Pearson Edexcel International Advanced Level in Mathematics Mathematical Formulae and Statistical Tables For use in Pearson Edexcel International Advanced Subsidiary and International Advanced Level examinations Core Mathematics C12 – C34 Further Pure Mathematics F1 – F3 Mechanics M1 – M3 Statistics S1 – S3 For use from January 2014 This copy is the property of Pearson. It is not to be removed from the examination room or marked in any way.
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Pearson Edexcel International Advanced Level in Mathematics
Mathematical Formulae and Statistical Tables
For use in Pearson Edexcel International Advanced Subsidiary and International Advanced Levelexaminations
Core Mathematics C12 – C34Further Pure Mathematics F1 – F3Mechanics M1 – M3Statistics S1 – S3
For use from January 2014
This copy is the property of Pearson. It is not to be removed from the examination room
4 Logarithms and exponentials4 Trigonometric identities4 Differentiation5 Integration
6 Further Pure Mathematics F1
6 Summations6 Numerical solution of equations6 Conics6 Matrix transformations
7 Further Pure Mathematics F2
7 Area of a sector7 Complex numbers7 Maclaurin’s and Taylor’s Series
8 Further Pure Mathematics F3
8 Vectors9 Hyperbolic functions9 Conics10 Differentiation10 Integration11 Arc length11 Surface area of revolution
2 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
12 Mechanics M1
There are no formulae given for M1 in addition to those candidates are expected to know.
12 Mechanics M2
12 Centres of mass
12 Mechanics M3
12 Motion in a circle12 Centres of mass12 Universal law of gravitation
13 Statistics S1
13 Probability13 Discrete distributions13 Continuous distributions14 Correlation and regression15 The Normal distribution function16 Percentage points of the Normal distribution
17 Statistics S2
17 Discrete distributions17 Continuous distributions18 Binomial cumulative distribution function23 Poisson cumulative distribution function
24 Statistics S3
24 Expectation algebra24 Sampling distributions24 Correlation and regression24 Non-parametric tests25 Percentagepointsoftheχ2 distribution26 Critical values for correlation coefficients27 Random numbers
There are no formulae provided for Decision Mathematics unit D1.
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 C34 might be expected to use formulae first introduced in C12).
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.
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 3
Core Mathematics C12
Mensuration
Surface area of sphere = 4π r 2
Area of curved surface of cone = π r × slant height
Arithmetic series
un = a + (n – 1)d
Sn = 12
n(a + l) = 12
n[2a + (n – 1)d]
Geometric series
un = arn – 1
Sn = a r
r
n( )11
−−
S¥ = ar1 −
for | r | < 1
Binomial series
( )a b ana b
na b
nra bn n n n n r r+ = +
+
+ +
+ +− − −
1 21 2 2 bb nn ( )∈
where nr
nr n r
nr
= =−
C !!( )!
( ) ( ) ( ) ( ) ( ,1 1 11 2
1 11 2
12+ = + + −×
+ + − − +× × ×
+ <x nx n n x n n n rr
x x nn r
∈∈)
Logarithms and exponentials
loglogloga
b
b
x xa
=
Cosine rule
a2 = b2 + c2 – 2bc cos A
Numerical integration
The trapezium rule:b
a
y x⌠
⌡ d ≈ 1
2 h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where h b an
= −
4 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
Core Mathematics C34
Candidates sitting C34 may also require those formulae listed under Core Mathematics C12.
Logarithms and exponentials
e lnx a xa=
Trigonometric identities
sin( ) sin cos cos sinA B A B A B± ≡ ±
cos( ) cos cos sin sinA B A B A B± ≡
12
tan tantan( ) ( ( ) )1 tan tan
A BA B A B k πA B
±± ≡ ± ≠ +∓
sin sin sin cosA B A B A B+ ≡ + −22 2
sin sin cos sinA B A B A B− ≡ + −22 2
cos cos cos cosA B A B A B+ ≡ + −22 2
cos cos sin sinA B A B A B− ≡ − + −22 2
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
f( )g( )
xx
f g f gg
′ ′( ) ( ) ( ) ( )( ( ))
x x x xx−
2
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 5
Integration (+ constant)
f(x) ⌠⌡ f( ) dx x
sec2 kx 1k
tan kx
tan x ln sec x
cot x ln sin x
cosec x − +ln cosec cot , ln tan( )x x x12
sec x 1 12 4ln sec tan , ln tan( )x x x π+ +
⌠⌡
⌠⌡= −u v
xx uv v u
xxd
dd d
dd
6 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
Further Pure Mathematics F1
Candidates sitting F1 may also require those formulae listed under Core Mathematics C12.
Summations
r n n nr
n2
1
16 1 2 1
=∑ = + +( )( )
r n nr
n3
1
14
2 21=
∑ = +( )
Numerical solution of equations
The Newton-Raphson iteration for solving f (x) = 0 : x x xxn nn
n+ = −
′1f( )f ( )
Conics
Parabola RectangularHyperbola
StandardForm y2 = 4ax xy = c2
ParametricForm (at2, 2at) ct c
t,
Foci (a, 0) Not required
Directrices x = –a Not required
Matrix transformations
Anticlockwise rotation through θabout O: cos sinsin cosθ θθ θ
−
Reflection in the line y = (tan θ )x: cos 2 sin 2sin 2 cos 2
θ θθ θ
−
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 7
Further Pure Mathematics F2
Candidates sitting F2 may also require those formulae listed under Further Pure Mathematics F1, and Core Mathematics C12 and C34.
