9709 mathematics learner guide 2015 levels/mathematics (9709... · 2016. 2. 5. · units p1 and p3...
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Learner GuideCambridge International AS and A LevelMathematics
9709
Cambridge Advanced
Cambridge International Examinations retains the copyright on all its publications. Registered Centres are permitted to copy material from this booklet for their own internal use. However, we cannot give permission to Centres to photocopy any material that is acknowledged to a third party even for internal use within a Centre.
© 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
How to use this guide
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
How to use this guide
4 Cambridge International AS and A Level Mathematics 9709
Section 1: How will you be tested?
5Cambridge International AS and A Level Mathematics 9709
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.
Section 1: How will you be tested?
6 Cambridge International AS and A Level Mathematics 9709
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
Paper 2 P2 Pure Mathematics 2
50 1 hour 15 mins AS Level Mathematics
Paper 3 P3 Pure Mathematics 3
75 1 hour 45 mins A Level Mathematics
Paper 4 M1 Mechanics 1
50 1 hour 15 mins AS Level Mathematics A Level Mathematics
Paper 5 M2 Mechanics 2
50 1 hour 15 mins A Level Mathematics
Paper 6 S1 Probability and
Statistics 1
50 1 hour 15 mins AS Level Mathematics A Level Mathematics
Paper 7 S2 Probability and
Statistics 2
50 1 hour 15 mins A Level Mathematics
Section 2: Examination advice
7Cambridge International AS and A Level Mathematics 9709
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.
Section 2: Examination advice
8 Cambridge International AS and A Level Mathematics 9709
• 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.
Section 2: Examination advice
9Cambridge International AS and A Level Mathematics 9709
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 2: Examination advice
10 Cambridge International AS and A Level Mathematics 9709
Section 3: What will be tested?
11Cambridge International AS and A Level Mathematics 9709
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 3: What will be tested?
12 Cambridge International AS and A Level Mathematics 9709
Section 4: What you need to know
13Cambridge International AS and A Level Mathematics 9709
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.
Section 4: What you need to know
14 Cambridge International AS and A Level Mathematics 9709
Pur
e M
athe
mat
ics
1: U
nit
P1To
pic
Ski
llK
no
wle
dg
eA
pp
licat
ion
Ch
eckl
ist
Qu
adra
tics
Com
plet
e th
e sq
uare
Find
dis
crim
inan
t
Sol
ve q
uadr
atic
equ
atio
ns
Sol
ve li
near
and
qua
drat
ic
ineq
ualit
ies
Sol
ve b
y su
bstit
utio
n a
pair
of
sim
ulta
neou
s eq
uatio
ns o
ne li
near
an
d on
e qu
adra
tic
Loca
te v
erte
x of
gra
ph
Ske
tch
grap
h of
qua
drat
ic
Use
in c
onne
ctio
n w
ith t
he n
umbe
r of
real
root
s
Rec
ogni
se a
nd s
olve
equ
atio
ns t
hat
can
be p
ut in
to q
uadr
atic
form
Sol
ve p
robl
ems,
e.g
. inv
olvi
ng t
he
inte
rsec
tion
of a
line
and
a c
urve
Fun
ctio
ns
Find
ran
ge o
f a
give
n fu
nctio
n
Find
fg(x
) for
giv
en f
and
g
Iden
tify
one-
one
func
tions
Find
inve
rse
of o
ne-o
ne f
unct
ion
Und
erst
and
term
s:
fu
nctio
n, d
omai
n an
d ra
nge
on
e-on
e fu
nctio
n, in
vers
e fu
nctio
n
co
mpo
sitio
n of
fun
ctio
ns
Und
erst
and
how
the
gra
phs
of a
on
e-on
e fu
nctio
n an
d its
inve
rse
are
rela
ted
Ske
tch
grap
hs o
f a
one-
one
func
tion
and
its in
vers
e
Res
tric
t a
dom
ain
to e
nsur
e th
at a
fu
nctio
n is
one
-one
Co
ord
inat
e g
eom
etry
Giv
en e
nd p
oint
s of
a li
ne:
fi n
d th
e le
ngth
fi n
d th
e gr
adie
nt
fi n
d th
e m
id-p
oint
Find
equ
atio
n of
a li
ne, e
.g. u
sing
2
poin
ts o
n it,
or
1 po
int
and
the
grad
ient
Gra
dien
ts o
f pa
ralle
l lin
es a
re e
qual
Per
pend
icul
ar g
radi
ents
: m1m
2 = –
1
Kno
w t
he fo
rms
y =
mx
+ c
and
y –
y 1
= m
(x –
x1)
Und
erst
and
the
rela
tions
hip
betw
een
a gr
aph
and
its e
quat
ion
Use
the
se re
latio
nshi
ps in
sol
ving
pr
oble
ms
Inte
rpre
t an
d us
e lin
ear
equa
tions
Sol
ve p
robl
ems
that
rela
te p
oint
s of
inte
rsec
tion
of g
raph
s to
so
lutio
n of
equ
atio
ns (i
nclu
ding
the
co
rres
pond
ence
bet
wee
n a
line
bein
g a
tang
ent
to c
urve
and
an
equa
tion
havi
ng a
repe
ated
root
)
Section 4: What you need to know
15Cambridge International AS and A Level Mathematics 9709
Pur
e M
athe
mat
ics
1: U
nit
P1To
pic
Ski
llK
no
wle
dg
eA
pp
licat
ion
Ch
eckl
ist
Cir
cula
r m
easu
reC
onve
rt b
etw
een
radi
ans
and
degr
ees
Use
form
ulae
s =
r θ
and
A =
½r 2
θ to
ca
lcul
ate
arc
leng
th a
nd s
ecto
r ar
ea
Defi
niti
on o
f a
radi
anS
olve
pro
blem
s in
volv
ing
arc
leng
ths
and
area
s of
sec
tors
and
seg
men
ts
Trig
on
om
etry
Ske
tch
and
reco
gnis
e gr
aphs
of
the
sine
, cos
ine
and
tang
ent
func
tions
(a
ngle
s of
any
siz
e, d
egre
es o
r ra
dian
s)
Sol
ve t
rig e
quat
ions
giv
ing
all
solu
tions
in s
peci
fi ed
inte
rval
Kno
w e
xact
val
ues
for
sin,
cos
, tan
of
30º
, 45º
, 60º
Und
erst
and
nota
tion
sin–1
x, c
os–1
x,
tan–1
x as
den
otin
g pr
inci
pal v
alue
s
Iden
titie
s:
co
ssi
nta
nii
i=
si
n2 θ
+ c
os2 θ
= 1
Use
trig
onom
etric
gra
phs
Use
exa
ct v
alue
s of
rela
ted
angl
es
e.g.
