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 applicants should turn to page 14. In this question we shall consider the function $f ( x )$ defined by
$$f ( x ) = x ^ { 2 } - 2 p x + 3$$
where $p$ is a constant.
(i) Show that the function $f ( x )$ has one stationary value in the range $0 < x < 1$ if $0 < p < 1$, and no stationary values in that range otherwise.
In the remainder of the question we shall be interested in the smallest value attained by $f ( x )$ in the range $0 \leqslant x \leqslant 1$. Of course, this value, which we shall call $m$, will depend on $p$.
(ii) Show that if $p \geqslant 1$ then $m = 4 - 2 p$.
(iii) What is the value of $m$ if $p \leqslant 0$ ?
(iv) Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.
(v)Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $- 2 \leqslant p \leqslant 2$. [Figure]

3. Let
$$f ( x ) = \left( c - \frac { 1 } { c } - x \right) \left( 4 - 3 x ^ { 2 } \right)$$
where $c$ is a positive constant and $x$ varies over the real numbers.
(i) Show that $f ( x )$ has one maximum and one minimum.
(ii) Show that the difference between the values of $f ( x )$ at its turning points is
$$\frac { 4 } { 9 } \left( c + \frac { 1 } { c } \right) ^ { 3 }$$
(iii) What is the least value that the difference in (ii) can have for $c > 0$ ?

1. The absolute value $| x |$ of a real number $x$ is defined by

$$| x | = \begin{cases} x & \text { if } x \geqslant 0 \\ - x & \text { if } x < 0 \end{cases}$$
(So, for example, $| 3 | = 3$ and $| - 5 | = 5$.)
(i) Sketch the locus of points ( $x , y$ ) in the first quadrant ( $x \geqslant 0 , y \geqslant 0$ ) of the $x - y$ plane that satisfy the equation $x + y = 1$.
(ii) Hence, sketch the set of points $( x , y )$ in the whole $x - y$ plane that satisfy the equation $| x | + | y | = 1$.
(iii) Give the equations of the straight lines that contain the sides of the figure that you have drawn in (ii).
(iv) If $| x | + | y | = 1$, what are the maximum and minimum values of $\sqrt { x ^ { 2 } + y ^ { 2 } }$ ? At which points $( x , y )$ are they attained?
[Figure]
(i)
[Figure]
(ii)

1. The game of Hexaglide is played on a board with $3 \times 3$ squares; White and Black each have 3 pieces, and they begin as shown in the first diagram. White moves first, and the players take turns to move one of their pieces forwards or backwards to an adjacent empty square. (Pieces never move sideways or diagonally, and they are never captured or removed from the board.) [Figure] [Figure] [Figure]
   (i) In a practice game, White plays without any Black pieces on the board, and, from his usual starting position, reaches the position shown in the second diagram. How many different sequences of moves end with this position if White makes (a) 6 moves, (b) 7 moves, (c) 8 moves?

For parts (ii) and (iii) of this question, give 'yes' or 'no' answers in the grids opposite; for both parts, you need not show your working or explain your answers. In the real game, White and Black both play, and a player wins if he can trap his opponent's pieces so that they cannot move: the third diagram shows a win for White.
(ii) (a) Is it possible to reach the position shown in the third diagram?
(b) Is it possible to reach a position where Black has won?
(c) Can White play so as to ensure that either he wins or the game goes on forever?
(d) Can Black play so as to ensure that either he wins or the game goes on forever?
(iii) In an advanced version of the game, the board has $4 \times 4$ squares, and each player has 4 pieces. What would be the answers to the four questions in part (ii) in this case?
[Figure]
(ii)
[Figure]
(iii)

4. In this question we shall consider the function $f ( x )$ defined by
$$f ( x ) = x ^ { 2 } - 2 p x + 3$$
where $p$ is a constant.
(i) Show that the function $f ( x )$ has one stationary value in the range $0 < x < 1$ if $0 < p < 1$, and no stationary values in that range otherwise.
In the remainder of the question, we shall be interested in the smallest value attained by $f ( x )$ in the range $0 \leq x \leq 1$. Of course, this value, which we shall call $m$, will depend on $p$.
(ii) Show that if $p \geq 1$ then $m = 4 - 2 p$.
(iii) What is the value of $m$ if $p \leq 0$ ?
(iv) Obtain a formula for $m$ in terms of $p$, valid for $0 < p < 1$.
(v) Using the axes opposite, sketch the graph of $m$ as a function of $p$ in the range $- 2 \leq p \leq 2$. [Figure]
5. [Figure]
The diagram represents an array of $N$ electric lights arranged in a circle. Initially, each light may be set to be ON or OFF in an arbitrary way. After one second the settings are updated according to the following rule which determine the new state of a bulb in terms of the initial states of that bulb and the one just next to it in the clockwise direction. if initially bulb $n$ and bulb $n + 1$ are in the same state (i.e. either both OFF or both ON) then after 1 second bulb $n$ will be OFF; if initially bulb $n$ and bulb $n + 1$ are in different states (i.e. one OFF the other ON) then after 1 second bulb $n$ will be ON ; Of course, if $n = N$, we replace $n + 1$ with 1 in the above. Subsequently, the settings are updated each second by reapplying the same rule.
(i) Explain why after one second there cannot be exactly one bulb ON .
(ii) More generally, explain why after one second there cannot be an odd number of bulbs ON.
(iii) Show that the state of bulb $n$ after 2 seconds is completely determined by the initial states of bulbs $n$ and $n + 2$ (with appropriate modifications when $n = N$ or $n = N - 1 )$.
(iv) The initial states of which bulbs determine the state of bulb $n$ after 4 seconds?
(v) Show that if $N = 8$ then, irrespective of the initial settings, all bulbs will eventually be OFF. How long will this take?

