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Papers (32)
2025
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2024
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2023
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2022
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2021
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2020
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2019
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2018
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2017
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2016
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2015
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2014
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2013
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2012
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2011
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2010
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2009
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2008
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2007
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2005
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2004
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2003
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2002
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2001
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2000
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1998
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1997
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1996
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Unknown
specimen_1 7 specimen_2 7 specimen_a 7 specimen_b 7
2016 mat_2016.pdf

6 maths questions

Q2 Composite & Inverse Functions Evaluate Composition from Algebraic Definitions View
2. For ALL APPLICANTS.
Let
$$A ( x ) = 2 x + 1 , \quad B ( x ) = 3 x + 2 .$$
(i) Show that $A ( B ( x ) ) = B ( A ( x ) )$.
(ii) Let $n$ be a positive integer. Determine $A ^ { n } ( x )$ where
$$A ^ { n } ( x ) = \underbrace { A ( A ( A \cdots A } _ { n \text { times } } ( x ) \cdots )$$
Put your answer in the simplest form possible.
A function $F ( x ) = 108 x + c$ (where $c$ is a positive integer) is produced by repeatedly applying the functions $A ( x )$ and $B ( x )$ in some order.
(iii) In how many different orders can $A ( x )$ and $B ( x )$ be applied to produce $F ( x )$ ? Justify your answer.
(iv) What are the possible values of $c$ ? Justify your answer.
(v) Are there positive integers $m _ { 1 } , \ldots , m _ { k } , n _ { 1 } , \ldots , n _ { k }$ such that
$$A ^ { m _ { 1 } } B ^ { n _ { 1 } } ( x ) + A ^ { m _ { 2 } } B ^ { n _ { 2 } } ( x ) + \cdots + A ^ { m _ { k } } B ^ { n _ { k } } ( x ) = 214 x + 92 \quad \text { for all } x ?$$
Justify your answer.
If you require additional space please use the pages at the end of the booklet
Q3 Indefinite & Definite Integrals Integral Equation with Symmetry or Substitution 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.
In this question we fix a real number $\alpha$ which will be the same throughout. We say that a function $f$ is bilateral if
$$f ( x ) = f ( 2 \alpha - x )$$
for all $x$.
(i) Show that if $f ( x ) = ( x - \alpha ) ^ { 2 }$ for all $x$ then the function $f$ is bilateral.
(ii) On the other hand show that if $f ( x ) = x - \alpha$ for all $x$ then the function $f$ is not bilateral.
(iii) Show that if $n$ is a non-negative integer and $a$ and $b$ are any real numbers then
$$\int _ { a } ^ { b } x ^ { n } \mathrm {~d} x = - \int _ { b } ^ { a } x ^ { n } \mathrm {~d} x$$
(iv) Hence show that if $f$ is a polynomial (and $a$ and $b$ are any reals) then
$$\int _ { a } ^ { b } f ( x ) \mathrm { d } x = - \int _ { b } ^ { a } f ( x ) \mathrm { d } x$$
(v) Suppose that $f$ is any bilateral function. By considering the area under the graph of $y = f ( x )$ explain why for any $t \geqslant \alpha$ we have
$$\int _ { \alpha } ^ { t } f ( x ) \mathrm { d } x = \int _ { 2 \alpha - t } ^ { \alpha } f ( x ) \mathrm { d } x$$
If $f$ is a function then we write $G$ for the function defined by
$$G ( t ) = \int _ { \alpha } ^ { t } f ( x ) \mathrm { d } x$$
for all $t$.
(vi) Suppose now that $f$ is any bilateral polynomial. Show that
$$G ( t ) = - G ( 2 \alpha - t )$$
for all $t$.
(vii) Suppose $f$ is a bilateral polynomial such that $G$ is also bilateral. Show that $G ( x ) = 0$ for all $x$.
If you require additional space please use the pages at the end of the booklet
Q4 Circles Circles Tangent to Each Other or to Axes 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.
The line $l$ passes through the origin at angle $2 \alpha$ above the $x$-axis, where $2 \alpha < \frac { \pi } { 2 }$. [Figure]
Circles $C _ { 1 }$ of radius 1 and $C _ { 2 }$ of radius 3 are drawn between $l$ and the $x$-axis, just touching both lines.
(i) What is the centre of circle $C _ { 1 }$ ?
(ii) What is the equation of circle $C _ { 1 }$ ?
(iii) For what value of $\alpha$ do circles $C _ { 1 }$ and $C _ { 2 }$ touch?
(iv) For this value of $\alpha$ (for which the circles $C _ { 1 }$ and $C _ { 2 }$ touch) a third circle, $C _ { 3 }$, larger than $C _ { 2 }$, is to be drawn between $l$ and the $x$-axis. $C _ { 3 }$ just touches both lines and also touches $C _ { 2 }$. What is the radius of this circle $C _ { 3 }$ ?
(v) For the same value of $\alpha$, what is the area of the region bounded by the $x$-axis and the circles $C _ { 1 }$ and $C _ { 2 }$ ?
If you require additional space please use the pages at the end of the booklet
Q5 Sequences and series, recurrence and convergence Summation of sequence terms View
5. For ALL APPLICANTS.
This question concerns the sum $s _ { n }$ defined by
$$s _ { n } = 2 + 8 + 24 + \cdots + n 2 ^ { n }$$
(i) Let $f ( n ) = ( A n + B ) 2 ^ { n } + C$ for constants $A , B$ and $C$ yet to be determined, and suppose $s _ { n } = f ( n )$ for all $n \geqslant 1$. By setting $n = 1,2,3$, find three equations that must be satisfied by $A , B$ and $C$.
(ii) Solve the equations from part (i) to obtain values for $A , B$ and $C$.
(iii) Using these values, show that if $s _ { k } = f ( k )$ for some $k \geqslant 1$ then $s _ { k + 1 } = f ( k + 1 )$.
You may now assume that $f ( n ) = s _ { n }$ for all $n \geqslant 1$.
(iv) Find simplified expressions for the following sums:
$$\begin{aligned} & t _ { n } = n + 2 ( n - 1 ) + 4 ( n - 2 ) + 8 ( n - 3 ) + \cdots + 2 ^ { n - 1 } 1 , \\ & u _ { n } = \frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } . \end{aligned}$$
(v) Find the sum
$$\sum _ { k = 1 } ^ { n } s _ { k }$$
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.

