
(b) [Figure]
(d)

4. (a) Find the values of
(i) $\int _ { - 1 } ^ { 1 } \left( x ^ { 2 } - x \right) \mathrm { d } x$,
(ii) $\int _ { - 1 } ^ { 1 } \left( x ^ { 3 } + x ^ { 2 } - 2 x \right) \mathrm { d } x$.
(b) Sketch the graph of $y = x ^ { 2 } - x$ and indicate which difference in areas is represented by your answer to (a)(i).
(c) Find the total area (measured positively) that lies between the graphs of $y = x ^ { 2 } - x$ and $y = x ^ { 3 } + x ^ { 2 } - 2 x$ between $x = - 1$ and $x = 1$.
(d) The answers to (a)(i) and (a)(ii) are related in a particular way. Explain how the relationship can be seen without working out any integrals.

The curves $y = \frac { 1 } { 2 } \pi \cos x$ and $x = \frac { 1 } { 2 } \pi \cos y$ intersect at the three points $\left( 0 , \frac { 1 } { 2 } \pi \right) , ( a , b )$, $\left( \frac { 1 } { 2 } \pi , 0 \right)$, as shown in the figure below.\ (a) Explain why $a = b = \frac { 1 } { 2 } \pi \cos b$.\ (b) Show that $\pi \sin b = \sqrt { \pi ^ { 2 } - 4 b ^ { 2 } }$.\ (c) Show that the area of the shaded region is
$$\sqrt { \pi ^ { 2 } - 4 b ^ { 2 } } - \frac { \pi } { 2 } - b ^ { 2 }$$

(a) Find the coordinates of the points at which the two curves $y = 6 x ^ { 2 }$ and $y = x ^ { 4 } - 16$ intersect.
(b) Give a rough sketch of the two curves (in the same diagram) for the range $- 3 \leq x \leq 3$.
(c) Find the area of the region enclosed by the two curves.

3. Let
$$f ( x ) = \left\{ \begin{array} { c c } x + 1 & \text { for } 0 \leq x \leq 1 \\ 2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leq x \leq 2 \end{array} \right.$$
(a) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leq x \leq 2$, labelling any turning points and the values attained at $x = 0,1,2$.
(b) For $1 \leq t \leq 2$, define
$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$
Express $g ( t )$ as a cubic in $t$.
(c) Calculate and factorize $g ^ { \prime } ( t )$.
(d) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leq t \leq 2$ ? [Figure]
4. [Figure] [Figure]
The triangle $A B C$, drawn above, has sides $B C , C A$ and $A B$ of length $a , b$ and $c$ respectively, and the angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$.
(a) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$.
Deduce the sine rule
$$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$
(b) In the triangle above, let $P , Q$ and $R$ respectively be the feet of the perpendiculars from $A$ to $B C , B$ to $C A$, and $C$ to $A B$, as shown.
Prove that
$$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$
For what triangles $A B C$, with angles $\alpha , \beta , \gamma$, does the equation
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$
hold?

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 14.
The graphs of $y = x ^ { 3 } - x$ and $y = m ( x - a )$ are drawn on the axes below. Here $m > 0$ and $a \leqslant - 1$.
The line $y = m ( x - a )$ meets the $x$-axis at $A = ( a , 0 )$, touches the cubic $y = x ^ { 3 } - x$ at $B$ and intersects again with the cubic at $C$. The $x$-coordinates of $B$ and $C$ are respectively $b$ and $c$. [Figure]
(i) Use the fact that the line and cubic touch when $x = b$, to show that $m = 3 b ^ { 2 } - 1$.
(ii) Show further that
$$a = \frac { 2 b ^ { 3 } } { 3 b ^ { 2 } - 1 }$$
(iii) If $a = - 10 ^ { 6 }$, what is the approximate value of $b$ ?
(iv) Using the fact that
$$x ^ { 3 } - x - m ( x - a ) = ( x - b ) ^ { 2 } ( x - c )$$
(which you need not prove), show that $c = - 2 b$.
(v) $R$ is the finite region bounded above by the line $y = m ( x - a )$ and bounded below by the cubic $y = x ^ { 3 } - x$. For what value of $a$ is the area of $R$ largest?
Show that the largest possible area of $R$ is $\frac { 27 } { 4 }$.

