1
Support for UCL
Mathematics
offer holders with
the Sixth Term
Examination Paper
and the Advanced
Extension Award
Sixth Term Examination Paper (STEP)
and Advanced Extension Award (AEA)
examinations test advanced mathematical
thinking and problem solving. Both
examinations encourage you to:
xplore different possible strategies to
›e
analyse and answer questions
se the techniques you know in unusual
›u
and interesting ways
evelop confidence, stamina and fluency
›d
in working through unstructured problems
Preparing for STEP and AEA examinations
will reward you with stronger problem
solving skills. These are essential in many
disciplines, and invaluable for studying
mathematics at university.
2
niversity College London provides
›U
Mathematics offer holders with a
comprehensive package to help
with preparation for STEP and
AEA examinations.
During 2015/16, all offer holders have free
access to:
•Online Resources – extensive notes,
questions and solutions to support
independent study
•Live Online Lectures – a series of seventeen
live online lectures giving detailed advice on
how to tackle STEP and AEA questions; to
attend these all you need is a computer or
tablet connected to the internet
•Study Days – two study days at University
College London giving you a chance to work
face-to-face with expert tutors and other
students on STEP and AEA questions.
Here is a selection of feedback on similar
support provided to previous applicants:
Thank you so much for the
online lectures, I found them
invaluable and they were
no doubt a vital part of my
preparation for STEP.
I found the whole
package incredibly
useful.
I found the resources
such as notes and
solutions really helpful.
The study day that I
attended was brilliant.
I took away a lot from
the day.
Online resources to support your independent study
s a UCL offer holder you get
›A
access to a set of online resources
to support your independent
study for STEP and AEA and to
complement the live online lectures
and study days.
These are provided by Mathematics in Education and
Industry (MEI), a charity that supports mathematics
teaching and learning. MEI also provides highly-acclaimed
online resources to support A level Mathematics and
Further Mathematics.
Arranged by mathematical theme these include:
• full worked solutions to all STEP and AEA papers
since 2005
• additional notes
• additional questions and solutions (not from STEP
and AEA past papers)
• video worked solutions.
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The Mathematics Department at UCL will send your
username and password for the resources to you
by email.
You log in to the resources at www.integralmaths.org
Live online lectures
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opic-by-topic, this series of
›T
live online lectures provides
comprehensive coverage of the
pure mathematics ideas, skills and
techniques useful in STEP and AEA.
Throughout the series there is an
emphasis on:
•providing insight into the type of thinking
needed to solve problems successfully
•building a knowledge-base of useful
techniques, facts and ‘tricks’
•producing high-quality written explanations
of solutions.
STEP and AEA past paper questions will
be considered in every session.
The Mathematics Department at UCL will send your
personal hyperlink to attend the online lectures to you
by e-mail. Similar online sessions were provided to
UCL mathematics applicants during 2014. Feedback
comments included:
Essential
I really enjoyed the online
lectures. It was helpful to
see how to make connections
between different areas of
maths and how to produce
elegant solutions.
Everything was
perfectly clear and very
informative, thank you.
Great sessions
and really helpful!
Live online lectures
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Session title
2016 dates
1
What are STEP and AEA? The problem-solving mindset
28 January
2
Using algebraic identities and solving equations
4 February
3
Digits, divisibility and logarithms
11 February
4
Counting and placement
25 February
5
Quadratics, cubics and other polynomials
3 March
6
Handling inequalities
10 March
7
Geometrical problems
17 March
8
Factors, primes and irrationals
24 March
9
Summing series
14 April
10
The binomial expansion and approximations
21 April
11
Sequences and induction
28 April
12
Advanced techniques in trigonometry
5 May
13
Analysing the behaviour of functions – limits and curve sketching
12 May
14
Advanced techniques in integration
19 May
15
Differential equations
26 May
16
Vectors and coordinate geometry
2 June
17
Examination technique and advice
9 June
Live online lectures
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Live online lecture 1
What are STEP and AEA?
The problem-solving mindset
Live online lecture 2
Using algebraic identities and solving
equations
To find an effective way to approach a problem often
requires a willingness to experiment and play around with
the information you have been given:
In this session, you’ll look at the importance of precision
and care in algebra, particularly when finding roots of
equations. This will include suggestions about how to
set out working to minimise the chance of mistakes.
