151728 Cambridge Learner Guide for as and a Level Mathematics

Learner GuideCambridge International AS and A LevelMathematics

9709

Cambridge Advanced

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© Cambridge International Examinations 2015

Version 2

Contents

How to use this guide ....................................................................................................... 3Section 1: How will you be tested?Section 2: Examination adviceSection 3: What will be tested?Section 4: What you need to knowSection 5: Useful websites

Section 1: How will you be tested? ..................................................................................... 5About the examination papersAbout the papers

Section 2: Examination advice ............................................................................................. 7How to use this adviceGeneral adviceAdvice for P1, P2 and P3Advice for M1 and M2Advice for S1 and S2

Section 3: What will be tested? ......................................................................................... 11

Section 4: What you need to know ................................................................................... 13How to use the tablePure Mathematics 1: Unit P1Pure Mathematics 2: Unit P2Pure Mathematics 3: Unit P3Mechanics 1: Unit M1Mechanics 2: Unit M2Probability and Statistics 1: Unit S1Probability and Statistics 2: Unit S2

Section 5: Useful websites ................................................................................................34

2 Cambridge International AS and A Level Mathematics 9709

How to use this guide

3Cambridge International AS and A Level Mathematics 9709


This guide describes what you need to know about your Cambridge International AS and A Level Mathematics examination.

It will help you plan your revision programme and it will explain what we are looking for in your answers. It can also be used to help you revise, by using the topic lists in Section 4 to check what you know and which topic areas you have covered.

The guide contains the following sections.

Section 1: How will you be tested?This section will give you information about the different examination papers that are available.

Section 2: Examination adviceThis section gives you advice to help you do as well as you can. Some of the ideas are general advice and some are based on the common mistakes that learners make in exams.

Section 3: What will be tested?This section describes the areas of knowledge, understanding and skills that you will be tested on.

Section 4: What you need to knowThis section shows the syllabus content in a simple way so that you can check:

• what you need to know about each topic;

• how much of the syllabus you have covered.

Not all the information in this checklist will be relevant to you.

You will need to select what you need to know in Sections 1, 2 and 4 by fi nding out from your teacher which examination papers you are taking.

Section 5: Useful websites

Section 1: How will you be tested?



About the examination papersThis syllabus is made up of seven units: three of them are Pure Mathematics units (P1, P2 and P3), two are Mechanics units (M1 and M2) and two are Probability and Statistics units (S1 and S2). Each unit has its own examination paper.

You need to take two units for the Cambridge International AS Level qualifi cation or four units for the full Cambridge International A Level qualifi cation. The table below shows which units are compulsory and which are optional, and shows also the weighting of each unit towards the overall qualifi cation.

You will need to check with your teacher which units you will be taking.

Certifi cation title Compulsory units Optional units

Cambridge International AS Level Mathematics

P1 (60%)

P2 (40%) orM1 (40%) orS1 (40%)

Cambridge International A Level Mathematics

P1 (30%) and P3 (30%)

M1 (20%) and S1 (20%) orM1 (20%) and M2 (20%) orS1 (20%) and S2 (20%)

The following combinations of units are possible.

AS Level

• P1 and P2

• P1 and M1

• P1 and S1

A Level

• P1, P3, M1 and S1

• P1, P3, M1 and M2

• P1, P3, S1 and S2

If you are taking the full A Level qualifi cation, you can take the examination papers for all four units in one session, or alternatively you can take two of them (P1 and M1 or P1 and S1) at an earlier session for an AS qualifi cation and then take your other two units later.

The AS Level combination of units P1 and P2 cannot be used as the fi rst half of a full A Level qualifi cation, so you should not be taking P2 if you intend to do the full Cambridge International A Level Mathematics qualifi cation.



About the papersOnce you have checked with your teacher which units you are doing you can use the table below to fi nd basic information about each unit.

All units are assessed by a written examination, externally set and marked. You must answer all questions and all relevant working must be clearly shown. Each paper will contain both shorter and longer questions, with the questions being arranged approximately in order of increasing mark allocation (i.e. questions with smaller numbers of marks will come earlier in the paper and those with larger numbers of marks will come later in the paper.)

Units P1 and P3 will contain about 10 questions and the other units about 7 questions.

Component Unit name Total marks Duration Qualifi cation use

Paper 1 P1 Pure Mathematics 1

75 1 hour 45 mins AS Level Mathematics A Level Mathematics


50 1 hour 15 mins AS Level Mathematics


75 1 hour 45 mins A Level Mathematics

Paper 4 M1 Mechanics 1


Paper 5 M2 Mechanics 2


Paper 6 S1 Probability and

Statistics 1


Paper 7 S2 Probability and

Statistics 2


Section 2: Examination advice



How to use this adviceThis section highlights some common mistakes made by learners. They are collected under various subheadings to help you when you revise a particular topic or area.

General advice• You should give numerical answers correct to three signifi cant fi gures in questions where no accuracy is

specifi ed, except for angles in degrees when one decimal place accuracy is required.

• To achieve three-fi gure accuracy in your answer you will have to work with at least four fi gures in your working throughout the question. For a calculation with several stages, it is usually best to use all the fi gures that your calculator shows; however you do not need to write all these fi gures down in your working.

• Giving too many signifi cant fi gures in an answer is not usually penalised. However, if the question does specify an accuracy level then you must keep to it for your fi nal answer, and giving too many signifi cant fi gures here will be penalised.

• There are some questions which ask for answers in exact form. In these questions you must not use your calculator to evaluate answers. Exact answers may include fractions, square roots and constants such as p, for example, and you should give them in as simple a form as possible.

• You are expected to use a scientifi c calculator in all examination papers. Computers, graphical calculators and calculators capable of algebraic manipulation are not permitted.

• Make sure you check that your calculator is in degree mode for questions that have angles in degrees, and in radian mode when you require or are given angles in radians.

• You should always show your working, as marks are usually awarded for using a correct method even if you make a mistake somewhere.