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 14.
Let $0 < k < 2$. Below is sketched a graph of $y = f _ { k } ( x )$ where $f _ { k } ( x ) = x ( x - k ) ( x - 2 )$. Let $A ( k )$ denote the area of the shaded region. [Figure]
(i) Without evaluating them, write down an expression for $A ( k )$ in terms of two integrals.
(ii) Explain why $A ( k )$ is a polynomial in $k$ of degree 4 or less. [You are not required to calculate $A ( k )$ explicitly.]
(iii) Verify that $f _ { k } ( 1 + t ) = - f _ { 2 - k } ( 1 - t )$ for any $t$.
(iv) How can the graph of $y = f _ { k } ( x )$ be transformed to the graph of $y = f _ { 2 - k } ( x )$ ?
Deduce that $A ( k ) = A ( 2 - k )$.
(v) Explain why there are constants $a , b , c$ such that
$$A ( k ) = a ( k - 1 ) ^ { 4 } + b ( k - 1 ) ^ { 2 } + c .$$
[You are not required to calculate $a , b , c$ explicitly.]

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.
(i) Let $a > 0$. On the axes opposite, sketch the graph of
$$y = \frac { a + x } { a - x } \quad \text { for } \quad - a < x < a .$$
(ii) Let $0 < \theta < \pi / 2$. In the diagram below is the half-disc given by $x ^ { 2 } + y ^ { 2 } \leqslant 1$ and $y \geqslant 0$. The shaded region $A$ consists of those points with $- \cos \theta \leqslant x \leqslant \sin \theta$. The region $B$ is the remainder of the half-disc.
Find the area of $A$. [Figure]
(iii) Assuming only that $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1$, show that $\sin \theta \cos \theta \leqslant 1 / 2$.
(iv) What is the largest that the ratio
$$\frac { \text { area of } A } { \text { area of } B }$$
can be, as $\theta$ varies? [Figure]

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 14.
For each positive integer $k$, let $f _ { k } ( x ) = x ^ { 1 / k }$ for $x \geqslant 0$.
(i) On the same axes (provided below), labelling each curve clearly, sketch $y = f _ { k } ( x )$ for $k = 1,2,3$, indicating the intersection points.
(ii) Between the two points of intersection in (i), the curves $y = f _ { k } ( x )$ enclose several regions. What is the area of the region between $y = f _ { k } ( x )$ and $y = f _ { k + 1 } ( x )$ ? Verify that the area of the region between $y = f _ { 1 } ( x )$ and $y = f _ { 2 } ( x )$ is $\frac { 1 } { 6 }$.
Let $c$ be a constant where $0 < c < 1$.
(iii) Find the $x$-coordinates of the points of intersection of the line $y = c$ with $y = f _ { 1 } ( x )$ and of $y = c$ with $y = f _ { 2 } ( x )$.
(iv) The constant $c$ is chosen so that the line $y = c$ divides the region between $y = f _ { 1 } ( x )$ and $y = f _ { 2 } ( x )$ into two regions of equal area. Show that $c$ satisfies the cubic equation $4 c ^ { 3 } - 6 c ^ { 2 } + 1 = 0$. Hence find $c$. [Figure]

