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
2014 mat_2014.pdf

6 maths questions

Q2 Roots of polynomials Multiplicity and derivative analysis of roots View
2. For ALL APPLICANTS.
Let $a$ and $b$ be real numbers. Consider the cubic equation
$$x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 } = 0$$
(i) Show that if $x = 1$ is a solution of ( $*$ ) then
$$1 - \sqrt { 2 } \leqslant b \leqslant 1 + \sqrt { 2 }$$
(ii) Show that there is no value of $b$ for which $x = 1$ is a repeated root of ( $*$ ).
(iii) Given that $x = 1$ is a solution, find the value of $b$ for which $( * )$ has a repeated root. For this value of $b$, does the cubic
$$y = x ^ { 3 } + 2 b x ^ { 2 } - a ^ { 2 } x - b ^ { 2 }$$
have a maximum or minimum at its repeated root?
Q3 Exponential Functions Functional Equation with Exponentials 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 14.
The function $f ( x )$ is defined for all real numbers and has the following properties, valid for all $x$ and $y$ :
(A) $\quad f ( x + y ) = f ( x ) f ( y )$.
(B) $\quad \mathrm { d } f / \mathrm { d } x = f ( x )$.
(C) $\quad f ( x ) > 0$.
Throughout this question, these should be the only properties of $f$ that you use; no marks will be awarded for any use of the exponential function.
Let $a = f ( 1 )$.
(i) Show that $f ( 0 ) = 1$.
(ii) Let
$$I = \int _ { 0 } ^ { 1 } f ( x ) \mathrm { d } x$$
Show that $I = a - 1$.
(iii) The trapezium rule with $n$ steps is used to produce an estimate $I _ { n }$ for the integral $I$. Show that
$$I _ { n } = \frac { 1 } { 2 n } \left( \frac { b + 1 } { b - 1 } \right) ( a - 1 )$$
where $b = f ( 1 / n )$.
(iv) Given that $I _ { n } \geqslant I$ for all $n$, show that
$$a \leqslant \left( 1 + \frac { 2 } { 2 n - 1 } \right) ^ { n }$$
Q4 Sine and Cosine Rules Multi-step composite figure 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, Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
In the diagram below is sketched a semicircle with centre $B$ and radius 1 . Three points $A , C , D$ lie on the semicircle as shown with $\alpha$ denoting angle $C A B$ and $\beta$ denoting angle $D A B$. The triangles $A B C$ and $A B D$ intersect in a triangle $A B X$.
Throughout the question we shall consider the value of $\alpha$ fixed. Assume for now that $0 < \alpha \leqslant \beta \leqslant \pi / 2$. [Figure]
(i) Show that the area of the triangle $A B C$ equals
$$\frac { 1 } { 2 } \sin ( 2 \alpha ) .$$
(ii) Let
$$F = \frac { \text { area of triangle } A B X } { \text { area of triangle } A B C }$$
Without calculation, explain why, for every $k$ in the range $0 \leqslant k \leqslant 1$, there is a unique value of $\beta$ such that $F = k$.
(iii) Find the value of $\beta$ such that $F = 1 / 2$.
(iv) Show that
$$F = \frac { \sin ( 2 \beta ) \sin \alpha } { \sin ( 2 \beta - \alpha ) \sin ( 2 \alpha ) }$$
(v) Suppose now that $0 < \beta < \alpha \leqslant \pi / 2$. Write down, without further calculation, an expression for the area of $A B X$ and hence a formula for $F$.
Q5 Permutations & Arrangements Combinatorial Proof or Identity Derivation View
5. For ALL APPLICANTS.
Poets use rhyme schemes to describe which lines of a poem rhyme. Each line is denoted by a letter of the alphabet, with the same letter given to two lines that rhyme. To say that a poem has the rhyming scheme ABABCDED , indicates that the first and third lines rhyme, the second and fourth lines rhyme, and the sixth and eighth lines rhyme, but no others.
More precisely, the first line of the poem is given the letter A . If a subsequent line rhymes with an earlier line, it is given the same letter; otherwise, it is given the first unused letter. (For the purposes of this question, you can assume that we have an infinite supply of "letters", not just the 26 letters of the alphabet.)
The purpose of this question is to investigate how many different rhyme schemes there are for poems of $n$ lines. We write $r _ { n }$ for this number.
(i) There are five different rhyming schemes for poems of three lines (so $r _ { 3 } = 5$ ). List them.
Let $c _ { n , k }$ denote the number of rhyme schemes for poems with lines $n$ that use exactly $k$ different letters. For example $c _ { 3,2 } = 3$ corresponding to the rhyming schemes AAB , ABA and ABB .
