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
2008 mat_2008.pdf

6 maths questions

Q2 Sequences and series, recurrence and convergence Direct term computation from recurrence View
2. For ALL APPLICANTS.
(i) Find a pair of positive integers, $x _ { 1 }$ and $y _ { 1 }$, that solve the equation
$$\left( x _ { 1 } \right) ^ { 2 } - 2 \left( y _ { 1 } \right) ^ { 2 } = 1$$
(ii) Given integers $a , b$, we define two sequences $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ and $y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots$ by setting
$$x _ { n + 1 } = 3 x _ { n } + 4 y _ { n } , \quad y _ { n + 1 } = a x _ { n } + b y _ { n } , \quad \text { for } n \geqslant 1$$
Find positive values for $a , b$ such that
$$\left( x _ { n + 1 } \right) ^ { 2 } - 2 \left( y _ { n + 1 } \right) ^ { 2 } = \left( x _ { n } \right) ^ { 2 } - 2 \left( y _ { n } \right) ^ { 2 } .$$
(iii) Find a pair of integers $X , Y$ which satisfy $X ^ { 2 } - 2 Y ^ { 2 } = 1$ such that $X > Y > 50$.
(iv) (Using the values of $a$ and $b$ found in part (ii)) what is the approximate value of $x _ { n } / y _ { n }$ as $n$ increases?
Q3 Function Transformations 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 applicants should turn to page 14.
(i) The graph $y = f ( x )$ of a certain function has been plotted below. [Figure]
On the next three pairs of axes (A), (B), (C) are graphs of
$$y = f ( - x ) , \quad f ( x - 1 ) , \quad - f ( x )$$
in some order. Say which axes correspond to which graphs. [Figure]
(A) [Figure]
(B) [Figure]
(C)
(ii) Sketch, on the axes opposite, graphs of both of the following functions
$$y = 2 ^ { - x ^ { 2 } } \quad \text { and } \quad y = 2 ^ { 2 x - x ^ { 2 } }$$
Carefully label any stationary points.
(iii) Let $c$ be a real number and define the following integral
$$I ( c ) = \int _ { 0 } ^ { 1 } 2 ^ { - ( x - c ) ^ { 2 } } \mathrm {~d} x$$
State the value(s) of $c$ for which $I ( c )$ is largest. Briefly explain your reasoning. [Note you are not being asked to calculate this maximum value.] [Figure]
Q4 Circles Inscribed/Circumscribed Circle Computations View
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
$$\frac { \text { area of circle } C } { \text { area of triangle } O P Q } = 2 \pi$$
Q5 Combinations & Selection View
5. For ALL APPLICANTS.
The Millennium school has 1000 students and 1000 student lockers. The lockers are in a line in a long corridor and are numbered from 1 to 1000.
Initially all the lockers are closed (but unlocked). The first student walks along the corridor and opens every locker. The second student then walks along the corridor and closes every second locker, i.e. closes lockers 2, 4, 6, etc. At that point there are 500 lockers that are open and 500 that are closed.
The third student then walks along the corridor, changing the state of every third locker. Thus s/he closes locker 3 (which had been left open by the first student), opens locker 6 (closed by the second student), closes locker 9 , etc.
All the remaining students now walk by in order, with the $k$ th student changing the state of every $k$ th locker, and this continues until all 1000 students have walked along the corridor.
(i) How many lockers are closed immediately after the third student has walked along the corridor? Explain your reasoning.
(ii) How many lockers are closed immediately after the fourth student has walked along the corridor? Explain your reasoning.
(iii) At the end (after all 1000 students have passed), what is the state of locker 100 ? Explain your reasoning.
(iv) After the hundredth student has walked along the corridor, what is the state of locker 1000 ? Explain your reasoning.
Q6 Proof View
6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.
(i) $\mathrm { A } , \mathrm { B }$ and C are three people. One of them is a liar who always tells lies, another is a saint who always tells the truth, and the third is a switcher who sometimes tells the truth and sometimes lies. They make the following statements:
A: I am the liar.
B : A is the liar. C: I am not the liar. Who is the liar among $\mathrm { A } , \mathrm { B }$ and C ? Briefly explain your answer.
(ii) P , Q and R are three more people, one a liar, one a saint, and the third a contrarian who tells a lie if he is the first to speak or if the preceding speaker told the truth, but otherwise tells the truth. They make the following statements: $\mathrm { P } : \mathrm { Q }$ is the liar. $\mathrm { Q } : \mathrm { R }$ is the liar. $\mathrm { R } : \mathrm { P }$ is the liar. Who is the liar among P, Q and R? Briefly explain your answer.
(iii) $\mathrm { X } , \mathrm { Y }$ and Z are three more people, one a liar, one a switcher and one contrarian. They make the following statements: $\mathrm { X } : \mathrm { Y }$ is the liar. $\mathrm { Y } : \mathrm { Z }$ is the liar. $\mathrm { Z } : \mathrm { X }$ is the liar. $\mathrm { X } : \mathrm { Y }$ is the liar. $\mathrm { Y } : \mathrm { X }$ is the liar. Who is the liar among $\mathrm { X } , \mathrm { Y }$ and Z ? Briefly explain your answer.
Q7 Proof View
7. For APPLICANTS IN COMPUTER SCIENCE ONLY.
Ox-words are sequences of letters $a$ and $b$ that are constructed according to the following rules: I. The sequence consisting of no letters is an Ox-word. II. If the sequence $W$ is an Ox-word, then the sequence that begins with $a$, followed by $W$ and ending in $b$, written $a W b$, is an Ox-word. III. If the sequences $U$ and $V$ are Ox-words, then the sequence $U$ followed by $V$, written $U V$, is an Ox-word.
All Ox-words are constructed using these rules. The length of an Ox-word is the number of letters that occur in it. For example $a a b b$ and $a b a b$ are Ox-words of length 4.
(i) Show that every Ox -word has an even length.
(ii) List all Ox-words of length 6 .
(iii) Let $W$ be an Ox-word. Is the number of occurrences of $a$ in $W$ necessarily equal to the number of occurrences of $b$ in $W$ ? Justify your answer.
You may now assume that every Ox-word (of positive length) can be written uniquely in the form $a W b W ^ { \prime }$ where $W$ and $W ^ { \prime }$ are Ox-words.
(iv) For $n \geqslant 0$, let $C _ { n }$ be the number of Ox-words of length $2 n$. Find an expression for $C _ { n + 1 }$ in terms of $C _ { 0 } , C _ { 1 } , \cdots , C _ { n }$. Explain your reasoning.