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.
A horse is attached by a rope to the corner of a square field of side length 1 .
(i) What length of rope allows the horse to reach precisely half the area of the field?
Another horse is placed in the field, attached to the corner diagonally opposite from the first horse. Each horse has a length of rope such that each can reach half the field.
(ii) Explain why the area that both can reach is the same as the area neither can reach. [Figure]
(iii) The angle $\alpha$ is marked in the diagram above. Show that $\alpha = \cos ^ { - 1 } \left( \frac { \sqrt { \pi } } { 2 } \right)$ and hence show that the area neither can reach is $\frac { 4 } { \pi } \cos ^ { - 1 } \left( \frac { \sqrt { \pi } } { 2 } \right) - \sqrt { \frac { 4 - \pi } { \pi } }$. Note that $\cos ^ { - 1 }$ can also be written as arccos.
A third horse is placed in the field, and the three horses are rearranged. One horse is now attached to the midpoint of the bottom side of the field, and another horse is now attached to the midpoint of the left side of the field. The third horse is attached to the upper right corner.
(iv) Given each horse can access an equal area of the field and that none of the areas overlap, what length of rope must each horse have to minimise the area that no horse can reach?
The horses on the bottom and left midpoints of the field are each replaced by a goat; each goat is attached by a rope of length $g$ to the same midpoint as in part (iii). The remaining horse is attached to the upper right corner with rope length $h$.
(v) Given that $0 \leqslant h \leqslant 1$, and that none of the animals' areas can overlap, show that $\frac { \sqrt { 5 } - 2 } { 2 } \leqslant g \leqslant \frac { 1 } { 2 \sqrt { 2 } }$ holds if the area that the animals can reach is maximised.

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 14.
Let $a , b , m$ be positive numbers with $0 < a < b$. In the diagram below are sketched the parabola with equation $y = ( x - a ) ( b - x )$ and the line $y = m x$. The line is tangential to the parabola. $R$ is the region bounded by the $x$-axis, the line and the parabola. $S$ is the region bounded by the parabola and the $x$-axis. [Figure]
(i) For $c > 0$, evaluate
$$\int _ { 0 } ^ { c } x ( c - x ) \mathrm { d } x$$
Without further calculation, explain why the area of region $S$ equals $\frac { ( b - a ) ^ { 3 } } { 6 }$.
(ii) The line $y = m x$ meets the parabola tangentially as drawn in the diagram. Show that $m = ( \sqrt { b } - \sqrt { a } ) ^ { 2 }$.
(iii) Assume now that $a = 1$ and write $b = \beta ^ { 2 }$ where $\beta > 1$. Given that the area of $R$ equals $( 2 \beta + 1 ) ( \beta - 1 ) ^ { 2 } / 6$, show that the areas of regions $R$ and $S$ are equal precisely when
$$( \beta - 1 ) ^ { 2 } \left( \beta ^ { 4 } + 2 \beta ^ { 3 } - 4 \beta - 2 \right) = 0$$
Explain why there is a solution $\beta$ to ( $*$ ) in the range $\beta > 1$.
Without further calculation, deduce that for any $a > 0$ there exists $b > a$ such that the area of region $S$ equals the area of region $R$.

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 20.
Charlie is trying to cut a cake. The cake is a square with side length 2 , and its corners are at $( 0,0 ) , ( 2,0 ) , ( 2,2 )$, and $( 0,2 )$. Charlie's first cut is a straight line segment from the point $( x , y )$ to $( x , 0 )$, where $0 \leqslant x \leqslant 2$ and $0 \leqslant y \leqslant 2$.
Charlie plans to make a second straight cut from the point $( x , y )$ to a point $( 0 , k )$ somewhere on the left-hand edge of the cake. This will make a slice of cake which is bounded to the left of the first cut and bounded below the second cut. [Figure]
(i) Find the area of the slice of cake in terms of $x , y$, and $k$. Check your expression by verifying that if $x = 1$ and $y = 1$, then choosing $k = 1$ gives a slice of cake with area 1 .
(ii) Find another point ( $x , y$ ) on the cake such that choosing $k = 1$ gives a slice of cake with area 1 .
(iii) Show that it is only possible to choose a value of $k$ that gives a slice of cake with area 1 if both $x y \leqslant 2$ and $x ( 2 + y ) \geqslant 2$.
(iv) Sketch the region $R$ of the cake for which both inequalities in part (iii) hold, indicating any relevant points on the edges of the cake.
(v) Charlie may instead plan to make the second straight cut from $( x , y )$ to a point $( m , 2 )$ on the top edge of the cake in order to make a slice bounded to the left of the two cuts. Find two necessary and sufficient inequalities for $x$ and $y$ which must both hold in order for this to give a slice of area 1 for some value of $m$. Sketch the region of the cake for which both inequalities hold.
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Let the function
$$f ( x ) = x ^ { 3 } - | x | + 2$$
a) ( 0.75 points) Study the continuity and differentiability of $f$ at $x = 0$.
b) (1 point) Determine the relative extrema of $f ( x )$ on the real line.
c) ( 0.75 points) Calculate the area of the region bounded by the graph of $f$, the x-axis $y = 0$, and the lines $x = - 1$ and $x = 1$.