An example which illustrates the need for care is the
following, without care it’s easy to miss some of the roots:
• looking at a special case of a problem might be helpful
• keeping a track of your ideas and organising your
thoughts is important
• you should make good use of notation and diagrams.
In this session examples will be used to illustrate
these ideas. You’ll also see how problems can have
many different solutions and discuss the notion of an
elegant solution.
Describe fully the roots of sin2x = sinx
Advice will be provided about algebraic techniques
including spotting how to factorise expressions. Can you
see how the following factorises?
a2b + abc + a2c +ac 2 + b 2a +b2c + abc + bc 2
Some standard but perhaps less well-known algebraic
relationships will be considered. The following identity,
for example, can be very useful:
Live online lecture 3
Digits, divisibility and logarithms
This session will look at how to deduce properties of a
number from its digits; how to spot factors of a number;
and techniques involving logarithms, including how logs
can help you to find the first digit of a number.
You need to remember that the digits of a number are not
the same as the number itself.
The two-digit number whose digits are a then b which
you would write down as ‘ab’ is actually 10a + b. The
three digit number ‘abc ’ is actually 100a + 10a + b.
The use of indices and logarithms will be considered.
How would you decide which of the following is correct?
log2 3 < log4 8;log2 3 = log4 8;log2 3 > log4 8
a3 – b3 ≡ (a – b)(a2 + ab + b 2)
› Example problem
› Example problem
› Example problem
What is the area
of the square in
this diagram?
For which positive integers n do there exists non-zero
integers a and b such that a2 – b 2 = n
Find all the positive integers solutions m and n of
m! + 3 = n2?
Live online lectures
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Live online lecture 4
Counting and Placement
Live online lecture 5
Quadratics, cubics and other polynomials
Combinatorics is a branch of mathematics partly
concerning the study of finite collections of objects.
Aspects of combinatorics include counting the number of
objects of a given kind or size, deciding when the objects
can be arranged so that certain criteria can be met;
and finding the “largest”, “smallest”, or “optimal” objects
Examples of this are:
Factorising and expanding polynomials; using techniques
such as ‘completing the square’; using the factor and
remainder theorem and employing substitutions to allow
the use of polynomial techniques will all be considered in
this session.
How many three digit numbers are there whose digits
sum to 25?
Arrange the numbers 0 – 20 (inclusive) into seven
separate groups each of three numbers so that when
the numbers in each are added together, they make
seven consecutive numbers?
Confidence with problems like this is important - not only
for problems which are directly of a combinatorial nature
but also to keep track of the various cases that might
develop in a piece of mathematical reasoning.
The fundamental theorem of algebra will be discussed
as well as ways to identify the number of real roots
a polynomial equation has. It’s also useful to know
the relationship between roots and coefficients of
polynomials. For example:
If ∝, β, γ are roots of the polynomial
x3 + ax 2 + bx + c = 0 then
a = –(∝ + β + γ ), b = ∝β + βγ + γ∝, c = – ∝βγ
It’s useful to use curve sketching alongside algebraic
techniques when dealing with polynomials. These ideas
will be covered in this lecture.
› Example problem
› Example problem
Imagine a three dimensional version of noughts and
crosses played in a 3 × 3 × 3 cell cube. How many
different winning lines are there?
Calculate the coefficient of x40 in the expansion of
(x 2 –1)14 (x4 + x 2 + 1)4
Live online lectures
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Live online lecture 6
Handling inequalities
Live online lecture 7
Geometrical problems
Live online lecture 8
Factors, primes and irrationals
Why is it that there are some inequalities that you would
look to solve and other inequalities that you would look
to prove?
Being able to use geometrical properties to deduce
information about a figure is important. Solving
geometrical problems can also require:
It’s clear that the statement
• good diagrams
Knowing its prime factorisation reveals a lot of information
about a number. In particular it’s a useful way to
understand how many factors a number has, what they
are and the number’s relationship to other numbers via
least common factors and highest common multiples.
It will help you to answer questions like:
2x + 4 > 10
is true for some values of x and false for other values of
x and so you would look to solve it.
However the statement
x 2 + y 2 ≥ 2xy
is true for all real values of x and y and you would look
to prove that that is the case.