• It is particularly important to show all your working in a question where there is a given answer. Marks will not be gained if the examiner is not convinced of the steps in your working out.

• Read questions carefully. Misreading a question can cost you marks.

• Write clearly. Make sure all numbers are clear, for example make sure your ‘1’s don’t look like ‘7’s.

• If you need to change a word or a number, or even a sign (+ to – for example) it is best to cross out and re-write it. Don’t try to write over the top of your previous work. If your alteration is not clear you will not get the marks.

• When an answer is given (i.e. the question says ‘show that…’), it is often because the answer is needed in the next part of the question. So a given answer may mean that you can carry on with a question even if you haven’t been able to obtain the answer yourself. When you need to use a given answer in a later part of a question, you should always use the result exactly as given in the question, even if you have obtained a different answer yourself. You should never continue with your own wrong answer, as this will lead to a further loss of marks.

• Make sure you are familiar with all the standard mathematical notation that is expected for this syllabus. Your teacher will be able to advise you on what is expected.



• Although there is no choice of questions in any of the Cambridge International AS and A Level papers, you do not have to answer the questions in the order they are printed in the question paper. You don’t want to spend time struggling on one question and then not have time for a question that you could have done and gained marks for.

• Check the number of marks for each question/part question. This gives an indication of how long you should be spending. Part questions with only one or two marks should not involve you doing complicated calculations or writing long explanations. You don’t want to spend too long on some questions and then run out of time at the end. A rough guide is that a rate of ’a mark a minute’ would leave you a good amount of time at the end for checking your work and for trying to complete any parts of questions that you hadn’t been able to do at fi rst.

• As long as you are not running out of time, don’t ‘give up’ on a question just because your working or answers are starting to look wrong. There are always marks available for the method used. So even if you have made a mistake somewhere you could still gain method marks.

• Don’t cross out anything until you have replaced it – even if you know it’s not correct there may still be method marks to gain.

• There are no marks available for just stating a method or a formula. The method has to be applied to the particular question, or the formula used by substituting values in.

• Always look to see if your answer is ‘reasonable’. For example if you had a probability answer of 1.2, you would know that you had made a mistake and you would need to go back and check your solution.

• Make sure that you have answered everything that a question asks. Sometimes one sentence asks two things (e.g. ‘Show that … and hence fi nd …’) and it is easy to concentrate only on the fi rst request and forget about the second one.

• Check the formula book before the examination. You must be aware which formulas are given and which ones you will need to learn.

• Make sure you practise lots of past examination papers so that you know the format and the type of questions. You could also time yourself when doing them so that you can judge how quickly you will need to work in the actual examination.

• Presentation of your work is important – don’t present your solutions in the examination in double column format.

• Make sure you know the difference between three signifi cant fi gures and three decimal places, and make sure you round answers rather than truncate them.

Advice for P1, P2 and P3• Make sure you know all the formulas that you need (even ones from Cambridge IGCSE). If you use an

incorrect formula you will score no marks.

• Check to see if your answer is required in exact form. If this is the case in a trigonometry question exact values of sin 60°, etc. will need to be used.

• Make sure you know the exact form for sin 60°, etc. They are not in the formula booklet.

Additional advice for P2 and P3• Calculus questions involving trigonometric functions use values in radians, so make sure your calculator

is in the correct mode if you need to use it.



Advice for M1 and M2• When using a numerical value for ‘g’ you must use 10 m s–2 (unless the question states otherwise).

• Always draw clear force diagrams when appropriate, whether the question asks for them or not.

• Make sure you are familiar with common words and phrases such as ‘initial’, ‘resultant’, and know the difference between ‘mass’ and ‘weight’. Go through some past examination papers and highlight common words and phrases. Make sure you know what they mean.

Advice for S1 and S2• Questions that ask for probabilities must have answers between 0 and 1. If you get an answer greater

than 1 for a probability you should check your working to try to fi nd the mistake.

• Answers for probabilities can often be given as either fractions or decimals. For answers in decimal form, it is important that you know the difference between three signifi cant fi gures and three decimal places. For example 0.03456 to three signifi cant fi gures is 0.0346, but to three decimal places is 0.035. A fi nal answer of 0.035 would not score the accuracy mark as it is not correct to the level of accuracy required.

• Sometimes you may be asked to answer ‘in the context of the question’. This means that you cannot just give a ‘textbook’ defi nition that could apply to any situation; your answer must relate to the situation given in the question. So, for example, you should not just say ‘The events must be independent’, but ‘The scores when the die is thrown must be independent’ or ‘The times taken by the people must be independent of each other’ or whatever it is that the question is about.

• When answering a question about a normal distribution, it is useful to draw a diagram. This can prevent you from making an error; e.g. if you are fi nding a probability, a diagram will indicate whether your answer should be greater than or less than 0.5.

Additional advice for S2• When carrying out a hypothesis test, you should always state the conclusion in the context of the

question. You should not state conclusions in a way that implies that a hypothesis test has proved something; e.g. it is better to say ‘There is evidence that the mean weight of the fruit has increased’ than ‘The test shows that the mean weight of the fruit has increased’.

• When calculating a confi dence interval, it may not always be sensible to apply the usual ‘three signifi cant fi gure’ rule about accuracy. For example, if an interval for a population mean turns out to be (99.974, 100.316), it makes no sense to round each end to 100, and you should give two or three decimal places in a case like this, so that the width of the interval is shown with reasonable accuracy.

Section 3: What will be tested?



The full syllabus, which your teacher will have, says that the assessment covers ‘technique with application’.

This means that you will be tested both on your knowledge of all the mathematical topics, methods, etc. that are set out in the syllabus content, and also on your ability to apply your knowledge in solving problems.

The syllabus also lists fi ve ‘assessment objectives’ that set out the kinds of thing that examination questions will be designed to test.

These assessment objectives indicate that:

• you need to be familiar with all the usual notation and terminology used in your units, as well understanding the mathematical ideas

• you need to work accurately, e.g. in solving equations, etc.

