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GCE

Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables

For use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinations =Core Mathematics C1 C4 Further Pure Mathematics FP1 FP3 Mechanics M1 M5 Statistics S1 S4

For use from June 2009

This copy is the property of Edexcel. It is not to be removed from the examination room or marked in any way.

=

Edexcel AS/A level Mathematics Formulae List: C1 C4, FP1 FP3 Contents Page Issue 1 September 2009 1

TABLE OF CONTENTS

Page

4 Core Mathematics C1

4 Mensuration 4 Arithmetic series 5 Core Mathematics C2

5 Cosine rule 5 Binomial series 5 Logarithms and exponentials 5 Geometric series 5 Numerical integration 6 Core Mathematics C3

6 Logarithms and exponentials 6 Trigonometric identities 6 Differentiation 7 Core Mathematics C4

7 Integration 8 Further Pure Mathematics FP1

8 Summations 8 Numerical solution of equations 8 Conics 8 Matrix transformations 9 Further Pure Mathematics FP2

9 Area of sector 9 Maclaurins and Taylors Series 10 Further Pure Mathematics FP3

10 Vectors 11 Hyperbolics 12 Differentiation 12 Integration 13 Arc length 13 Surface area of revolution

2 Edexcel AS/A level Mathematics Formulae List: M1M5, S1S4 Contents Page Issue 1 September 2009

14 Mechanics M1

14 There are no formulae given for M1 in addition to those candidates are expected to know. 14 Mechanics M2

14 Centres of mass 14 Mechanics M3

14 Motion in a circle 14 Centres of mass 14 Universal law of gravitation 15 Mechanics M4

15 There are no formulae given for M4 in addition to those candidates are expected to know. 15 Mechanics M5

15 Moments of inertia 15 Moments as vectors 16 Statistics S1

16 Probability 16 Discrete distributions 16 Continuous distributions 17 Correlation and regression 18 The Normal distribution function 19 Percentage points of the Normal distribution 20 Statistics S2

20 Discrete distributions 20 Continuous distributions 21 Binomial cumulative distribution function 26 Poisson cumulative distribution function 27 Statistics S3

27 Expectation algebra 27 Sampling distributions 27 Correlation and regression 27 Non-parametric tests 28 Percentage points of the 2 distribution 29 Critical values for correlation coefficients 30 Random numbers 31 Statistics S4

31 Sampling distributions 32 Percentage points of Students t distribution 33 Percentage points of the F distribution There are no formulae provided for Decision Mathematics units D1 and D2.

Edexcel AS/A level Mathematics Formulae List Issue 1- September 2009 3

The formulae in this booklet have been arranged according to the unit in which they are first introduced. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit (e.g. candidates sitting C3 might be expected to use formulae first introduced in C1 or C2). It may also be the case that candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification.

4 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 September 2009

Core Mathematics C1

Mensuration

Surface area of sphere = 4 r 2

Area of curved surface of cone = r slant height

Arithmetic series

un = a + (n 1)d

Sn = 21 n(a + l) =

21 n[2a + (n 1)d]

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 Issue 1 September 2009 5

Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1.

Cosine rule

a2 = b2 + c2 2bc cos A

Binomial series

2

1

)( 221 nrrnnnnn bbarn

ban

ban

aba ++

++

+

+=+ KK (n )

where )!(!

!C rnr

nrn

rn

==

6 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 Issue 1 September 2009

Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2.

Logarithms and exponentials

xax a=lne

Trigonometric identities

BABABA sincoscossin)(sin = BABABA sinsincoscos)(cos m=

))(( tantan1tantan)(tan 21 +

= kBABABABA

m

2cos

2sin2sinsin BABABA +=+

2sin

2cos2sinsin BABABA +=

2cos

2cos2coscos BABABA +=+

2sin

2sin2coscos BABABA +=

Differentiation

f(x) f (x)

tan kx k sec2 kx

sec x sec x tan x

cot x cosec2 x

cosec x cosec x cot x

)g()f(

xx

))(g(

)(g)f( )g()(f2x

xxxx

Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 Issue 1 September 2009 7

Core Mathematics C4 Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3.