In this session a variety of techniques for solving and
proving inequalities will be considered.
• experimenting by adding elements such as lines or
circles to diagrams to reveal relationships
• using algebra alongside geometry where necessary.
In this diagram
what is the sum
of the four
shaded angles?
Find the number of factors of 180. How many of these
factors are divisible by 10?
The classic proof that the square root of 2 is irrational
uses prime factorisation.
In this session you’ll look at questions involving these
ideas as well as discussing what it means to say that a
number is irrational. Mathematical reasoning involving
irrational numbers will be considered in detail.
This problem is reasonably straightforward but in
producing a solution it would be important to be
clear about which geometric properties you are using
at all times.
› Example problem
› Example problem
› Example problem
Solve the inequality
Show that the shortest distance from the point (∝, β )
to the line
|m∝ + c – β |
y = mx + c has the value
√m2 + 1
If a, b, c, d are rational numbers and if p is irrational
prove that a + bp = c + d p implies that
a = c and b = d
cos∅ +1 ≤1 where 0 ≤ ∅ < 2π and sin∅ ≠ 0
sin∅
Live online lectures
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Live online lecture 9
Summing series
Live online lecture 10
The binomial expansion and approximations
Live online lecture 11
Sequences and induction
General knowledge about summing series such as the
formula for the sum of the first n whole numbers; the
formula for the sum of the squares of the first n whole
numbers; arithmetic and geometric series can be useful
in STEP and AEA problems.
The binomial expansion provides you with a quick way to
expand brackets like:
Can you identify all of these sequences?
This knowledge combines with topics already covered to
enable you to answer questions like:
Prove that
1
1
1
1
+
+
+
...
<
n + 1 (n + 1)(n + 2) (n + 1)(n + 2)(n + 3)
n
Sigma notation will also be covered – can you show that
3
i
∑ ∑j
i=1 j=1
2
= 20?
(a + b) (a + b) (a + b) (a + b) (a + b) (a + b)
It also provides you with series with finitely many terms
which approximate functions like:
f (x ) =
1
3
(1+2x ) 2
Computers use series such as those provided by the
binomial expansion to evaluate certain functions. This
lecture will take a look at how the binomial expansion can
be used to produce approximations and how judgements
about their accuracy can be made.
A pocket calculator may evaluate the following as 1:
1
√1–1.5x10 –12
1, 8, 27, 64, 125, ...
1, 3, 6, 10, 15, 21, ...
1, 1, 2, 3, 5, 8, 13, 21, 34, ...
1, 2, 6, 24, 120, 720, ...
As well as discussing sequences in general, this session
will cover situations where sequences are defined or
properties of sequences are deduced recursively (from
one term to the next). This will lead naturally to the idea of
proof by induction.
Proof by induction is one way to prove a statement like
“11n – 6 is divisible by 5 for all positive integers n.”
The binomial expansion can be used to estimate how
close to 1 this value really is.
› Example problem
› Example problem
› Example problem
The first and second terms of an arithmetic series are
100 and 95 respectively. The sum to n terms of the
series is Sn
Use the binomial expansion of √9 – x as far as the
term in x 2 with
x= 1
Prove, by induction, that for all positive integers n
greater than or equal to 8 there exist non-negative
integers x, y such that n = 3x + 5y
Find the largest positive value of Sn
64
to find a rational number which approximates √23
Live online lectures
Live online lecture 12
Advanced techniques in trigonometry
Recalling the graphs of trigonometric functions and
being able to use trigonometric identities fluently is very
important in mathematical problem-solving. Can you
see why:
cos(2x) = cos4 x – sin4 x ?
Familiarity with key facts is also important – it’s useful to
be able to recall values such as:
( 54π (, sin(11π), tan(– 34π (
cos
quickly.
Trigonometry in integration and differentiation and in
geometrical problems will also be considered.
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Live online lecture 13
Analysing the behaviour of functions
– limits and curve sketching
Curve sketching is a very important skill in all
mathematical subjects – from Economics to Engineering.