• you may have to decide for yourself how to go about solving a problem, without being told exactly what procedures you need to use

• you may need to use more than one aspect of your knowledge to solve a problem

• your work needs to be clearly and logically set out.

You should ask your teacher if you require more detailed information on this section.

Section 4: What you need to know



This section has a table for each of the seven units. Each table lists the things you may be tested on in the examination for that unit.

How to use the tableYou can use the table throughout your mathematics course to check the topic areas you have covered.

You can also use it as a revision aid. You can go through the list at various stages in the course, and put:

• a RED dot against a topic if you are really unsure

• an ORANGE dot if you are fairly sure of the topic, but need some further practice

• a GREEN dot if you are fully confi dent.

As you progress through the course and prepare for the examination you should be giving yourself more and more green dots!

Remember that we are looking for you to know certain skills and facts and to be able to apply these in different situations.

It is therefore important to learn the facts and practise the skills fi rst, in isolation; but then you need to fi nd more complex questions that use the skills you are learning, so that you can practise applying what you have learnt to solving problems. Working through past examination papers is invaluable here.

You will see that the syllabus has been divided into ‘skills’, ‘knowledge’, and ‘application’. However, you should remember that each skill and knowledge item could be tested in questions requiring use of that particular skill or knowledge in a situation that may not be specifi cally mentioned here.



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ineq

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olve

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g

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and

term

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fu

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how

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a li

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d th

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fi n

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2

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1

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its e

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to c

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m o

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m o

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rogr

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ms

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,(

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abou

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to s

ketc

h gr

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gra

tio

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s.

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fi nite

inte

gral

s

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as re

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entia

tion

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inte

grat

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to s

olve

pro

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volv

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fi ndi

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con

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in

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ar

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us e

quat

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use

of:

|a

| = |b

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2 = b

2

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– a

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mai

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s

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ndi

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ic

and

exp

on

enti

al

fun

ctio

ns

Sol

ve e

quat

ions

of

form

ax =

b a

nd

corr

espo

ndin

g in

equa

litie

sR

elat

ions

hip

betw

een

loga

rithm

s an

d in

dice

s

Law

s of

loga

rithm

s

Defi

niti

on a

nd p

rope

rtie

s of

ex a

nd

ln x

Use

of

law

s, e

.g. i

n so

lvin

g eq

uatio

ns

Use

gra

phs

of e

x and

ln x

Tran

sfor

mat

ion

to li

near

form

, and

us

e of

gra

dien

t an

d in

terc

ept



Pur

e M

athe

mat

ics

2: U

nit

P2

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Trig

on

om

etry

The

sec,

cos

ec a

nd c

ot f

unct

ions

an

d th

eir

rela

tions

hip

to c

os, s

in a

nd

tan

Iden

titie

s:

sec2 θ

= 1

+ t

an2 θ

co

sec2 θ

= 1

+ c

ot2 θ

Exp

ansi

ons

of:

si

n(A

± B

)

co

s(A

± B

)

ta

n(A

± B

)

Form

ulae

for:

si

n 2A

co

s 2A

ta

n 2A

Use

pro

pert

ies

and

grap

hs o

f al

l si

x tr

ig f

unct

ions

for

angl

es o

f an

y m

agni

tude

Use

of

thes

e in

eva

luat

ing

and

sim

plify

ing

expr

essi

ons,

and

in

sol

ving

equ

atio

ns, i

nclu

ding

ex

pres

sing

a s

in θ

+ b

cos

θ in

the

fo

rms

R s

in(θ

± α

) and

R c

os (θ

± α

)

Dif

fere

nti

atio

nD

iffer

entia

te e

x and

ln x

,

sin

x, c

os x

and

tan

x, t

oget

her

with

con

stan

t m

ultip

les,

sum

s,

diff

eren

ces

and

com

posi

tes

Diff

eren

tiate

pro

duct

s an

d qu

otie

nts

Par

amet

ric d

iffer

entia

tion

Impl

icit

diff

eren

tiatio

n

App

licat

ions

of

diff

eren

tiatio

n in

clud

e al

l tho

se in

uni

t P

1



Pur

e M

athe

mat

ics

2: U

nit

P2

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Inte

gra

tio

nIn

tegr

ate:

• eax

+ b

• (a

x +

b)–1

• si

n(ax

+ b

)

• co

s(ax

+ b

)

• se

c2 (ax

+ b

)

Trap

eziu

m r

ule

Car

ry o

ut in

tegr

atio

ns u

sing

ap

prop

riate

trig

iden

titie

s

App

licat

ions

of

inte

grat

ion

incl

ude

all

thos

e in

uni

t P

1

Use

of

sket

ch g

raph

s to

iden

tify

over

/und

er e

stim

atio

n

Nu

mer

ical

so

luti

on

of

equ

atio

ns

Loca

te ro

ot g

raph

ical

ly o

r by

sig

n ch

ange

Car

ry o

ut it

erat

ion

x n +

1 =

F(x

n)

Idea

of

sequ

ence

of

appr

oxim

atio

ns

whi

ch c

onve

rge

to a

root

of

an

equa

tion,

and

not

atio

n fo

r th

is

Und

erst

and

rela

tion

betw

een

itera

tive

form

ula

and

equa

tion

bein

g so

lved

Find

app

roxi

mat

e ro

ots

to a

giv

en

degr

ee o

f ac

cura

cy



Pur

e M

athe

mat

ics

3: U

nit

P3

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Ass

um

ed

kno

wle

dg

eC

onte

nt o

f un

it P

1 is

ass

umed

, and

m

ay b

e re

quire

d in

sol

ving

pro

blem

s on

P3

topi

cs

Alg

ebra

Sol

ve m

odul

us e

quat

ions

and

in

equa

litie

s, in

clud

ing

use

of:

|a

| = |b

| ⇔ a

2 = b

2

|x

– a

| < b

⇔ a

– b

< x

< a

+ b

Car

ry o

ut a

lgeb

raic

pol

ynom

ial

divi

sion

Find

par

tial f

ract

ions

Mea

ning

of

|x|

Mea

ning

of

quot

ient

and

rem

aind

er

Fact

or a

nd re

mai

nder

the

orem

s

Kno

w a

ppro

pria

te fo

rms

of p

artia

l fr

actio

ns fo

r de

nom

inat

ors:

(a

x +

b)(c

x +

d)(e

x +

f)

(a

x +

b)(c

x +

d)2

(a

x +

b)(x

2 + c

2 )

Exp

ansi

on o

f (1

+ x

)n for

rat

iona

l n

and

|x| <

1

Use

of

theo

rem

s in

fi nd

ing

fact

ors,

so

lvin

g po

lyno

mia

l equ

atio

ns, fi

ndi

ng

unkn

own

coef

fi cie

nts

etc

Use

of

fi rst

few

term

s, e

.g. f

or

appr

oxim

atio

ns

Dea

ling

with

(a +

bx)

n

Log

arit

hm

ic

and

exp

on

enti

al

fun

ctio

ns

Sol

ve e

quat

ions

of

form

ax =

b a

nd

corr

espo

ndin

g in

equa

litie

sR

elat

ions

hip

betw

een

loga

rithm

s an

d in

dice

s

Law

s of

loga

rithm

s

Defi

niti

on a

nd p

rope

rtie

s of

ex a

nd

ln x

Use

of

law

s, e

.g. i

n so

lvin

g eq

uatio

ns

Use

gra

phs

of e

x and

ln x

Tran

sfor

mat

ion

to li

near

form

, and

us

e of

gra

dien

t an

d in

terc

ept



Pur

e M

athe

mat

ics

3: U

nit

P3

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Trig

on

om

etry

The

sec,

cos

ec a

nd c

ot f

unct

ions

an

d th

eir

rela

tions

hip

to c

os, s

in a

nd

tan

Iden

titie

s:

sec2 θ

= 1

+ t

an2 θ

co

sec2 θ

= 1

+ c

ot2 θ

Exp

ansi

ons

of:

si

n(A

± B

)

co

s(A

± B

)

ta

n(A

± B

)

Form

ulae

for:

si

n 2A

co

s 2A

ta

n 2A

Use

pro

pert

ies

and

grap

hs o

f al

l si

x tr

ig f

unct

ions

for

angl

es o

f an

y m

agni

tude

Use

of

thes

e in

eva

luat

ing

and

sim

plify

ing

expr

essi

ons,

and

in

sol

ving

equ

atio

ns, i

nclu

ding

ex

pres

sing

a s

in θ

+ b

cos

θ in

the

fo

rms

R s

in(θ

± α

) and

R c

os(θ

± α

)

Dif

fere

nti

atio

nD

iffer

entia

te e

x and

ln x

, sin

x, c

os x

an

d ta

n x,

toge

ther

with

con

stan

t m

ultip

les,

sum

s, d

iffer

ence

s an

d co

mpo

site

s

Diff

eren

tiate

pro

duct

s an

d qu

otie

nts

Par

amet

ric d

iffer

entia

tion

Impl

icit

diff

eren

tiatio

n

App

licat

ions

of

diff

eren

tiatio

n in

clud

e al

l tho

se in

uni

t P

1



Pur

e M

athe

mat

ics

3: U

nit

P3

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Inte

gra

tio

nIn

tegr

ate:

• eax

+ b

• (a

x +

b)–1

• si

n(ax

+ b

)

• co

s(ax

+ b

)

• se

c2 (ax

+ b

)

Inte

grat

e by

mea

ns o

f de

com

posi

tion

into

par

tial f

ract

ions

Rec

ogni

se a

nd in

tegr

ate

()()

ff xk

xl

Inte

grat

ion

by p

arts

Inte

grat

ion

by s

ubst

itutio

n

Trap

eziu

m r

ule

Car

ry o

ut in

tegr

atio

ns u

sing

:•

appr

opria

te t

rig id

entit

ies

• pa

rtia

l fra

ctio

ns

App

licat

ions

of

inte

grat

ion

incl

ude

all

thos

e in

uni

t P

1

Use

of

sket

ch g

raph

s to

iden

tify

over

/und

er e

stim

atio

n

Nu

mer

ical

so

luti

on

of

equ

atio

ns

Loca

te ro

ot g

raph

ical

ly o

r by

sig

n ch

ange

Car

ry o

ut it

erat

ion

x n +

1 =

F(x

n)

Idea

of

sequ

ence

of

appr

oxim

atio

ns

whi

ch c

onve

rge

to a

root

of

an

equa

tion,

and

not

atio

n fo

r th

is

Und

erst

and

rela

tion

betw

een

itera

tive

form

ula

and

equa

tion

bein

g so

lved

Find

app

roxi

mat

e ro

ots

to a

giv

en

degr

ee o

f ac

cura

cy



Pur

e M

athe

mat

ics

3: U

nit

P3

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Vec

tors

Find

:•

the

angl

e be

twee

n tw

o lin

es

• th

e po

int

of in

ters

ectio

n of

tw

o lin

es w

hen

it ex

ists

• th

e ve

ctor

equ

atio

n of

a li

ne

• th

e pe

rpen

dicu

lar

dist

ance

fro

m

a po

int

to a

line

Find

:th

e eq

uatio

n of

a p

lane

the

angl

e be

twee

n tw

o pl

anes

the

perp

endi

cula

r di

stan

ce f

rom

a

poin

t to

a p

lane

Und

erst

and

r =

a +

tb

as

the

equa

tion

of a

str

aigh

t lin

e

Und

erst

and

ax +

by

+ c

z =

d a

nd

(r –

a).n

= 0

as

the

equa

tion

of a

pl

ane

Det

erm

ine

whe

ther

tw

o lin

es a

re

para

llel,

inte

rsec

t or

are

ske

w

Use

equ

atio

ns o

f lin

es a

nd p

lane

s to

so

lve

prob

lem

s, in

clud

ing:

• fi n

ding

the

ang

le b

etw

een

a lin

e an

d a

plan

e

• de

term

inin

g w

heth

er a

line

lies

in

a p

lane

, is

para

llel t

o it,

or

inte

rsec

ts it

• fi n

ding

the

poi

nt o

f in

ters

ectio

n of

a li

ne a

nd a

pla

ne w

hen

it ex

ists

• fi n

ding

the

line

of

inte

rsec

tion

of

two

non-

para

llel p

lane

s

Dif

fere

nti

al

equ

atio

ns

Form

a d

iffer

entia

l equ

atio

n fr

om

info

rmat

ion

abou

t a

rate

of

chan

ge

Sol

ve a

fi rs

t or

der

diff

eren

tial

equa

tion

by s

epar

atin

g va

riabl

es

Gen

eral

and

par

ticul

ar s

olut

ions

Use

initi

al c

ondi

tions

and

inte

rpre

t so

lutio

ns in

con

text

Co

mp

lex

nu

mb

ers

Add

, sub

trac

t, m

ultip

ly a

nd d

ivid

e tw

o co

mpl

ex n

umbe

rs in

car

tesi

an

form

Mul

tiply

and

div

ide

two

com

plex

nu

mbe

rs in

pol

ar fo

rm

Find

the

tw

o sq

uare

root

s of

a

com

plex

num

ber

Illus

trat

e eq

uatio

ns a

nd in

equa

litie

s as

loci

in a

n A

rgan

d di

agra

m

Mea

ning

of

term

s:

real

and

imag

inar

y pa

rts

m

odul

us a

nd a

rgum

ent

co

njug

ate

ca

rtes

ian

and

pola

r fo

rms

A

rgan

d di

agra

m

Equa

lity

of c

ompl

ex n

umbe

rs

Roo

ts in

con

juga

te p

airs

Geo

met

rical

inte

rpre

tatio

n of

co

njug

atio

n, a

dditi

on, s

ubtr

actio

n,

mul

tiplic

atio

n an

d di

visi

on



Mec

hani

cs 1

: Uni

t M

1To

pic

Ski

llK

no

wle

dg

eA

pp

licat

ion

Ch

eckl

ist

Trig

on

om

etry

kn

ow

led

ge

req

uir

ed

sin(

90°

– θ)

≡ c

os θ

cos(

90°

– θ)

≡ s

in θ

tan θ

≡ co

ssi

nii

sin2 θ

+ c

os2 θ

≡ 1

Forc

es a

nd

eq

uili

bri

um

Iden

tify

forc

es a

ctin

g

Find

and

use

com

pone

nts

and

resu

ltant

s

Equi

libriu

m o

f fo

rces

Use

‘sm

ooth

’ con

tact

mod

el

Forc

es a

s ve

ctor

s

Mea

ning

of

term

s:

cont

act

forc

e

no

rmal

com

pone

nt

fr

ictio

nal c

ompo

nent

lim

iting

fric

tion

lim

iting

equ

ilibr

ium

co

effi c

ient

of

fric

tion

sm

ooth

con

tact

New

ton’

s th

ird la

w

Use

con

ditio

ns fo

r eq

uilib

rium

in

prob

lem

s in

volv

ing

fi ndi

ng fo

rces

, an

gles

, etc

.

Lim

itatio

ns o

f ‘s

moo

th’ c

onta

ct

mod

el

Use

F =

μR

and

F <

μR

as

appr

opria

te in

sol

ving

pro

blem

s in

volv

ing

fric

tion

Use

thi

s la

w in

sol

ving

pro

blem

s



Mec

hani

cs 1

: Uni

t M

1To

pic

Ski

llK

no

wle

dg

eA

pp

licat

ion

Ch

eckl

ist

Kin

emat

ics

of

mo

tio

n in

a

stra

igh

t lin

e

Ske

tch

disp

lace

men

t-tim

e an

d ve

loci

ty-t

ime

grap

hsD

ista

nce

and

spee

d as

sca

lars

, and

di

spla

cem

ent,

vel

ocity

, acc

eler

atio

n as

vec

tors

(in

one

dim

ensi

on)

Velo

city

as

rate

of

chan

ge o

f di

spla

cem

ent,

and

acc

eler

atio

n as

ra

te o

f ch

ange

of

velo

city

Form

ulae

for

mot

ion

with

con

stan

t ac

cele

ratio

n

Use

of

posi

tive

and

nega

tive

dire

ctio

ns fo

r di

spla

cem

ent,

vel

ocity

an

d ac

cele

ratio

n

Inte

rpre

t gr

aphs

and

use

in s

olvi

ng

prob

lem

s th

e fa

cts

that

:

area

und

er v

-t g

raph

repr

esen

ts

disp

lace

men

t

gr

adie

nt o

f s-

t gra

ph re

pres

ents

ve

loci

ty

gr

adie

nt o

f v-

t gra

ph re

pres

ents

ac

cele

ratio

n

Use

diff

eren

tiatio

n an

d in

tegr

atio

n to

sol

ve p

robl

ems

invo

lvin

g di

spla

cem

ent,

vel

ocity

and

ac

cele

ratio

n

Use

sta

ndar

d S

UVA

T fo

rmul

ae

in p

robl

ems

invo

lvin

g m

otio

n in

a s

trai

ght

line

with

con

stan

t ac

cele

ratio

n



Mec

hani

cs 1

: Uni

t M

1To

pic

Ski

llK

no

wle

dg

eA

pp

licat

ion

Ch

eckl

ist

New

ton

’s la

ws

of

mo

tio

nN

ewto

n’s

seco

nd la

w

Mea

ning

of

term

s:

mas

s

w

eigh

t

Use

New

ton’

s la

ws

in p

robl

ems

invo

lvin

g m

otio

n in

a s

trai

ght

line

unde

r co

nsta

nt fo

rces

Use

the

rela

tions

hip

betw

een

mas

s an

d w

eigh

t

Sol

ve c

onst

ant

acce

lera

tion

prob

lem

s in

volv

ing

wei

ght:

• pa

rtic

le m

ovin

g ve

rtic

ally

• pa

rtic

le m

ovin

g on

an

incl

ined

pl

ane

• tw

o pa

rtic

les

conn

ecte

d by

a

light

str

ing

pass

ing

over

a

smoo

th p

ulle

y

En

erg

y, w

ork

an

d p

ow

erC

alcu

late

wor

k do

ne b

y co

nsta

nt

forc

eC

once

pts

of g

ravi

tatio

nal p

oten

tial

ener

gy a

nd k

inet

ic e

nerg

y, a

nd

form

ulae

mgh

and

½m

v 2

Pow

er a

s ra

te o

f w

orki

ng P

= F

v

Sol

ve p

robl

ems

invo

lvin

g th

e w

ork-

ener

gy p

rinci

ple,

and

use

co

nser

vatio

n of

ene

rgy

whe

re

appr

opria

te

Use

the

rela

tions

hip

betw

een

pow

er,

forc

e an

d ve

loci

ty w

ith N

ewto

n’s

seco

nd la

w to

sol

ve p

robl

ems

abou

t ac

cele

ratio

n et

c.



Mec

hani

cs 2

: Uni

t M

2To

pic

Ski

llK

no

wle

dg

wA

Ap

plic

atio

nC

hec

klis

t

Ass

um

ed

kno

wle

dg

eC

onte

nt o

f un

it M

1 is

ass

umed

, and

m

ay b

e re

quire

d in

sol

ving

pro

blem

s on

M2

topi

cs

Mo

tio

n o

f a

pro

ject

ileU

se h

oriz

onta

l and

ver

tical

equ

atio

ns

of m

otio

n

Der

ive

the

cart

esia

n eq

uatio

n of

the

tr

ajec

tory

Con

stan

t ac

cele

ratio

n m

odel

Lim

itatio

ns o

f co

nsta

nt a

ccel

erat

ion

mod

el

Sol

ve p

robl

ems,

e.g

. fi n

ding

:

the

velo

city

at

a gi

ven

poin

t or

in

stan

t

th

e ra

nge

th

e gr

eate

st h

eigh

t

Use

the

tra

ject

ory

equa

tion

to s

olve

pr

oble

ms,

incl

udin

g fi n

ding

the

initi

al

velo

city

and

ang

le o

f pr

ojec

tion

Eq

uili

bri

um

of

a ri

gid

bo

dy

Loca

te c

entr

e of

mas

s of

a s

ingl

e un

iform

bod

y:

usin

g sy

mm

etry

us

ing

data

fro

m fo

rmul

a lis

t

Cal

cula

te t

he m

omen

t of

a fo

rce

Loca

te c

entr

e of

mas

s of

a

com

posi

te b

ody

usin

g m

omen

ts

Con

cept

of

cent

re o

f m

ass

Con

ditio

ns fo

r eq

uilib

rium

Sol

ve e

quili

briu

m p

robl

ems,

in

clud

ing

topp

ling/

slid

ing

Un

ifo

rm m

oti

on

in

a c

ircl

eC

once

pt o

f an

gula

r sp

eed

v =

rω

Acc

eler

atio

n is

tow

ards

cen

tre

Use

of

ω 2 r

or

v 2/r

as a

ppro

pria

te in

so

lvin

g pr

oble

ms

abou

t a

part

icle

m

ovin

g w

ith c

onst

ant

spee

d in

a

horiz

onta

l circ

le



Mec

hani

cs 2

: Uni

t M

2To

pic

Ski

llK

no

wle

dg

wA

Ap

plic

atio

nC

hec

klis

t

Ho

oke

’s la

wM

eani

ng o

f te

rms:

m

odul

us o

f el

astic

ity

el

astic

pot

entia

l ene

rgy

Use

of T

lxm

= a

nd E

lx 22

m=

in s

olvi

ng

prob

lem

s in

volv

ing

elas

tic s

trin

g or

spr

ings

, inc

ludi

ng t

hose

whe

re

cons

ider

atio

ns o

f w

ork

and

ener

gy

are

need

ed

Lin

ear

mo

tio

n

un

der

a v

aria

ble

fo

rce

Use

ddtv

or

ddv

xv a

s ap

prop

riate

in

appl

ying

New

ton’

s se

cond

law

ddv

tx=

and

dd

dda

tvv

xv=

=S

et u

p an

d so

lve

diff

eren

tial

equa

tions

for

prob

lem

s w

here

a

part

icle

mov

es u

nder

the

act

ion

of

varia

ble

forc

es



Pro

babi

lity

and

Stat

istic

s 1:

Uni

t S1

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Rep

rese

nta

tio

n

of

dat

aS

elec

t su

itabl

e w

ays

of p

rese

ntin

g ra

w d

ata

Con

stru

ct a

nd in

terp

ret:

• st

em a

nd le

af d

iagr

am

• bo

x &

whi

sker

plo

ts

• hi

stog

ram

s

• cu

mul

ativ

e fr

eque

ncy

grap

hs

Est

imat

e m

edia

n an

d qu

artil

es f

rom

cu

mul

ativ

e fr

eque

ncy

grap

h

Cal

cula

te m

ean

and

stan

dard

de

viat

ion

usin

g:•

indi

vidu

al d

ata

item

s

• gr

oupe

d da

ta

• gi

ven

tota

ls Σ

x an

d Σx

2

• gi

ven

Σ(x

– a

) and

Σ(x

– a

)2

Mea

sure

s of

cen

tral

tend

ency

:

mea

n, m

edia

n, m

ode

Mea

sure

s of

var

iatio

n:•

rang

e

• in

terq

uart

ile r

ange

• st

anda

rd d

evia

tion

Dis

cuss

adv

anta

ges

and/

or d

isad

vant

ages

of

part

icul

ar

repr

esen

tatio

ns

Cal

cula

te a

nd u

se t

hese

mea

sure

s,

e.g.