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.
As shown in the diagram below: $C$ is the parabola with equation $y = x ^ { 2 } ; P$ is the point $( 0,1 ) ; Q$ is the point ( $a , a ^ { 2 }$ ) on $C ; L$ is the normal to $C$ which passes through $Q$. [Figure]
(i) Find the equation of $L$.
(ii) For what values of $a$ does $L$ pass through $P$ ?
(iii) Determine $| Q P | ^ { 2 }$ as a function of $a$, where $| Q P |$ denotes the distance from $P$ to $Q$.
(iv) Find the values of $a$ for which $| Q P |$ is smallest.
(v) Find a point $R$, in the $x y$-plane but not on $C$, such that $| R Q |$ is smallest for a unique value of $a$. Briefly justify your answer.

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 $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where the coefficients $a , b$ and $c$ are real numbers. The figure below shows a section of the graph of $y = f ( x )$. The curve has two distinct turning points; these are located at $A$ and $B$, as shown. (Note that the axes have been omitted deliberately.) [Figure]
(i) Find a condition on the coefficients $a , b , c$ such that the curve has two distinct turning points if, and only if, this condition is satisfied.
It may be assumed from now on that the condition on the coefficients in (i) is satisfied.
(ii) Let $x _ { 1 }$ and $x _ { 2 }$ denote the $x$ coordinates of $A$ and $B$, respectively. Show that
$$x _ { 2 } - x _ { 1 } = \frac { 2 } { 3 } \sqrt { a ^ { 2 } - 3 b }$$
(iii) Suppose now that the graph of $y = f ( x )$ is translated so that the turning point at $A$ now lies at the origin. Let $g ( x )$ be the cubic function such that $y = g ( x )$ has the translated graph. Show that
$$g ( x ) = x ^ { 2 } \left( x - \sqrt { a ^ { 2 } - 3 b } \right)$$
(iv) Let $R$ be the area of the region enclosed by the $x$-axis and the graph $y = g ( x )$. Show that if $a$ and $b$ are rational then $R$ is also rational.
(v) Is it possible for $R$ to be a non-zero rational number when $a$ and $b$ are both irrational? Justify your answer.

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.
The degree of a polynomial is the highest exponent that appears among its terms. For example, $2 x ^ { 6 } - 3 x ^ { 2 } + 1$ is a polynomial of degree 6 .
(i) A polynomial $p ( x )$ has a turning point at ( 0,0 ). Explain why $p ( 0 ) = 0$ and why $p ^ { \prime } ( 0 ) = 0$, and explain why there is a polynomial $q ( x )$ such that
$$p ( x ) = x ^ { 2 } q ( x ) .$$
(ii) A polynomial $r ( x )$ has a turning point at ( $a , 0$ ) for some real number $a$. Write down an expression for $r ( x )$ that is of a similar form to the expression (*) above. Justify your answer in terms of a transformation of a graph.
(iii) You are now given that $f ( x )$ is a polynomial of degree 4 , and that it has two turning points at $( a , 0 )$ and at $( - a , 0 )$ for some positive number $a$.
(a) Write down the most general possible expression for $f ( x )$. Justify your answer.
(b) Describe a symmetry of the graph of $f ( x )$, and prove algebraically that $f ( x )$ does have this symmetry.
(c) Write down the $x$-coordinate of the third turning point of $f ( x )$.
(iv) Is there a polynomial of degree 4 which has turning points at $( 0,0 )$, at $( 1,3 )$, and at $( 2,0 )$ ? Justify your answer.
(v) Is there a polynomial of degree 4 which has turning points at ( 1,6 ), at ( 2,3 ), and at $( 4,6 )$ ? Justify your answer.
This page has been intentionally left blank