Algebraic techniques such as factorising, long division
and the binomial expansion can be very useful to
determine the behaviour of functions. Of particular interest
in this session will be limiting values of functions and their
behaviour around asymptotes. For example it’s clear that
the function:
f (x) =
x2 – 1
x–1
is not defined at x = 1, but there is such a thing as:
lim =
x⟶1
x –1
?
x–1
2
This session will cover techniques for analysing questions
like this as well as for general curve sketching.
Live online lecture 14
Advanced techniques in integration
Proficiency in integration involves being able to choose
correctly from a range of possible techniques to solve a
given problem. It’s important to gain as much experience
as you can with methods like integration by substitution
and integration by parts because this will give you a much
better instinct for what will and what won’t work.
Familiarity with trigonometric and other relationships can
also be very important in problems with integration. For
example it’s quite easy to see that:
π
2
π
2
0
0
∫cos xdx = ∫sin xdx
and that
π
(
π
∫cos x –
π
2
2
π
2
( dx = ∫cos xdx
0
Being able to recall relationships like these and
spot when to use them can be important in questions
involving integration.
› Example problem
› Example problem
› Example problem
Solve the equation
What is
tan
lim =
Calculate
1
x4
∫
2 dx
0 1 + x
( ∅2 ( = tan2 ∅
x⟶27
x – 27
1
x3 – 3
?
Explain why this is the case.
Live online lectures
11
Live online lecture 15
Differential equations
Live online lecture 16
Vectors and coordinate geometry
Live online lecture 17
Examination technique and advice
Differential equations describe how mathematical systems
change. There are a variety of techniques for solving
differential equations and examples of these will be given
in this session.
Practising vectors and coordinate geometry questions is
very worthwhile. Confidence can be built quite quickly.
A theme of this session will be situations which require
working with general coordinates or vectors rather than
with coordinates and vectors given explicitly. For example:
In the final session we’ll look at some examination
questions alongside examiners’ comments about them.
This will give a clearer understanding of what is required
to produce an answer to a STEP or AEA question that
will attract as much credit as possible. Final pointers on
examination technique will also be provided.
Techniques such as substitution that can be used to
reduce some differential equations to a form that can be
dealt with using standard methods will be discussed.
Sometimes problems will require a real life situation to be
modelled by appropriate differential equations. Doing this
is a skill in its own right and examples of this will be given
in this lecture.
Being able to interpret the solution of a differential
equation is also very important. Sketching solutions to
differential equations will be considered.
A and C are the opposite vertices of a square ABCD,
and have coordinates (a,b) and (c,d ), respectively.
In terms of a, b, c and d, what are the coordinates
of the other two vertices?
The session will also cover a number skills that it is useful
to have to hand when attempting vectors questions.
For example it’s useful to be able to quickly deal with
situations like this:
If point A has position vector a and point B has position
vector b relative to the origin find the position vector of
the point P on the line AB which is 2 of the way from
3
A to B, in terms of a and b.
› Example problem
› Example problem
Find two solutions of the differential equation
dy 2
dy
3
+ 2x
=1
dx
dx
If A is the point (a, 0, 0), B is the point (0, b, 0) and C is
the point (0, 0, c ) show that
( (
cos (ACB ) =
c2
√(a 2 + c 2) (b 2 + c 2)
Study days at UCL
Two STEP & AEA study days designed
to complement the online support
will take place at UCL in 2015.
They provide a great opportunity to
come to UCL to learn and practise
mathematics. The emphasis will be
on gaining experience in answering
past paper questions but there will
also be teaching sessions on some
selected topics.
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1st Study Day 8 April 2016
all UCL offer holders are welcome to attend
The focus of the day will be curve sketching and
trigonometry. We will cover the curve sketching
techniques needed in the STEP/AEA papers but not
necessarily covered in A-level Maths and Further Maths,
general solutions of trigonometric equations, and inverse
trigonometric functions.
2nd Study Day 2 June 2016
for offer holders who make UCL their firm choice
on UCAS
The focus of the day will be calculus. After an extensive
revision of integration techniques, we will examine different
types of exam questions on integration. We will also look
at differentiation and differential equations.
To book a place at a study day email:
admissions@math.ucl.ac.uk
I really enjoyed the
trigonometry and the
graph questions and
once again found that my
confidence had improved.
I enjoyed meeting people and
working through the questions
at the study day and I thought
that the tutor was very good.