to c

ompa

re a

nd c

ontr

ast

data

Sol

ve p

robl

ems

invo

lvin

g m

eans

and

st

anda

rd d

evia

tions

Per

mu

tati

on

s an

d

com

bin

atio

ns

Mea

ning

of

term

s:•

perm

utat

ion

• co

mbi

natio

n

• se

lect

ion

• ar

rang

emen

t

Sol

ve p

robl

ems

invo

lvin

g se

lect

ions

Sol

ve p

robl

ems

abou

t ar

rang

emen

ts

in a

line

incl

udin

g th

ose

with

:•

repe

titio

n

• re

stric

tion



Pro

babi

lity

and

Stat

istic

s 1:

Uni

t S1

Top

icS

kill

Kn

ow

led

ge

Ap

plic

atio

nC

hec

klis

t

Pro

bab

ility

Eva

luat

e pr

obab

ilitie

s by

:•

coun

ting

even

ts in

the

sam

ple

spac

e

• ca

lcul

atio

n us

ing

perm

utat

ions

an

d co

mbi

natio

ns

Use

add

ition

and

mul

tiplic

atio

n of

pr

obab

ilitie

s as

app

ropr

iate

Cal

cula

te c

ondi

tiona

l pro

babi

litie

s

Mea

ning

of

term

s:•

excl

usiv

e ev

ents

• in

depe

nden

t ev

ents

• co

nditi

onal

pro

babi

lity

Sol

ve p

robl

ems

invo

lvin

g co

nditi

onal

pr

obab

ilitie

s, e

.g. u

sing

tre

e di

agra

ms

Dis

cret

e ra

nd

om

va

riab

les

Con

stru

ct a

pro

babi

lity

dist

ribut

ion

tabl

e

Cal

cula

te E

(X) a

nd V

ar(X

)

For

the

bino

mia

l dis

trib

utio

n:•

calc

ulat

e pr

obab

ilitie

s

• us

e fo

rmul

ae n

p an

d np

q

Not

atio

n B

(n, p

)R

ecog

nise

situ

atio

ns w

here

the

bi

nom

ial d

istr

ibut

ion

is a

sui

tabl

e m

odel

, and

sol

ve p

robl

ems

invo

lvin

g bi

nom

ial p

roba

bilit

ies

Th

e n

orm

al

dis

trib

uti

on

Use

nor

mal

dis

trib

utio

n ta

bles

Idea

of

cont

inuo

us r

ando

m v

aria

ble,

ge

nera

l sha

pe o

f no

rmal

cur

ve a

nd

nota

tion

N(µ

, σ2 )

Con

ditio

n fo

r B

(n, p

) to

be

appr

oxim

ated

by

N(n

p, n

pq)

Use

of

the

norm

al d

istr

ibut

ion

as a

m

odel

Sol

ve p

robl

ems

invo

lvin

g a

norm

al

dist

ribut

ion,

incl

udin

g:

fi ndi

ng p

roba

bilit

ies

fi n

ding

µ a

nd/o

r σ

Sol

ve p

robl

ems

invo

lvin

g th

e us

e of

a

norm

al d

istr

ibut

ion,

with

con

tinui

ty

corr

ectio

n, to

app

roxi

mat

e a

bino

mia

l di

strib

utio

n



Pro

babi

lity

and

Stat

istic

s 2:

Uni

t S

2To

pic

Ski

llK

no

wle

dg

eA

pp

licat

ion

Ch

eckl

ist

Ass

um

ed

kno

wle

dg

eC

onte

nt o

f un

it S1

is a

ssum

ed, a

nd

may

be

requ

ired

in s

olvi

ng p

robl

ems

on S

2 to

pics

Th

e P

ois

son

d

istr

ibu

tio

nC

alcu

late

Poi

sson

pro

babi

litie

sN

otat

ion

Po(µ)

Mea

n an

d va

rianc

e of

Po(µ)

Con

ditio

ns fo

r ra

ndom

eve

nts

to

have

a P

oiss

on d

istr

ibut

ion

Con

ditio

ns fo

r B

(n, p

) to

be

appr

oxim

ated

by

Po(

np)

Con

ditio

ns fo

r P

o(µ)

to b

e ap

prox

imat

ed b

y N

(µ, µ

)

Use

the

Poi

sson

dis

trib

utio

n as

a

mod

el, a

nd s

olve

pro

blem

s in

volv

ing

Poi

sson

pro

babi

litie

s

Sol

ve p

robl

ems

whi

ch in

volv

e ap

prox

imat

ing

bino

mia

l pro

babi

litie

s us

ing

a P

oiss

on d

istr

ibut

ion

and/

or

appr

oxim

atin

g P

oiss

on p

roba

bilit

ies

usin

g a

norm

al d

istr

ibut

ion

with

co

ntin

uity

cor

rect

ion



Pro

babi

lity

and

Stat

istic

s 2:

Uni

t S

2To

pic

Ski

llK

no

wle

dg

eA

pp

licat

ion

Ch

eckl

ist

Lin

ear

com

bin

atio

ns

of

ran

do

m

vari

able

s

For

a ra

ndom

var

iabl

e X

:

E(aX

+ b

) = a

E(X

) + b

Va

r(aX

+ b

) = a

2 Var

(X)

For

rand

om v

aria

bles

X a

nd Y

:

E(aX

+ b

Y) =

aE(

X) +

bE(

Y)

For

inde

pend

ent

X an

d Y

:

Var(

aX +

bY

) = a

2 Var

(X) +

b2 V

ar(Y

)

For

a no

rmal

var

iabl

e X

:

aX +

b h

as a

nor

mal

dis

trib

utio

n

For

inde

pend

ent

norm

al v

aria

bles

X

and

Y:

aX

+ b

Y ha

s a

norm

al d

istr

ibut

ion

For

inde

pend

ent

Poi

sson

var

iabl

es X

an

d Y

:

X +

Y h

as a

Poi

sson

dis

trib

utio

n

Sol

ve p

robl

ems

usin

g re

sults

abo

ut

com

bina

tions

of

rand

om v

aria

bles

Co

nti

nu

ou

s ra

nd

om

va

riab

les

Use

a d

ensi

ty f

unct

ion

to fi

nd:

pr

obab

ilitie

s

th

e m

ean

th

e va

rianc

e

Und

erst

and

the

conc

ept

of a

co

ntin

uous

ran

dom

var

iabl

e.