Let the function $f ( x ) = \frac { x } { x ^ { 2 } + 1 }$ a) ( 0.5 points) Check whether $f ( x )$ satisfies the hypotheses of Bolzano's Theorem on the interval $[ - 1,1 ]$ b) (1 point) Calculate and classify the relative extrema of $f ( x )$ in $\mathbb { R }$. c) (1 point) Determine the area between the graph of the function $f ( x )$ and the x-axis on the interval $[ - 1,1 ]$.

Let the function $f ( x ) = x \sqrt [ 3 ] { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$.
a) ( 0.75 points) Find $\lim _ { x \rightarrow 1 } \frac { f ( x ) } { ( x - 1 ) ^ { 2 / 3 } }$.
b) ( 1.75 points) Find the area, in the first quadrant, between the line $y = x$ and the graph of the function $f ( x )$.

On the coordinate plane, let $\Gamma$ be the graph of the cubic function $f(x) = x^{3} - 9x^{2} + 15x - 4$. The tangent line $L$ to $\Gamma$ at point $P(1, 3)$ was found in question 16. Continuing from 16, find the area of the bounded region enclosed by $\Gamma$ and $L$.

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Prove that for all $a \in \left[-\frac{1}{2}, 1\right]$, the area of $\Gamma$ is always 2. (Non-multiple choice question, 2 points)

What is the total area enclosed between the curve $y = x ^ { 2 } - 1$, the $x$-axis and the lines $x = - 2$ and $x = 2$ ?
A $\frac { 4 } { 3 }$ B $\frac { 8 } { 3 }$ C 4 D $\frac { 16 } { 3 }$ E 12 F 16

A line $l$ has equation $y = 6 - 2 x$
A second line is perpendicular to $l$ and passes through the point $( - 6,0 )$.
Find the area of the region enclosed by the two lines and the $x$-axis.
A $16 \frac { 1 } { 5 }$
B 18
C $21 \frac { 3 } { 5 }$
D 27
E $\quad 40 \frac { 1 } { 2 }$

The function $f ( x )$ is increasing and $f ( 0 ) = 0$.
The positive constants $a$ and $b$ are such that $a < b$.
The area of the region enclosed by the curve $y = f ( x )$, the $x$-axis and the lines $x = a$ and $x = b$ is denoted by $R$.
The function $g ( x )$ is defined by $g ( x ) = f ( x ) + 2 f ( b )$.
Which of the following is an expression for the area enclosed by the curve $y = g ( x )$, the $x$-axis and the lines $x = a$ and $x = b$ ?
A $\quad R + ( b - a ) f ( b )$
B $R + 2 ( b - a ) f ( b )$
C $\quad R + 2 f ( b ) - f ( a )$
D $R + 2 f ( b )$
E $\quad R + ( f ( b ) ) ^ { 2 }$
F $\quad R + ( f ( b ) ) ^ { 2 } - ( f ( a ) ) ^ { 2 }$
G $\quad R + 2 ( f ( b ) - f ( a ) ) f ( b )$