OCR A Level Further Mathematics B (MEI)

163 Pages·2016·5.03 MB·English

OCR Level 3 Advanced GCE in Further
Mathematics B (MEI) (H645)
Specification
Version 1: First assessment 2019
This draft qualification has not yet been accredited by Ofqual. It is published to enable teachers to
have early sight of our proposed approach to A Level in Further tMathematics B (MEI). Further
changes may be required and no assurance can be given at this time that the proposed
qualification will be made available in its current form, or thfat it will be accredited in time for first
teaching in 2017 and first award in 2019.
a

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© OCR 2016 B10007/10 QN Awaiting Accreditation
A Level in Further Mathematics B (MEI)
Contents
1 Why choose an OCR A Level in Further Mathematics B (MEI)? 3
1a. Why choose an OCR qualification? 3
1b. Why choose an OCR A Level in Further Mathematics B (MEI) 4
1c. What are the key features of this specification? 5
1d. How do I find out more information? 5
2 The specification overview 6
2a. OCR A Level in Further Mathematics B (MEI) (H645) 6
2b. Content of A Level in Further Mathematics B (MEI) (H645) 7
2c. Content of Mathematical processes 14
2d. Content of Core pure (Y420) – mandatory unit 16
2e. Content of Mechanics major (Y421) – major option 32
2f. Content of Statistics major (Y422) – major option 48
2g. Content of Mechanics minor (Y431) – minor option 68
2h. Content of Statistics minor (Y432) – minor option 77
2i. Content of Modelling with algorithms (Y433) – minor optiotn 89
2j. Content of Numerical methods (Y434) – minor option 100
f
2k. Content of Extra pure (Y435) – minor option 109
2l. Content of Further pure with technology (Y436) – minor option 124
a
2m. Prior knowledge, learning and progression 130
3 Assessment of A Level in Further Mathematics B (MEI) 131
3a. Forms of assessment 131
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3b. Assessment objectives (AO) 132
3c. Assessment availability 134
3d. Retaking the qualification D 134
3e. Assessment of extended response 134
3f. Synoptic assessment 134
3g. Calculating qualification results 135
4 Admin: what you need to know 136
4a. Pre-assessment 136
4b. Special consideration 137
4c. External assessment arrangements 137
4d. Results and certificates 138
4e. Post-results services 138
4f. Malpractice 138
5 Appendices 139
5a. Overlap with other qualifications 139
5b. Accessibility 140
5c. Mathematical notation 140
5d. Mathematical formulae, identities and statistical tables 145
2 © OCR 2016
A Level Further Mathematics B (MEI)
1 Why choose an OCR A Level in Further
Mathematics B (MEI)?
1a. Why choose an OCR qualification?
Choose OCR and you’ve got the reassurance We provide a range of support services
that you’re working with one of the UK’s designed to help you at every stage, from
leading exam boards. Our new A Level in preparation through to the delivery of our
Further Mathematics B (MEI) course has specifications. This includes:
been developed in consultation with teachers,
employers and Higher Education to provide
 A wide range of high-quality creative
learners with a qualification that’s relevant to
resources including:
them and meets their needs.

o Delivery Guides
We’re part of the Cambridge Assessment
t
o Transition Guides
Group, Europe’s largest assessment agency
and a department of the University of o Topic Exploration Packs
f
Cambridge. Cambridge Assessment plays a
o Lesson Elements
leading role in developing and delivering
a
o …and much more.
assessments throughout the world, operating
in over 150 countries.
 Access to subject specialists to support

you through the transition and
We work with a range of education providers,
r throughout the lifetime of the
including schools, colleges, workplaces and
specifications.
other institutions in both the public and
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private sectors. Over 13,000 centres choose  CPD/Training for teachers including
our A Levels, GCSEs and vocational
events to introduce the qualifications
qualifications including Cambridge Nationals,
and prepare you for first teaching.
Cambridge Technicals and Cambridge
Progression.  Active Results – our free results
analysis service to help you review the
Our Specifications performance of individual learners or
whole schools.
We believe in developing specifications that
 ExamCreator – our new online past
help you bring the subject to life and inspire
your students to achieve more. papers service that enables you to build
your own test papers from past OCR
We’ve created teacher-friendly specifications exam questions.
based on extensive research and

engagement with the teaching community.
All A level qualifications offered by OCR are
They’re designed to be straightforward and
accredited by Ofqual, the Regulator for
accessible so that you can tailor the delivery
qualifications offered in England. The
of the course to suit your needs. We aim to
accreditation number for OCR A Level in
encourage learners to become responsible
Further Mathematics B (MEI) is QNXXXXXX
for their own learning, confident in discussing
ideas, innovative and engaged.

