mat

Papers (32)
2025
mat_2025.pdf 23
2024
mat_2024.pdf 2
2023
mat_2023.pdf 5
2022
mat_2022.pdf 6
2021
mat_2021.pdf 6
2020
mat_2020.pdf 7
2019
mat_2019.pdf 6
2018
mat_2018.pdf 6
2017
mat_2017.pdf 7
2016
mat_2016.pdf 6
2015
mat_2015.pdf 6
2014
mat_2014.pdf 6
2013
mat_2013.pdf 6
2012
mat_2012.pdf 7
2011
mat_2011.pdf 7
2010
mat_2010.pdf 7
2009
mat_2009.pdf 6
2008
mat_2008.pdf 6
2007
mat_2007.pdf 6
2005
mat_2005.pdf 3
2004
mat_2004.pdf 3
2003
mat_2003.pdf 4
2002
mat_2002.pdf 3
2001
mat_2001.pdf 5
2000
mat_2000.pdf 2
1998
mat_1998.pdf 4
1997
mat_1997.pdf 5
1996
mat_1996.pdf 4
Unknown
specimen_1 7 specimen_2 7 specimen_a 7 specimen_b 7
2015 mat_2015.pdf

6 maths questions

Q2 Proof Proof Involving Combinatorial or Number-Theoretic Structure View
2. For ALL APPLICANTS.
(i) Expand and simplify
$$( a - b ) \left( a ^ { n } + a ^ { n - 1 } b + a ^ { n - 2 } b ^ { 2 } + \cdots + a b ^ { n - 1 } + b ^ { n } \right)$$
(ii) The prime number 3 has the property that it is one less than a square number. Are there any other prime numbers with this property? Justify your answer.
(iii) Find all the prime numbers that are one more than a cube number. Justify your answer.
(iv) Is $3 ^ { 2015 } - 2 ^ { 2015 }$ a prime number? Explain your reasoning carefully.
(v) Is there a positive integer $k$ for which $k ^ { 3 } + 2 k ^ { 2 } + 2 k + 1$ is a cube number? Explain your reasoning carefully.
If you require additional space please use the pages at the end of the booklet
Q3 Integration by Substitution Substitution within a Multi-Part Proof or Derivation View
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 16.
In this question we shall investigate when functions are close approximations to each other. We define $| x |$ to be equal to $x$ if $x \geqslant 0$ and to $- x$ if $x < 0$. With this notation we say that a function $f$ is an excellent approximation to a function $g$ if
$$| f ( x ) - g ( x ) | \leqslant \frac { 1 } { 320 } \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
we say that $f$ is a good approximation to a function $g$ if
$$| f ( x ) - g ( x ) | \leqslant \frac { 1 } { 100 } \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
For example, any function $f$ is an excellent approximation to itself. If $f$ is an excellent approximation to $g$ then $f$ is certainly a good approximation to $g$, but the converse need not hold.
(i) Give an example of two functions $f$ and $g$ such that $f$ is a good approximation to $g$ but $f$ is not an excellent approximation to $g$.
(ii) Show that if
$$f ( x ) = x \quad \text { and } \quad g ( x ) = x + \frac { \sin \left( 4 x ^ { 2 } \right) } { 400 }$$
then $f$ is an excellent approximation to $g$. For the remainder of the question we are going to a try to find a good approximation to the exponential function. This function, which we shall call $h$, satisfies the following equation
$$h ( x ) = 1 + \int _ { 0 } ^ { x } h ( t ) \mathrm { d } t \quad \text { whenever } \quad x \geqslant 0$$
You may not use any other properties of the exponential function during this question, and any attempt to do so will receive no marks.
Let
$$f ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 6 }$$
(iii) Show that if
$$g ( x ) = 1 + \int _ { 0 } ^ { x } f ( t ) \mathrm { d } t$$
then $f$ is an excellent approximation to $g$.
(iv) Show that for $x \geqslant 0$
$$h ( x ) - f ( x ) = g ( x ) - f ( x ) + \int _ { 0 } ^ { x } ( h ( t ) - f ( t ) ) \mathrm { d } t$$
(v) You are given that $h ( x ) - f ( x )$ has a maximum value on the interval $0 \leqslant x \leqslant 1 / 2$ at $x = x _ { 0 }$. Explain why
$$\int _ { 0 } ^ { x } ( h ( t ) - f ( t ) ) \mathrm { d } t \leqslant \frac { 1 } { 2 } \left( h \left( x _ { 0 } \right) - f \left( x _ { 0 } \right) \right) \quad \text { whenever } \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
(vi) You are also given that $f ( x ) \leqslant h ( x )$ for all $0 \leqslant x \leqslant \frac { 1 } { 2 }$. Show that $f$ is a good approximation to $h$ when $0 \leqslant x \leqslant \frac { 1 } { 2 }$.
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If you require additional space please use the pages at the end of the booklet
Q4 Circles Circle Equation Derivation View
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 16.
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 $\tan ( \theta ) = x$.
If you require additional space please use the pages at the end of the booklet
Q5 Proof View
5. For ALL APPLICANTS.
The following functions are defined for all integers $a , b$ and $c$ :
$$\begin{aligned} p ( x ) & = x + 1 \\ m ( x ) & = x - 1 \\ s ( x , y , z ) & = \begin{cases} y & \text { if } x \leqslant 0 \\ z & \text { if } x > 0 \end{cases} \end{aligned}$$
(i) Show that the value of