Integration (+ constant)

f(x) xx d)f(

sec2 kx k1 tan kx

xtan xsecln

xcot xsinln

xcosec )tan(ln,cotcosecln 21 xxx +

xsec )tan(ln,tansecln 4121 ++ xxx

= xxuvuvx

xvu d

ddd

dd

8 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 Issue 1 September 2009

Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2.

Summations

)12)(1(61

1

2 ++==

nnnrn

r

2241

1

3 )1( +==

nnrn

r

Numerical solution of equations

The Newton-Raphson iteration for solving 0)f( =x : )(f)f(

1n

nnn x

xxx

=+

Conics

Parabola Rectangular Hyperbola

Standard Form axy 4

2 = xy = c2

Parametric Form (at

2, 2at)

tcct,

Foci )0 ,(a Not required

Directrices ax = Not required

Matrix transformations

Anticlockwise rotation through about O:

cos sinsincos

Reflection in the line xy )(tan= :

2cos2sin2sin 2cos

In FP1, will be a multiple of 45.

Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 Issue 1 September 2009 9

Further Pure Mathematics FP2 Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1C4. Area of a sector

A = d

21 2r (polar coordinates)

Complex numbers

sinicosei += )sini(cos)}sini(cos{ nnrr nn +=+

The roots of 1=nz are given by nk

zi2

e

= , for 1 , ,2 ,1 ,0 = nk K

Maclaurins and Taylors Series

KK )0(f!

)0(f!2

)0(f)0f()f( )(2

+++++= rr

rxxxx

KK )(f!

)( )(f!2

)()(f)()f()f( )(2

+

++

++= araxaaxaaxax r

r

KK )(f!

)(f!2

)(f)f()f( )(2

+++++=+ arxaxaxaxa r

r

xrxxxx

rx allfor

!

!21)exp(e

2

KK +++++==

)11( )1( 32

)1(ln 132

10 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 Issue 1 September 2009

Further Pure Mathematics FP3 Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1C4.

Vectors

The resolved part of a in the direction of b is b

a.b

The point dividing AB in the ratio : is

++ ba

Vector product:

===

1221

3113

2332

321

321 sinbabababababa

bbbaaakji

nbaba

)()()(

321

321

321

bac.acb.cba. ===cccbbbaaa

If A is the point with position vector kjia 321 aaa ++= and the direction vector b is given by

kjib 321 bbb ++= , then the straight line through A with direction vector b has cartesian equation

)( 3

3

2

2

1

1 =

=

=

baz

bay

bax

The plane through A with normal vector kjin 321 nnn ++= has cartesian equation

a.n==+++ ddznynxn where0321

The plane through non-collinear points A, B and C has vector equation

cbaacabar ++=++= )1()()( The plane through the point with position vector a and parallel to b and c has equation

cbar ts ++=

The perpendicular distance of ) , ,( from 0321 =+++ dznynxn is 23

22

21

321

nnn

dnnn

++

+++ .

Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 Issue 1September 2009 11

Hyperbolic functions 1sinhcosh 22 = xx

xxx coshsinh22sinh = xxx 22 sinhcosh2cosh +=

)1( 1lnarcosh }{ 2 += xxxx }{ 1lnarsinh 2 ++= xxx

)1( 11lnartanh 21

12 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 Issue 1September 2009

Differentiation

f(x) f (x)

xarcsin 21

1

x

xarccos 21

1

x

xarctan 211x+

xsinh xcosh xcosh xsinh xtanh x2sech

xarsinh 21

1

x+

xarcosh 1

12 x

artanh x 211x

Integration (+ constant; 0>a where relevant)

f(x) xx d)f(