© OCR 2016 3
A Level Further Mathematics B (MEI)
1b. Why choose an OCR A Level in Further Mathematics B (MEI)
OCR A level in Further Mathematics B (MEI) teachers and representatives from Higher
has been developed by Mathematics in Education to decide how best to meet the
Education and Industry (MEI) and is long-term needs of learners.
assessed by OCR. This is a well-established MEI provides advice and CPD relating to all
partnership which provides a firm foundation the curriculum and teaching aspects of the
for curriculum and qualification development. course. It also provides teaching resources,
MEI is a long-established, independent which for this specification can be found on
curriculum development body; in developing the website (www.mei.org.uk).
this specification, MEI has consulted with

Aims and learning outcomes
OCR A Level in Further Mathematics B (MEI) problemts in context and mathematical
will encourage learners to: models that may be applied to solve them
f
 draw diagrams and sketch graphs to help
 understand mathematics and
explore mathematical situations and
mathematical processes in ways that a
interpret solutions
promote confidence, foster enjoyment and
provide a strong foundation for progress to  make deductions and inferences and draw
further study conclusions by using mathematical
r reasoning
 extend their range of mathematical skills
and techniques  interpret solutions and communicate their
D interpretation effectively in the context of
 understand coherence and progression in
the problem
mathematics and how different areas of
mathematics are connected  read and comprehend mathematical
arguments, including justifications of
 apply mathematics in other fields of study
methods and formulae, and communicate
and be aware of the relevance of
their understanding
mathematics to the world of work and to
situations in society in general  read and comprehend articles concerning
applications of mathematics and
 use their mathematical knowledge to make
communicate their understanding
logical and reasoned decisions in solving
problems both within pure mathematics  use technology such as calculators and
and in a variety of contexts, and computers effectively, and recognise when
communicate the mathematical rationale such use may be inappropriate
for these decisions clearly
 take increasing responsibility for their own
 reason logically and recognise incorrect learning and the evaluation of their own
reasoning mathematical development.
 generalise mathematically
 construct mathematical proofs
OCR A Level in Further Mathematics B (MEI)
 use their mathematical skills and is designed for students with an enthusiasm
techniques to solve challenging problems for mathematics, many of whom will go on to
which require them to decide on the degrees in mathematics, engineering, the
solution strategy sciences and economics. or any subject
where mathematics is developed further than
 recognise when mathematics can be used
in A level Mathematics.
to analyse and solve a problem in context

 represent situations mathematically and
understand the relationship between
4 © OCR 2016
A Level Further Mathematics B (MEI)
OCR A Level in Further Mathematics B (MEI) includes different options that can enable
is both deeper and broader than A level students to specialise in areas of
mathematics. AS and A level Further mathematics that are particularly relevant to
Mathematics build from GCSE level and AS their interests and future aspirations. A level
and A level Mathematics. As well as building Further Mathematics prepares students for
on algebra and calculus introduced in A level further study and employment in highly
Mathematics, the A level Further mathematical disciplines that require
Mathematics core content introduces knowledge and understanding of
complex numbers and matrices, fundamental sophisticated mathematical ideas and
mathematical ideas with wide applications in techniques.
mathematics, engineering, physical sciences
and computing. The non-core content
1c. What are the key features of this specification?
OCR A Level in Further Mathematics B (MEI) learners having access to appropriate
has been designed to help learners to fulfil software in the examination
their potential in mathematics and to support
 is assestsed in a way which is designed to
teachers in enabling them to do this. The
enable all learners to show what they are
qualification: abfle to do
 encourages learners to develop a deep  is clearly laid out with detailed guidance
a
understanding of mathematics and an regarding what learners need to be able to
ability to use it in a variety of contexts do
 allows a choice of options to enable  is resourced and supported by MEI in line
teachers to create the most appropriate r with the aims and learning outcomes of
course for their students: choosing a major the qualification.
and a minor option focuses on depth of
D This specification is designed to be co-
study; choosing three minor options
focuses on breadth of study teachable with A level Mathematics B (MEI)
and with AS level Further Mathematics B
 encourages learners to use appropriate
(MEI). Clear labelling of the material allows
technology to deepen their mathematical
teachers to know which parts of the A level
understanding and extend the range of
Further Mathematics specification could be
problems which they are able to solve
taught in the first year of the course,
 includes an option (Further pure with alongside AS Mathematics and AS Further
technology) which is assessed with Mathematics.
1d. How do I find out more information?
If you are already using OCR specifications Want to find out more?
you can contact us at: www.ocr.org.uk
Get in touch with one of OCR’s Subject
If you are not already a registered OCR Specialists:
centre then you can find out more information
on the benefits of becoming one at: Email: [email protected]
www.ocr.org.uk
Customer Contact Centre: 01223 553998
If you are not yet an approved centre and
Teacher support: www.ocr.org.uk
would like to become one go to:
www.ocr.org.uk Advice is also available from MEI; contact
details can be found on www.mei.org.uk