cos
150
º
Use
the
se id
entit
ies,
e.g
. to
prov
e ot
her
iden
titie
s or
to s
olve
equ
atio
ns
Vec
tors
Add
and
sub
trac
t ve
ctor
s, a
nd
mul
tiply
a v
ecto
r by
a s
cala
r
Find
mag
nitu
de o
f a
vect
or a
nd t
he
scal
ar p
rodu
ct o
f tw
o ve
ctor
s
Sta
ndar
d ve
ctor
not
atio
ns
Mea
ning
of
term
s:
unit
vect
or
di
spla
cem
ent
vect
or
po
sitio
n ve
ctor
Geo
met
rical
inte
rpre
tatio
n
Use
sca
lar
prod
ucts
to fi
nd a
ngle
s an
d to
sol
ve p
robl
ems
invo
lvin
g pe
rpen
dicu
lar
lines
Ser
ies
Exp
and
(a +
b)n f
or p
ositi
ve in
tege
r n
Find
:
nth
term
of
AP
su
m o
f A
P
nt
h te
rm G
P
su
m o
f G
P
su
m to
infi n
ity o
f co
nver
gent
GP
Not
atio
n n!
and
n rcm
Rec
ogni
se a
rithm
etic
pro
gres
sion
s an
d ge
omet
ric p
rogr
essi
ons
Con
ditio
n fo
r co
nver
genc
e of
ge
omet
ric p
rogr
essi
on
Use
arit
hmet
ic p
rogr
essi
on a
nd/o
r ge
omet
ric p
rogr
essi
on fo
rmul
ae in
so
lvin
g pr
oble
ms
Section 4: What you need to know
16 Cambridge International AS and A Level Mathematics 9709
Pur
e M
athe
mat
ics
1: U
nit
P1To
pic
Ski
llK
no
wle
dg
eA
pp
licat
ion
Ch
eckl
ist
Dif
fere
nti
atio
nD
iffer
entia
te x
n for
rat
iona
l n,
toge
ther
with
con
stan
t m
ultip
les,
su
ms
and
diff
eren
ces
of f
unct
ions
Use
the
cha
in r
ule
on c
ompo
site
fu
nctio
ns
Find
sta
tiona
ry p
oint
s an
d id
entif
y m
axim
um/m
inim
um
Und
erst
and
grad
ient
of
a cu
rve
Not
atio
n ,
,(
),(
)dd
ddxy
xyf
xf
x2
2
lm
App
ly d
iffer
entia
tion
to:
gr
adie
nts
ta
ngen
ts a
nd n
orm
als
in
crea
sing
/dec
reas
ing
func
tions
co
nnec
ted
rate
s of
cha
nge
Use
info
rmat
ion
abou
t st
atio
nary
po
ints
to s
ketc
h gr
aphs
Inte
gra
tio
nIn
tegr
ate
(ax
+ b
)n (ra
tiona
l n ≠
–1)
to
geth
er w
ith c
onst
ant
mul
tiple
s,
sum
s an
d di
ffer
ence
s.
Eva
luat
e de
fi nite
inte
gral
s
Inte
grat
ion
as re
vers
e di
ffer
entia
tion
Use
inte
grat
ion
to s
olve
pro
blem
s in
volv
ing
fi ndi
ng a
con
stan
t of
in
tegr
atio
n
Sol
ve p
robl
ems
invo
lvin
g:
area
und
er a
cur
ve
ar
ea b
etw
een
two
curv
es
vo
lum
e of
revo
lutio
n ab
out
one
of t
he a
xes
Section 4: What you need to know
17Cambridge International AS and A Level Mathematics 9709
Pur
e M
athe
mat
ics
2: U
nit
P2
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
P2
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
Mea
ning
of
|x|
Mea
ning
of
quot
ient
and
rem
aind
er
Fact
or a
nd re
mai
nder
the
orem
s
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.
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
Section 4: What you need to know
18 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
19Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
20 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
21Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
22 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
23Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
24 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
25Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
26 Cambridge International AS and A Level Mathematics 9709
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.
Section 4: What you need to know
27Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
28 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
29Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
30 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
31Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
32 Cambridge International AS and A Level Mathematics 9709
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
Section 4: What you need to know
33Cambridge International AS and A Level Mathematics 9709
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
Section 5: Useful websites
34 Cambridge International AS and A Level Mathematics 9709
Section 5: Useful websites
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
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