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
2013 mat_2013.pdf

6 maths questions

Q2 Composite & Inverse Functions Recover a Function from a Composition or Functional Equation View
2. For ALL APPLICANTS.
(i) Let $k \neq \pm 1$. The function $f ( t )$ satisfies the identity
$$f ( t ) - k f ( 1 - t ) = t$$
for all values of $t$. By replacing $t$ with $1 - t$, determine $f ( t )$.
(ii) Consider the new identity
$$f ( t ) - f ( 1 - t ) = g ( t )$$
(a) Show that no function $f ( t )$ satisfies $( * )$ when $g ( t ) = t$.
(b) What condition must the function $g ( t )$ satisfy for there to be a solution $f ( t )$ to $( * )$ ?
(c) Find a solution $f ( t )$ to $( * )$ when $g ( t ) = ( 2 t - 1 ) ^ { 3 }$.
Q3 Areas by integration 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.
Let $0 < k < 2$. Below is sketched a graph of $y = f _ { k } ( x )$ where $f _ { k } ( x ) = x ( x - k ) ( x - 2 )$. Let $A ( k )$ denote the area of the shaded region. [Figure]
(i) Without evaluating them, write down an expression for $A ( k )$ in terms of two integrals.
(ii) Explain why $A ( k )$ is a polynomial in $k$ of degree 4 or less. [You are not required to calculate $A ( k )$ explicitly.]
(iii) Verify that $f _ { k } ( 1 + t ) = - f _ { 2 - k } ( 1 - t )$ for any $t$.
(iv) How can the graph of $y = f _ { k } ( x )$ be transformed to the graph of $y = f _ { 2 - k } ( x )$ ?
Deduce that $A ( k ) = A ( 2 - k )$.
(v) Explain why there are constants $a , b , c$ such that
$$A ( k ) = a ( k - 1 ) ^ { 4 } + b ( k - 1 ) ^ { 2 } + c .$$
[You are not required to calculate $a , b , c$ explicitly.]
Q4 Areas by integration 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.
(i) Let $a > 0$. On the axes opposite, sketch the graph of
$$y = \frac { a + x } { a - x } \quad \text { for } \quad - a < x < a .$$
(ii) Let $0 < \theta < \pi / 2$. In the diagram below is the half-disc given by $x ^ { 2 } + y ^ { 2 } \leqslant 1$ and $y \geqslant 0$. The shaded region $A$ consists of those points with $- \cos \theta \leqslant x \leqslant \sin \theta$. The region $B$ is the remainder of the half-disc.
Find the area of $A$. [Figure]
(iii) Assuming only that $\sin ^ { 2 } \theta + \cos ^ { 2 } \theta = 1$, show that $\sin \theta \cos \theta \leqslant 1 / 2$.
(iv) What is the largest that the ratio
$$\frac { \text { area of } A } { \text { area of } B }$$
can be, as $\theta$ varies? [Figure]
Q5 Permutations & Arrangements Forming Numbers with Digit Constraints View
5. For ALL APPLICANTS.
We define the digit sum of a non-negative integer to be the sum of its digits. For example, the digit sum of 123 is $1 + 2 + 3 = 6$.
(i) How many positive integers less than 100 have digit sum equal to 8 ?
Let $n$ be a positive integer with $n < 10$.
(ii) How many positive integers less than 100 have digit sum equal to $n$ ?
(iii) How many positive integers less than 1000 have digit sum equal to $n$ ?
(iv) How many positive integers between 500 and 999 have digit sum equal to 8 ?
(v) How many positive integers less than 1000 have digit sum equal to 8 , and one digit at least 5 ?
(vi) What is the total of the digit sums of the integers from 0 to 999 inclusive?
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, Bob and Charlie are well-known expert logicians; they always tell the truth. In each of the scenarios below, Charlie writes a whole number on Alice and Bob's foreheads. The difference between the two numbers is one: either Alice's number is one larger than Bob's, or Bob's number is one larger than Alice's. Each of Alice and Bob can see the number on the other's forehead, but can't see their own number.
(i) Charlie writes a number on Alice and Bob's foreheads, and says "Each of your numbers is at least 1 . The difference between the numbers is $1 . "$
Alice then says "I know my number." Explain why Alice's number must be 2 . What is Bob's number?
(ii) Charlie now writes new numbers on their foreheads, and says "Each of your numbers is between 1 and 10 inclusive. The difference between the numbers is 1 . Alice's number is a prime." (A prime number is a number greater than 1 that is divisible only by 1 and itself.)
Alice then says "I don't know my number." Bob then says "I don't know my number." What is Alice's number? Explain your answer.
(iii) Charlie now writes new numbers on their foreheads, and says "Each of your numbers is between 1 and 10 inclusive. The difference between the numbers is 1. "
Alice then says "I don't know my number. Is my number a square number?" Charlie then says "If I told you that, you would know your number." Bob then says "I don't know my number." What is Alice's number? 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.
AB-words are "words" formed from the letters $\mathbf { A }$ and $\mathbf { B }$ according to certain rules. The rules are applied starting with the empty word, containing no letters. The basic rules are:
(1) If the current word is $x$, then it can be replaced with the word that starts with $\mathbf { A }$, followed by $x$ and ending with $\mathbf { B }$, written $\mathbf { A } x \mathbf { B }$.
(2) If the current word ends with $\mathbf { B }$, the final $\mathbf { B }$ can be removed.
(i) Show how the word $\mathbf { A A A B }$ can be produced.
(ii) Describe precisely all the words that can be produced with these two rules. Justify your answer. You might like to write $\mathbf { A } ^ { i }$ for the word containing just $i$ consecutive copies of $\mathbf { A }$, and similarly for $\mathbf { B }$; for example $\mathbf { A } ^ { 3 } \mathbf { B } ^ { 2 } = \mathbf { A } \mathbf { A } \mathbf { A B B }$.
We now add a third rule:
(3) Reverse the word, and replace every $\mathbf { A }$ by $\mathbf { B }$, and every $\mathbf { B }$ by $\mathbf { A }$.
For example, applying this rule to $\mathbf { A A A B }$ would give $\mathbf { A B B B }$.
(iii) Describe precisely all the words that can be produced with these three rules. Justify your answer.
Finally, we add a fourth rule:
(4) Reverse the word.
(iv) Show that every word consisting of $\mathbf { A s }$ and $\mathbf { B s }$ can be formed using these four rules. Hint: show how, if we have produced the word $w$, we can produce (a) the word $\mathbf { A } w$, and (b) the word $\mathbf { B } w$; hence deduce the result.
End of last question
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