© OCR 2016 5
A Level Further Mathematics B (MEI)
2 The specification overview
2a. OCR A Level in Further Mathematics B (MEI) (H645)
To be awarded OCR’s A Level in Further Mathematics B (MEI) learners must take one of three
routes through the qualification, Route A, Route B or Route C. The qualification comprises of one
mandatory Core pure unit taken by all learners and then a combination of optional units.

Route A: Candidates must take the mandatory Core pure and Mechanics major units and then one
further optional minor unit. This unit must not be Mechanics minor.
Route B: Candidates must take the mandatory Core pure and Statistics major units and then one
further optional minor unit. This unit must not be Statistics minor.
Route C: Candidates must take the mandatory Core pure unit and then three further minor optional
units.

Learners may not enter for Mechanics major Y421 and Mechanics minor Y431 or Statistics major
Y422 and Statistics minor Y432.

Learners may take more than the required number of minor unitts to increase the breadth of their
course. For details of how their grade will be awarded, see section 3g.
f
Content Overview Assaessment Overview
Mandatory unit:
Content is in four areas:
Core pure 50%
r
 Mathematical processes (Y420)
 Core pure content1
of total
D 144 raw marks
 Major options
(180 scaled)
o Mechanics major (Y421)1 A level
2 hour 40 mins
o Statistics major (Y422)1
Written paper
 Minor options
o Mechanics minor (Y431)2 Major Option 33⅓%

o Statistics minor (Y432)2
120 raw marks
of total
o Modelling with algorithms (120 scaled)
(Y433)2
o Numerical methods (Y434)2 2 hour 15 mins A level
Written paper
o Extra pure (Y435)

o Further pure with technology
(Y436)

The content of Mathematical processes is Minor Option
assessed in every unit. 16⅔%
60 raw marks
(60 scaled)
1One third of the Core pure content, and
of total
one half of the content of each major
1 hour 15 mins
option can be co-taught with AS Further
Written paper
Mathematics. This material is labelled (a) A level
throughout section 2c to 2f. (1 hour 45 mins
Written paper for
2These minor options can be co-taught
Y436)
with AS Further Mathematics.
6 © OCR 2016 © OCR 2016
A Level Further Mathematics B (MEI)
2b. Content of A Level in Further Mathematics B (MEI) (H645)
The content is listed below, under four
This A level qualification builds on the skills,
headings:
knowledge and understanding set out in the
whole GCSE subject content for mathematics 1. Mathematical processes consisting of
and the subject content for AS and A level mathematical argument and language,
mathematics. Problem solving, proof and problem solving and mathematical
mathematical modelling will be assessed in modelling. This content is assessed in
further mathematics in the context of the every unit.
wider knowledge which students taking A
2. Core pure content
level further mathematics will have studied.
3. Major options
A Level Further Mathematics B (MEI) is a
• Mechanics major (Y421)
linear qualification. Learners enter for the
mandatory unit Core pure (Y420) and then a • Statistics major (Y422)
combination of optional units.
4. Minor options
t
Route A: Candidates must take the • Mechanics minor (Y431)
mandatory Core pure and Mechanics major f
• Statistics minor (Y432)
units and then one further optional minor unit.
This unit must not be Mechanics minor. a• Modelling with algorithms (Y433)

• Numerical methods (Y434)
Route B: Candidates must take the
mandatory Core pure and Statistics major • Extra pure (Y435)
units and then one further optional minor unrit.
• Further pure with technology
This unit must not be Statistics minor.
(Y436)

