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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
2009 mat_2009.pdf

6 maths questions

Q2 Sequences and series, recurrence and convergence Closed-form expression derivation View
2. For ALL APPLICANTS.
A list of real numbers $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is defined by $x _ { 1 } = 1 , x _ { 2 } = 3$ and then for $n \geqslant 3$ by
$$x _ { n } = 2 x _ { n - 1 } - x _ { n - 2 } + 1$$
So, for example,
$$x _ { 3 } = 2 x _ { 2 } - x _ { 1 } + 1 = 2 \times 3 - 1 + 1 = 6$$
(i) Find the values of $x _ { 4 }$ and $x _ { 5 }$.
(ii) Find values of real constants $A , B , C$ such that for $n = 1,2,3$,
$$x _ { n } = A + B n + C n ^ { 2 }$$
(iii) Assuming that equation ( $*$ ) holds true for all $n \geqslant 1$, find the smallest $n$ such that $x _ { n } \geqslant 800$.
(iv) A second list of real numbers $y _ { 1 } , y _ { 2 } , y _ { 3 } , \ldots$ is defined by $y _ { 1 } = 1$ and
$$y _ { n } = y _ { n - 1 } + 2 n$$
Find, explaining your reasoning, a formula for $y _ { n }$ which holds for $n \geqslant 2$. What is the approximate value of $x _ { n } / y _ { n }$ for large values of $n$ ?
Q3 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences 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. For a positive whole number $n$, the function $f _ { n } ( x )$ is defined by
$$f _ { n } ( x ) = \left( x ^ { 2 n - 1 } - 1 \right) ^ { 2 } .$$
(i) On the axes provided opposite, sketch the graph of $y = f _ { 2 } ( x )$ labelling where the graph meets the axes.
(ii) On the same axes sketch the graph of $y = f _ { n } ( x )$ where $n$ is a large positive integer.
(iii) Determine
$$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$
(iv) The positive constants $A$ and $B$ are such that
$$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x \leqslant 1 - \frac { A } { n + B } \text { for all } n \geqslant 1 .$$
Show that
$$( 3 n - 1 ) ( n + B ) \geqslant A ( 4 n - 1 ) n ,$$
and explain why $A \leqslant 3 / 4$.
(v) When $A = 3 / 4$, what is the smallest possible value of $B$ ? [Figure]
Q4 Stationary points and optimisation Geometric or applied optimisation problem 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.
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.
Q5 Combinations & Selection View
5. For ALL APPLICANTS.
Given an $n \times n$ grid of squares, where $n > 1$, a tour is a path drawn within the grid such that:
  • along its way the path moves, horizontally or vertically, from the centre of one square to the centre of an adjacent square;
  • the path starts and finishes in the same square;
  • the path visits the centre of every other square just once.

For example, below is a tour drawn in a $6 \times 6$ grid of squares which starts and finishes in the top-left square. [Figure]
For parts (i)-(iv) it is assumed that $n$ is even.
(i) With the aid of a diagram, show how a tour, which starts and finishes in the top-left square, can be drawn in any $n \times n$ grid.
(ii) Is a tour still possible if the start/finish point is changed to the centre of a different square? Justify your answer.
Suppose now that a robot is programmed to move along a tour of an $n \times n$ grid. The robot understands two commands:
  • command $R$ which turns the robot clockwise through a right angle;
  • command $F$ which moves the robot forward to the centre of the next square.

The robot has a program, a list of commands, which it performs in the given order to complete a tour; say that, in total, command $R$ appears $r$ times in the program and command $F$ appears $f$ times.
(iii) Initially the robot is in the top-left square pointing to the right. Assuming the first command is an $F$, what is the value of $f$ ? Explain also why $r + 1$ is a multiple of 4 .
(iv) Must the results of part (iii) still hold if the robot starts and finishes at the centre of a different square? Explain your reasoning.
(v) Show that a tour of an $n \times n$ grid is not possible when $n$ is odd.
Q6 Proof View
6. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.
(i) Alice, Bob, and Charlie make the following statements:
Alice: Bob is lying. Bob: Charlie is lying. Charlie: $1 + 1 = 2$. Who is telling the truth? Who is lying?. Explain your answer.
(ii) Now Alice, Bob, and Charlie make the following statements:
Alice: Bob is telling the truth. Bob: Alice is telling the truth. Charlie: Alice is lying. What are the possible numbers of people telling the truth? Explain your answer.
(iii) They now make the following statements:
Alice: Bob and Charlie are both lying. Bob: Alice is telling the truth or Charlie is lying (or both). Charlie: Alice and Bob are both telling the truth. Who is telling the truth and who is lying on this occasion? Explain your answer.
Q7 Proof View
7. For APPLICANTS IN COMPUTER SCIENCE ONLY.
Consider sequences of the letters M, X and W. Valid sequences are made up according to the rule that an M and a W can never be adjacent in the sequence. So M, XMXW, and XMMXW are examples of valid sequences, whereas the sequences MW and XWMX are not valid.
(i) Clearly, there are 3 valid sequences of length 1 . List all valid sequences of length 2 .
(ii) Let $g ( n )$ denote the number of valid sequences of length $n$. Further, let $m ( n ) , x ( n )$, $w ( n )$ denote the number of valid sequences of length $n$ that start with an M , an X , a W respectively.
Explain why
$$\begin{aligned} m ( n ) & = w ( n ) , \\ m ( n ) & = m ( n - 1 ) + x ( n - 1 ) \quad \text { for } n > 1 , \\ x ( n ) & = 2 m ( n - 1 ) + x ( n - 1 ) \quad \text { for } n > 1 , \end{aligned}$$
and write down a formula for $g ( n )$ in terms of $m ( n )$ and $x ( n )$. Hence compute $g ( 3 )$, and verify that $g ( 4 ) = 41$.
(iii) Given a sequence using these letters then we say that it is reflexive if the following operation on the sequence does not change it: reverse the letters in the sequence, and then replace each occurrence of M by W and vice versa. So MXW, WXXM and XWXMX are reflexive strings, but MXM and XMXX are not. Let $r ( n )$ be the number of valid, reflexive sequences of length $n$.
If a sequence is reflexive and has odd length, what must the middle letter be? Explain your answer.
Hence, show that
$$r ( n ) = \left\{ \begin{array} { c l } x \left( \frac { n + 1 } { 2 } \right) & \text { if } n \text { is odd } \\ x \left( \frac { n } { 2 } \right) & \text { if } n \text { is even. } \end{array} \right.$$