(ii) What is $c _ { n , 1 }$ for $n \geqslant 1$ ?
What is $c _ { n , n }$ ? Explain your answers.
(iii) Suppose that $1 < k < n$. By considering the final letter of a rhyming scheme, explain why
$$c _ { n , k } = k c _ { n - 1 , k } + c _ { n - 1 , k - 1 }$$
(iv) Write down an equation showing how to calculate $r _ { n }$ in terms of the $c _ { n , k }$. Hence calculate $r _ { 4 }$.
(v) Give a formula for $c _ { n , 2 }$ in terms of $n$ (for $n \geqslant 2$ ). Justify your answer.
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.
Alice plays a game 5 times with her friends Sam and Pam. In each game Alice chooses two integers $x$ and $y$ with $1 \leqslant x \leqslant y$. She whispers the sum $x + y$ to Sam, and the product $x \times y$ to Pam, so that neither knows what the other was told. Sam and Pam then have to try to work out what the numbers $x$ and $y$ are. Sam and Pam are well known expert logicians.
(i) In the first game, Pam says "I know $x$ and $y$."
What can we deduce about the values of $x$ and $y$ ? Explain your answer.
(ii) In the second game, Pam says "I don't know what $x$ and $y$ are."
Sam then says "I know $x$ and $y$." Suppose the sum is 4 . What are $x$ and $y$ ? Explain your answer.
(iii) In the third game, Pam says "I don't know what $x$ and $y$ are."
Sam then says "I don't know what $x$ and $y$ are." Pam then says "I now know $x$ and $y$." Suppose the product is 4 . What are $x$ and $y$ ? Explain your answer.
(iv) In the fourth game, Pam says "I don't know what $x$ and $y$ are."
Sam then says "I already knew that." Pam then says "I now know $x$ and $y$." Suppose the product is 8 . What are $x$ and $y$ ? Explain your answer.
(v) Finally, in the fifth game, Pam says "I don't know what $x$ and $y$ are."
Sam then says "I don't know what $x$ and $y$ are." Pam then says "I don't know what $x$ and $y$ are." Sam then says "I now know $x$ and $y$." Suppose the sum is 5 . What are $x$ and $y$ ? Explain your answer.
Q7 Proof View
7. For APPLICANTS IN $\left\{ \begin{array} { l } \text { COMPUTER SCIENCE } \\ \text { COMPUTER SCIENCE \& PHILOSOPHY } \end{array} \right\}$ ONLY.
A finite automaton is a mathematical model of a simple computing device. A small finite automaton is illustrated below. [Figure]
A finite automaton has some finite number of states; the above automaton has three states, labelled $s _ { 0 } , s _ { 1 }$ and $s _ { 2 }$. The initial state, $s _ { 0 }$, is indicated with an incoming arrow. The automaton receives inputs (e.g. via button presses), which might cause it to change state. In the example, the inputs are $a$ and $b$. The state changes are illustrated by arrows; for example, if the automaton is in state $s _ { 1 }$ and it receives input $b$, it changes to state $s _ { 0 }$; if it is in state $s _ { 2 }$ and receives either input, it remains in state $s _ { 2 }$. (For each state, there is precisely one out-going arrow for each input.)
Some of the states are defined to be accepting states; in the example, just $s _ { 1 }$ is defined to be an accepting state, represented by the double circle. A word is a sequence of inputs. The automaton accepts a word $w$ if that sequence of inputs leads to an accepting state from the initial state. For example, the above automaton accepts the word $a b a$.
(i) Write down a description of the set of words accepted by the above automaton. A clear but informal description will suffice.
(ii) Suppose we alter the above automaton by swapping accepting and non-accepting states; i.e. we make $s _ { 0 }$ and $s _ { 2 }$ accepting, and make $s _ { 1 }$ non-accepting. Write down a description of the set of words accepted by this new automaton. Again, a clear but informal description will suffice.
(iii) Draw an automaton that accepts all words containing an even number (possibly zero) of $a$ 's and any number of $b$ 's (and no other words).
(iv) Now draw an automaton that accepts all words containing an even number of $a$ 's or an odd number of $b$ 's (and no other words).
Let $a ^ { n }$ represent $n$ consecutive $a$ 's. Let $L$ be the set of all words of the form $a ^ { n } b ^ { n }$ where $n = 0,1,2 , \ldots$; i.e. all words composed of some number of $a$ 's followed by the same number of $b$ 's. We will show that no finite automaton accepts precisely this set of words.
(v) Suppose a particular finite automaton $A$ does accept precisely the words in $L$. Show that if $i \neq j$ then the words $a ^ { i }$ and $a ^ { j }$ must lead to different states of $A$. Hence show that this leads to a contradiction.
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