D
Route C: Candidates must take the
The mathematical processes should be
mandatory Core pure unit and then three
applied, along with associated mathematical
further minor optional units.
thinking and understanding, across every

permissible combination of units in this
Learners may not enter for Mechanics major
specification.
Y421 and Mechanics minor Y431 or Statistics
major Y422 and Statistics minor Y432. The applied optional units (Mechanics major,
Mechanics minor, Statistics major, Statistics
Learners may take more than the required minor and Modelling with algorithms) should
number of minor units to increase the breadth
be regarded as applications of pure maths as
of their course. For details of how their grade
well as ways of thinking about the world in
will be awarded, see section 3g.
their own right. The pure optional units (Extra

pure and Further pure with technology)
extend the content of the Core pure unit. The
Numerical methods unit extends the range of

non-analytic techniques for solving a wider
class of problems from within pure maths. In
all of these cases appropriate links should be

made with the content of A level Mathematics
and the content of the Core pure unit in this A
level Further Mathematics.

© OCR 2016 7
A Level Further Mathematics B (MEI)
Formulae and statistical tables If there is no particular instruction in a
question or the rubric for the paper about the
Some formulae will be available to learners in use of a calculator then allowable calculators
the examination, in a separate formulae can be used for any function they can
booklet. A list of these formulae can be found perform. Learners are expected to make it
in Section 5d. This list also contains the clear in their answers what calculations have
statistical tables which will be available to been performed in answering the question.
learners in the examination.
Use of technology
Use of calculators
It is expected that learners will have used
Calculators must comply with the published appropriate technology including
"instructions for conducting examinations", mathematical graphing tools and
which can be found at http://www.jcq.org.uk . spreadsheets when studying A Level Further
It is expected that calculators available in the Mathematics B (MEI). Several units have
examinations will include the following their own requirements for generic software
features: which learners will have used; the content
sections give more detail, including what is
expected in the examination. In general,
 An iterative function
learners are not expected to be familiar with
 The ability to perform calculations with t
particular software, nor will they be expected
matrices up to at least order 3×3
to use the syntax associated with particular
 The ability to compute summary f
software but examination questions may
statistics and access probabilities from
include output from software which learners
standard statistical distributions. a
will need to complete or interpret. However,

the Numerical methods unit will also assess
Some questions will instruct learners "Do not
learners’ ability to write some spreadsheet
use a calculator in this question." or "Do
formulae; the Further pure with technology
not use a calculator in this part of the r
unit will assess learners’ ability to use a
question." It is expected that they will show
spreadsheet, graph-drawing software, a
sufficient working to indicate that they have
D computer algebra system and a programming
followed this instruction.
language on a computer or calculator in the

examination.
Some questions will instruct learners to

"show your calculations"; the expectation is
Simplifying expressions
that a calculator will only be used for basic

arithmetic and for the evaluation of
It is expected that learners will simplify
elementary functions (powers, logarithmic,
algebraic and numerical expressions when
exponential and trigonometric). Learners
giving their final answers, even if the
should show sufficient working to show they
examination question does not explicitly ask
have followed this instruction.
them to do so.

3
In the Numerical methods unit only basic
 80 should be written as 40 3.
operations (+, -, ×, ÷) and elementary 2
functions (powers, logarithmic, exponential
1 3 2
and trigonometric) may be used; capabilities  should be written as .
for numerical differentiation, numerical 3 2 7
integration and solving equations should not
1 1
be used. This is not a restriction on the type  12x22 should be written as
2
of calculator which may be used in the
examination, but a restriction on which 1 1
functions may be used. either 12x2 or .
12x

Calculators with spreadsheets and graph-  ln2ln3ln1 should be written as
drawing functionality are permitted in the ln6.
examination for any unit, but this functionality
 The equation of a straight line should
is only required in the Further pure with
be given in the form ymxc or
technology unit, where it may be available on
either a computer or a calculator. axbycunless otherwise stated.

8 © OCR 2016 © OCR 2016
A Level Further Mathematics B (MEI)
The meanings of some instructions used alternative methods exist which will be given
in examination questions full credit, but that they may be more time-
consuming or complex.
In general, learners should show sufficient
detail of their working and reasoning to e.g. You are given that
indicate that a correct method is being used. fx2x3x2 7x6. Show that x1is
The following command words are used to
a factor of f(x). Hence find the three factors
indicate when more, or less, specific detail is
required. of fx.