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.
(i) Sketch the graph of $y = \sqrt { x } - \frac { x } { 4 }$ for $x \geqslant 0$, and find the coordinates of the turning point.
(ii) Describe in words how the graph of $y = \sqrt { 4 x + 1 } - x - 1$ for $x \geqslant - \frac { 1 } { 4 }$ is related to the graph that you sketched in part (i). Write down the coordinates of the turning point of this new graph.
Point $A$ is at $( - 1,0 )$ and point $B$ is at $( 1,0 )$. Curve $C$ is defined to be all points $P$ that satisfy the equation $| A P | \times | B P | = 1$, that is; the distance from $P$ to $A$, multiplied by the distance from $P$ to $B$, is 1 .
(iii) Find all points that lie on both the $x$-axis and also on the curve $C$.
(iv) Find an equation in the form $y = f ( x )$ for the part of the curve $C$ in the region where $x > 0$ and $y > 0$. You should explicitly determine the function $f ( x )$.
(v) Use part (ii) to determine the coordinates of any turning points of the curve $C$ in the region where $x > 0$ and $y > 0$.
(vi) Sketch the curve $C$, including any parts of the curve with $x < 0$ or $y < 0$ or both.
This page has been intentionally left blank

a) (1.5 points) In a laboratory experiment, 5 measurements of the same object have been made, which gave the following results: $\mathrm { m } _ { 1 } = 0.92 ; \mathrm { m } _ { 2 } = 0.94 ; \mathrm { m } _ { 3 } = 0.89 ; \mathrm { m } _ { 4 } = 0.90$; $\mathrm { m } _ { 5 } = 0.91$.
The result will be taken as the value of x such that the sum of the squares of the errors is minimized. That is, the value for which the function
$E ( x ) = \left( x - m _ { 1 } \right) ^ { 2 } + \left( x - m _ { 2 } \right) ^ { 2 } + \left( x - m _ { 3 } \right) ^ { 2 } + \left( x - m _ { 4 } \right) ^ { 2 } + \left( x - m _ { 5 } \right) ^ { 2 }$ reaches its minimum.
Calculate this value x .
b) (1 point) Apply the integration by parts method to calculate the integral $\int _ { 1 } ^ { 2 } x ^ { 2 } \ln ( x ) d x$, where ln denotes the natural logarithm.

Given the function $f ( x ) = \sqrt [ 3 ] { \left( x ^ { 2 } - 1 \right) ^ { 2 } }$, find:\ a) ( 0.25 points) Study whether it is even or odd.\ b) ( 0.75 points) Study its differentiability at the point $x = 1$.\ c) (1.5 points) Study its relative and absolute extrema.

A team of engineers conducts fuel consumption tests for a new hybrid vehicle. The fuel consumption in liters per 100 kilometers as a function of speed, measured in tens of kilometers per hour, is
$$c ( v ) = \left\{ \begin{array} { l l l } \frac { 5 v } { 3 } & \text { if } & 0 \leq v < 3 \\ 14 - 4 v + \frac { v ^ { 2 } } { 3 } & \text { if } & v \geq 3 \end{array} \right.$$
a) (1 point) If in a first test the vehicle must travel at more than 3 tens of kilometers per hour, at what speed should the vehicle travel to obtain minimum fuel consumption?\ b) (1.5 points) If in another test the vehicle must travel at a speed $v$ such that $1 \leq v \leq 8$, what will be the maximum and minimum possible fuel consumption of the vehicle?

Given the real function of a real variable $f ( x ) = x - \frac { 4 } { ( x - 1 ) ^ { 2 } }$, it is requested:
a) ( 0.75 points) Find the domain of definition of $f ( x )$ and determine, if they exist, the equations of the asymptotes of its graph.
b) (1 point) Determine the relative extrema of the function, as well as its intervals of increase and decrease.
c) ( 0.75 points) Calculate the equation of a tangent line to the graph of $f ( x )$ that is parallel to the line with equation $9 x - 8 y = 6$.

3. The temperature function for a certain desert region during a certain period is $f(t) = -t^2 + 10t + 11$, where $1 \leq t \leq 10$. The maximum temperature difference in this region during this period is
(1) 9
(2) 16
(3) 20
(4) 25
(5) 36

Given that a real-coefficient quadratic polynomial function $f ( x )$ satisfies $f ( - 1 ) = k , f ( 1 ) = 9 k , f ( 3 ) = - 15 k$, where $k > 0$. Let the $x$-coordinate of the vertex of the graph of $y = f ( x )$ be $a$. Select the correct option.
(1) $a \leq - 1$
(2) $- 1 < a < 1$
(3) $a = 1$
(4) $1 < a < 3$
(5) $3 \leq a$

Let $F ( x )$ be a polynomial with real coefficients and $F ^ { \prime } ( x ) = f ( x )$ . It is known that $f ^ { \prime } ( x ) > x ^ { 2 } + 1.1$ holds for all real numbers $x$. Select the correct options.
(1) $f ^ { \prime } ( x )$ is an increasing function
(2) $f ( x )$ is an increasing function
(3) $F ( x )$ is an increasing function
(4) $[ f ( x ) ] ^ { 2 }$ is an increasing function
(5) $f ( f ( x ) )$ is an increasing function