Pro

pert

ies

of a

den

sity

fun

ctio

n:

alw

ays

non-

nega

tive

to

tal a

rea

is 1

Sol

ve p

robl

ems

usin

g de

nsity

fu

nctio

n pr

oper

ties



Pro

babi

lity

and

Stat

istic

s 2:

Uni

t S

2To

pic

Ski

llK

no

wle

dg

eA

pp

licat

ion

Ch

eckl

ist

Sam

plin

g a

nd

es

tim

atio

nC

alcu

late

unb

iase

d es

timat

es o

f po

pula

tion

mea

n an

d va

rianc

e

Det

erm

ine

a co

nfi d

ence

inte

rval

for

a po

pula

tion

mea

n (n

orm

al p

opul

atio

n or

larg

e sa

mpl

e)

Det

erm

ine

a co

nfi d

ence

inte

rval

for

a po

pula

tion

prop

ortio

n (la

rge

sam

ple)

Mea

ning

of

term

s:

sam

ple

po

pula

tion

Nee

d fo

r ra

ndom

ness

in s

ampl

ing

Use

of

rand

om n

umbe

rs fo

r sa

mpl

ing

Idea

of

sam

ple

mea

n as

a r

ando

m

varia

ble

Dis

trib

utio

n of

sam

ple

mea

n fr

om

a no

rmal

pop

ulat

ion,

and

for

a la

rge

sam

ple

from

any

pop

ulat

ion

Exp

lain

why

a g

iven

sam

plin

g m

etho

d m

ay b

e un

satis

fact

ory

Sol

ve p

robl

ems

invo

lvin

g th

e us

e of

(

,/

)X

Nn

2+

nv

, inc

ludi

ng a

ppro

pria

te

use

of t

he C

entr

al L

imit

theo

rem

Inte

rpre

t co

nfi d

ence

inte

rval

s an

d so

lve

prob

lem

s in

volv

ing

confi

den

ce

inte

rval

s

Hyp

oth

esis

te

sts

Form

ulat

e hy

poth

eses

for

a te

st

Car

ry o

ut a

test

of

the

valu

e of

p in

a

bino

mia

l dis

trib

utio

n, u

sing

eith

er

bino

mia

l pro

babi

litie

s di

rect

ly o

r a

norm

al a

ppro

xim

atio

n, a

s ap

prop

riate

Car

ry o

ut a

test

of

the

mea

n of

a

Poi

sson

dis

trib

utio

n, u

sing

eith

er

Poi

sson

pro

babi

litie

s di

rect

ly o

r a

norm

al a

ppro

xim

atio

n, a

s ap

prop

riate

Car

ry o

ut a

test

of

a po

pula

tion

mea

n w

here

:•

the

popu

latio

n is

nor

mal

with

kn

own

varia

nce

• th

e sa

mpl

e si

ze is

larg

e

Cal

cula

te p

roba

bilit

ies

of T

ype

I and

Ty

pe II

err

ors

in t

he a

bove

test

s

Con

cept

of

a hy

poth

esis

test

Mea

ning

of

term

s:•

one-

tail

and

two

-tai

l

• nu

ll an

d al

tern

ativ

e hy

poth

eses

• si

gnifi

canc

e le

vel

• re

ject

ion

(or

criti

cal)

regi

on

• ac

cept

ance

regi

on

• te

st s

tatis

tic

• Ty

pe I

and

Type

II e

rror

s

Rel

ate

the

resu

lts o

f te

sts

to t

he

cont

ext

and

inte

rpre

t Ty

pe I

and

Type

II e

rror

pro

babi

litie

s




The websites listed below are useful resources to help you study for your Cambridge International AS and A Level Mathematics. The sites are not designed specifi cally for the 9709 syllabus, but the content is generally of the appropriate standard and is mostly suitable for your course.

www.s-cool.co.uk/a-level/mathsThis site covers some Pure Maths and Statistics topics, but not Mechanics. Revision material is arranged by topics, and includes explanations, revision summaries and exam-style questions with answers.

www.examsolutions.co.ukThis site has video tutorials on topics in Pure Maths, Mechanics and Statistics. There are also other resources, including examination questions (from UK syllabuses) with videos of worked solutions.

www.bbc.co.uk/bitesize/higher/mathsThis site has revision material and test questions covering some Pure Maths topics only.

www.cimt.plymouth.ac.ukThis site contains resources, projects, and publications, and is intended for both learners and teachers. The main items relevant to AS and A Level revision are the course materials and the interactive ‘A-level Audits’, which you can access from the ‘Resources’ option; these both cover Pure Maths, Mechanics and Statistics. The course materials are textbook style notes on selected topics, and can be found by following the Mathematics Enhancement Programme (MEP) link. The ‘audits’ are online tests containing questions of progressive diffi culty, and can be found by following the Test and Audits link.

Cambridge International Examinations1 Hills Road, Cambridge, CB1 2EU, United KingdomTel: +44 (0)1223 553554 Fax: +44 (0)1223 553558Email: [email protected] www.cie.org.uk

© Cambridge International Examinations 2015

Version 2

151728 Cambridge Learner Guide for as and a Level Mathematics

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Transcript of 151728 Cambridge Learner Guide for as and a Level Mathematics