Exact e.g. Show that cosxsinx2 1sin2x for
An exact answer is one where numbers are all x. Hence, or otherwise, find the derivative
not given in rounded form. The answer will
of cosxsinx2.
often contain an irrational number such as
3, e or π and these numbers should be
You may use the result
given in that form when an exact answer is
When this phrase is used it indicates a given
required.
result that learners would not always be
The use of the word ‘exact’ also tells learners
expected to know, but which may be useful in
that rigorous (exact) working is expected in
answering the question.
the answer to the question. t
The phrase should be taken as permissive;
e.g. Find the exact solution of lnx2.
use of the given result is not required.
The correct answer is e2 and not 7.389 056. f
Plot
a
Show that Learners should mark points accurately on
Learners are given a result and have to get to graph paper provided in the Printed Answer
the given result from the starting information. Booklet. They will either have been given the
Because they are given the result, the points or have had to calculate them. They
r
explanation has to be sufficiently detailed to may also need to join them with a curve or a
cover every step of their working. straight line, or draw a line of best fit through
e.g. Show that the curve yxlnx hDas a them.
1 1 e.g. Plot this additional point on the scatter
stationary point  ,  . diagram.
e e

Determine
Sketch (a graph)
This command word indicates that
Learners should draw a diagram, not
justification should be given for any results
necessarily to scale, showing the main
found, including working where appropriate.
features of a curve. These are likely to

include at least some of the following.
State, Write down
 Turning points
These command words indicate that neither
 Asymptotes
working nor justification is required.
 Intersection with the y-axis

 Intersection with the x-axis
Hence
 Behaviour for large x (+ or –)
When a question uses the word ‘hence’, it is

an indication that the next step should be
Any other important features should also be
based on what has gone before. The
shown.
intention is that learners should start from the
e.g. Sketch the curve with equation
indicated statement.
1

y 
Hence or otherwise is used when there are (x1)
multiple ways of answering a given question.
Draw
Learners starting from the indicated
Learners should draw to an accuracy
statement may well gain some information
appropriate to the problem. They are being
about the solution from doing so, and may
asked to make a sensible judgement about
already be some way towards the answer.
the level of accuracy which is appropriate.
The command phrase is used to direct
e.g. Draw a diagram showing the forces
learners towards using a particular piece of
acting on the particle.
information to start from or to a particular
e.g. Draw a line of best fit for the data.
method. It also indicates to learners that valid
© OCR 2016 9 9
A Level Further Mathematics B (MEI)
Mathematical Problem Solving Cycle
Mathematical problem solving is a core part of mathematics. The concept of a mathematical
problem solving cycle is outlined in competence statement FMp9 and developed more fully below.
The problem solving cycle gives a general strategy for dealing with problems which can be solved
using mathematical methods; it can be used for problems within mathematical contexts and for
problems in real-world contexts.

Problem specification and
analysis

Interpretation Information collection

t
Processing and
representation
f

a
Process Description
Problem The problem to be addressed needs to be formulated in a way which allows
specification mathematical methods to be used. It then needs to be analysed so that a plan
r
and analysis can be made as to how to go about it. The plan will almost always involve the
collection of information in some form. The information may already be available
(e.g. online) or it Dmay be necessary to carry out some form of experimental or
investigational work to gather it.
In some cases the plan will involve considering simple cases with a view to
generalising from them. In others, physical experiments may be needed. In
statistics, decisions need to be made at this early stage about what data will be
relevant and how they will be collected.
The analysis may involve considering whether there is an appropriate standard
model to use (e.g. the Normal distribution or the particle model) or whether the
problem is similar to one which has been solved before.

At the completion of the problem solving cycle, there needs to be
consideration of whether the original problem has been solved in a
satisfactory way or whether it is necessary to repeat the problem solving
cycle in order to gain a better solution. For example, the solution might not
be accurate enough or only apply in some cases.
Information This stage involves getting the necessary inputs for the mathematical
collection processing that will take place at the next stage. This may involve deciding
which are the important variables, finding key measurements or collecting data.
Processing This stage involves using suitable mathematical techniques, such as
and calculations, graphs or diagrams, in order to make sense of the information
representation collected in the previous stage. This stage ends with a provisional solution to
the problem.
Interpretation This stage of the process involves reporting the solution to the problem in a way
which relates to the original situation. Communication should be in clear plain
English which can be understood by someone who has an interest in the original
problem but is not an expert in mathematics. This should lead into reflection on
the solution to consider whether it is satisfactory or further work is needed.
10 © OCR 2016 © OCR 2016
A Level Further Mathematics B (MEI)