$$s ( s ( p ( 0 ) , m ( 0 ) , m ( m ( 0 ) ) ) , s ( p ( 0 ) , m ( 0 ) , p ( p ( 0 ) ) ) , s ( m ( 0 ) , p ( 0 ) , m ( p ( 0 ) ) ) )$$
is 2 .
Let $f$ be a function defined, for all integers $a$ and $b$, as follows:
$$f ( a , b ) = s ( b , p ( a ) , p ( f ( a , m ( b ) ) ) ) .$$
(ii) What is the value of $f ( 5,2 )$ ?
(iii) Give a simple formula for the value of $f ( a , b )$ for all integers $a$ and all positive integers $b$, and explain why this formula holds.
(iv) Define a function $g ( a , b )$ in a similar way to $f$, using only the functions $p , m$ and $s$, so that the value of $g ( a , b )$ is equal to the sum of $a$ and $b$ for all integers $a$ and all integers $b \leqslant 0$. Explain briefly why your function gives the correct value for all such values of $a$ and $b$.
If you require additional space please use the pages at the end of the booklet
Q6 Proof View
6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.
The world is divided into two species, vampires and werewolves. Vampires always tell the truth when talking about a vampire, but always lie when talking about a werewolf. Werewolves always tell the truth when talking about a werewolf, but always lie when talking about a vampire. (Note that this does not imply that creatures necessarily lie when speaking to creatures of the other species. Note also that "Zaccaria is a vampire" is a statement about Zaccaria, rather than necessarily about a vampire.)
These facts are well known to both sides, and creatures can tell instinctively which species an individual belongs to.
In your answers to the questions below, you may abbreviate "vampire" and "werewolf" to "V" and "W", respectively.
(i) Azrael says, "Beela is a werewolf." Explain why Azrael must be a werewolf, but that we cannot tell anything about Beela.
(ii) Cesare says, "Dita says 'Elith is a vampire.'" What can we infer about any of the three from this statement? Explain your answer.
(iii) Suppose $N$ creatures (where $N \geq 2$ ) are sitting around a circular table. Each tells their right-hand neighbour, "You lie about your right-hand neighbour." What can we infer about $N$ ? What can we infer about the arrangement of creatures around the table? Explain your answer.
(iv) Consider a similar situation to that in part (iii) (possibly for a different value of $N$ ), except that now each tells their right-hand neighbour, "Your right-hand neighbour lies about their right-hand neighbour." Again, what can we infer about $N$ and the arrangement of creatures around the table? Explain your answer.
If you require additional space please use the pages at the end of the booklet
Q7 Proof View
7. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.
In this question we will study a mechanism for producing a set of words. We will only consider words containing the letters $a$ and/or $b$, and that have length at least 1 . We will make use of variables, which we shall write as capital letters, including a special start variable called $S$. We will also use rules, which show how a variable can be replaced by a sequence of variables and/or letters. Starting with the start variable $S$, we repeatedly replace one of the variables according to one of the rules (in any order) until no variables remain.
For example suppose the rules are
$$S \rightarrow A B , \quad A \rightarrow A A , \quad A \rightarrow a , \quad B \rightarrow b b .$$
We can produce the word $a a b b$ as follows; at each point, the variable that is replaced is underlined:
$$\underline { S } \rightarrow \underline { A } B \rightarrow A \underline { A } B \rightarrow \underline { A } a B \rightarrow a a \underline { B } \rightarrow a a b b .$$
(i) Show that the above rules can be used to produce all words of the form $a ^ { n } b b$ with $n \geq 1$, where $a ^ { n }$ represents $n$ consecutive $a$ 's. Also briefly explain why the rules can be used to produce no other words.
(ii) Give a precise description of the words produced by the following rules.
$$S \rightarrow a b , \quad S \rightarrow a S b .$$
(iii) A palindrome is a word that reads the same forwards as backwards, for example $b b a a b b$. Give rules that produce all palindromes (and no other words).
(iv) Consider the words with the same number of $a$ 's as $b$ 's; for example, $a a b a b b$. Write down rules that produce these words (and no others).
(v) Suppose you are given a collection of rules that produces the words in $L _ { 1 }$, and another collection of rules that produces the words in $L _ { 2 }$. Show how to produce a single set of rules that produce all words in $L _ { 1 }$ or $L _ { 2 }$, or both (and no other words). Hint: you may introduce new variables if you want.
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