A real coefficient polynomial $f(x)$ has degree greater than 5, and its leading coefficient is positive. Moreover, $f(x)$ has local minima at $x = 1, 2, 4$ and local maxima at $x = 3, 5$. Based on the above, select the correct options.
(1) $f(1) < f(3)$
(2) There exist real numbers $a, b$ satisfying $1 < a < b < 2$ such that $f'(a) > 0$ and $f'(b) < 0$
(3) $f''(3) > 0$
(4) There exists a real number $c > 5$ such that $f'(c) > 0$
(5) The degree of $f(x)$ is greater than 7

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Regarding the statements about $f ^ { \prime \prime } ( - 3 )$ and $f ^ { \prime \prime } ( 1 )$, select the correct option. (Single choice)
(1) $f ^ { \prime \prime } ( - 3 ) = f ^ { \prime \prime } ( 1 ) = 0$
(2) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(3) $f ^ { \prime \prime } ( - 3 ) > 0$ and $f ^ { \prime \prime } ( 1 ) < 0$
(4) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) > 0$
(5) $f ^ { \prime \prime } ( - 3 ) < 0$ and $f ^ { \prime \prime } ( 1 ) < 0$

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$.
Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, find $f ^ { \prime } ( x )$.

A right circular cylinder is contained within a sphere of radius 5 cm in such a way that the whole of the circumferences of both ends of the cylinder are in contact with the sphere.
The diagram shows a planar cross section through the centre of the sphere and cylinder.
Find, in cubic centimetres, the maximum possible volume of the cylinder.
A $250 \pi$ B $500 \pi$ C $1000 \pi$ D $\frac { 250 \sqrt { 3 } } { 3 } \pi$ E $\frac { 500 \sqrt { 3 } } { 9 } \pi$ F $\frac { 1000 \sqrt { 3 } } { 9 } \pi$

The function $\frac { 1 - x } { \sqrt [ 3 ] { x ^ { 2 } } }$ is defined for all $x \neq 0$. The complete set of values of $x$ for which the function is decreasing is
A $x \leq - 2 , x > 0$ B $- 2 \leq x < 0$ C $x \leq 1 , x \neq 0$ D $x \geq 1$ E $- 2 \leq x \leq 1 , \quad x \neq 0$ F $x \leq - 2 , x \geq 1$

A curve $C$ has equation $y = f ( x )$ where
$$f ( x ) = p ^ { 3 } - 6 p ^ { 2 } x + 3 p x ^ { 2 } - x ^ { 3 }$$
and $p$ is real.
The gradient of the normal to the curve $C$ at the point where $x = - 1$ is $M$.
What is the greatest possible value of $M$ as $p$ varies?
A $- \frac { 3 } { 2 }$
B $- \frac { 2 } { 3 }$
C $- \frac { 1 } { 2 }$
D $\frac { 1 } { 4 }$
E $\frac { 2 } { 3 }$
F $\frac { 3 } { 2 }$

The functions $f$ and $g$ are given by $f ( x ) = 3 x ^ { 2 } + 12 x + 4$ and $g ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 9 x - 8$.
What is the complete set of values of $x$ for which one of the functions is increasing and the other decreasing?
A $x \geq - 1$
B $x \leq - 1$
C $\quad - 3 \leq x \leq - 2 , x \geq - 1$
D $x \leq - 2 , x \geq - 1$
E $\quad x \leq - 3 , - 2 \leq x \leq - 1$
F $x \leq - 3 , x \geq - 2$
G $\quad - 2 \leq x \leq - 1$

The function $\mathrm { f } ( x )$ has derivative $\mathrm { f } ^ { \prime } ( x )$.
The diagram below shows the graph of $y = f ^ { \prime } ( x )$.
Which point corresponds to a local minimum of $\mathrm { f } ( x )$ ?
[Figure]

The curve $C$ has equation $y = x ^ { 2 } + b x + 2$, where $b \geq 0$.
Find the value of $b$ that minimises the distance between the origin and the stationary point of the curve $C$.
A $\quad b = 0$
B $b = 1$
C $b = 2$
D $b = \frac { \sqrt { 6 } } { 2 }$
E $\quad b = \sqrt { 2 }$
F $\quad b = \sqrt { 6 }$

A curve has equation
$$y = (2q - x^2)(2qx + 3)$$
The gradient of the curve at $x = -1$ is a function of $q$. Find the value of $q$ which minimises the gradient of the curve at $x = -1$.

Find the maximum value of
$$4^{\sin x} - 4 \times 2^{\sin x} + \frac{17}{4}$$
for real $x$.