[Figure]
Four people $A , B , C , D$ are performing a dance, holding hands in the arrangement shown above. Each dancer is assigned a 1 or a 0 to determine their steps, and there must always be at least a 1 and a 0 in the group of dancers (dancers cannot all dance the same kind of steps). A dancer is off-beat if their assigned number plus the numbers assigned to the people holding hands with them is odd. The entire dance is an off-beat dance if every dancer is off-beat.
(i) In how many ways can the four dancers perform an off-beat dance? Explain your answer.
A new dance starts and two more people, $E$ and $F$, join the dance such that each dancer holds hands with their neighbours to form a ring.
(ii) In how many ways can the ring of six dancers perform an off-beat dance? Explain your answer.
(iii) In a ring of $n$ dancers explain why an off-beat dance can only occur if $n$ is a multiple of 3 .
(iv) For a new dance a ring of $n > 4$ dancers, each holds hands with dancers one person away from them round the ring (so $C$ holds hands with $A$ and $E$ and $D$ holds hands with $B$ and $F$ and so on). For which values of $n$ can the dance be off-beat?
On another planet the alien inhabitants have three (extendible) arms and still like to dance according to the rules above.
(v) If four aliens dance, each holding hands with each other, how many ways can they perform an off-beat dance?
(vi) Six aliens standing in a ring perform a new dance where each alien holds hands with their direct neighbours and the alien opposite them in the ring. In how many ways can they perform an off-beat dance?
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.
An $n$-fan consists of a row of $n$ points, the tips, in a straight line, together with another point, the hub, that is not on the line. The $n$ tips are joined to each other and to the hub with line segments. For example, the left-hand picture here shows a 6 -fan, [Figure] [Figure]
For a given $n$-fan, an $n$-span is a subset containing all $n + 1$ points and exactly $n$ of the line segments, chosen so that all the points are connected together, with a unique path between any two points. The right-hand picture shows one of many 6 -spans obtained from the given 6 -fan; in this 6 -span, the tips are in "groups" of 3,1 and 2 , with the top "group" containing 3 tips.
(i) Draw all three 2 -spans.
(ii) Draw all 3 -spans.
(iii) By considering the possible sizes of the top group of tips and how the group is connected to the hub, calculate the number of 4 -spans.
(iv) For $n \geqslant 1$ let $z _ { n }$ denote the number of $n$-spans. Give an expression for $z _ { n }$ in terms of $z _ { k }$, where $1 \leqslant k < n$. Use this expression to show that $z _ { 5 } = 55$.
(v) Use this relationship to calculate $z _ { 6 }$.
If you require additional space please use the pages at the end of the booklet
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