Area of a sector
21 d2
A r θ= ⌠⌡
(polar coordinates)
Complex numbers
ie cos i sinθ θ θ= + { (cos i sin )} (cos i sin )n nr θ θ r nθ nθ+ = +
The roots of zn = 1 are given by 2 i
eπknz = , for k = 0, 1, 2, …, n – 1
Maclaurin’s and Taylor’s Series
f( ) f( ) f ( )!
f ( )!
f ( )( )x x x xr
rr= + ′ + ′′ + + +0 0
20 0
2
f( ) f( ) ( ) f ( ) ( )!
f ( ) ( )!
f ( )( )x a x a a x a a x ar
ar
r= + − ′ + − ′′ + + − +2
2
f( ) f( ) f ( )!
f ( )!
f ( )( )a x a x a x a xr
ar
r+ = + ′ + ′′ + + +2
2
e exp( )! !
xr
x x x xr
x= = + + + + +12
2
for all
ln( ) ( ) ( )12 3
1 1 12 3
1+ = − + − + − + − < ≤+x x x x xr
xrr
sin! !
( )( )!
x x x x xr
xrr
= − + − + −+
++3 5 2 1
3 51
2 1 for all
cos! !
( )( )!
x x x xr
xrr
= − + − + − +12 4
12
2 4 2
for all
arctan x x x x xr
xrr
= − + − + −+
+ − ≤ ≤+3 5 2 1
3 51
2 11 1 ( ) ( )
8 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
Further Pure Mathematics F3
Candidates sitting F3 may also require those formulae listed under Further Pure Mathematics F1, and Core Mathematics C12 and C34.
Vectors
The resolved part of a in the direction of b is a.bb
The point dividing AB in the ratio λ : μ is μ λλ μ
++a b
Vector product: 2 3 3 2
1 2 3 3 1 1 3
1 2 3 1 2 2 1
ˆsina b a b
θ a a a a b a bb b b a b a b
− × = = = −
−
i j ka b a b n
a. b c b. c a c. a b( ) ( ) ( )× = = × = ×a a ab b bc c c
1 2 3
1 2 3
1 2 3
If A is the point with position vector a = a1i + a2 j + a3k and the direction vector b is given by b = b1i + b2 j + b3k, then the straight line through A with direction vector b has cartesian equation
31 2
1 2 3
( )z ax a y a λ
b b b−− −
= = =
The plane through A with normal vector n i j k= + +n n n1 2 3 has cartesian equation
n x n y n z d d1 2 3 0+ + + = = − where a.n
The plane through non-collinear points A, B and C has vector equation
( ) ( ) (1 )λ μ λ μ λ μ= + − + − = − − + +r a b a c a a b c
The plane through the point with position vector a and parallel to b and c has equation
r a b c= + +s t
The perpendicular distance of (α, β, γ) from n x n y n z d1 2 3 0+ + + = is 1 2 3
2 2 21 2 3
n α n β n γ d
n n n
+ + +
+ +.
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 9
Hyperbolic functions
cosh sinh2 2 1x x− ≡
sinh sinh cosh2 2x x x≡
cosh cosh sinh2 2 2x x x≡ +
arcosh x x x x≡ + − ≥ln ( ){ }2 1 1
arsinh x x x≡ + +ln{ }2 1
artanh x xx
x≡ +−
<12
11
1ln ( )
Conics
Ellipse Parabola Hyperbola RectangularHyperbola
StandardForm
xa
yb
2
2
2
2 1+ = y2 = 4ax xa
yb
2
2
2
2 1− = xy = c2
ParametricForm (α cos θ, b sin θ) (at2, 2at) (α sec θ, b tan θ)
(±a cosh θ, b sinh θ)ct c
t,
Eccentricity e < 1b2 = a2(1 – e2)
e = 1 e > 1b2 = a2(e2 – 1)
e = √2
Foci (±ae, 0) (a, 0) (±ae, 0) (±√2c, ±√2c)
Directrices x ae
= ± x = –a x ae
= ± x + y = ±√2c
Asymptotes none nonexa
yb
= ± x = 0, y = 0
10 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
Differentiation
f(x) f ′(x)
arcsin x 1
1 2− x
arccos x −−
11 2x
arctan x 11 2+ x
sinh x cosh x
cosh x sinh x
tanh x sech2 x
arsinh x 1
1 2+ x
arcosh x 112x −
artanh x 11 2− x
Integration (+ constant; a > 0 where relevant)
f(x) f( ) dx x⌠⌡
sinh x cosh x
cosh x sinh x
tanh x ln cosh x
1
2 2a x− arcsin x
ax a
<( )
12 2a x+
1a
xa
arctan
12 2x a−
arcosh xa
x x a x a
+ − >, ln ( ){ }2 2
12 2a x+
arsinh xa
x x a
+ +, ln{ }2 2
12 2a x−
12
1a
a xa x a
xa
x aln ( )+−
=
<artanh
12 2x a−
12a
x ax a
ln −+
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 11
Arc length
s yx
x= +
⌠
⌡ 1
2dd
d (cartesian coordinates)
s xt
yt
t=
+
⌠
⌡
dd
dd
d2 2
(parametric form)
Surface area of revolution
2d2 d 2 1 d
dxyS y s y xx
= = +
⌠⌠ ⌡ ⌡
π π
2 2d d2 d
d dx yy tt t
= + ⌠⌡
π
12 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
Mechanics M1
There are no formulae given for M1 in addition to those candidates are expected to know.
Candidates sitting M1 may also require those formulae listed under Core Mathematics C12.
Mechanics M2
Candidates sitting M2 may also require those formulae listed under Core Mathematics C12 and C34.
Centres of mass
For uniform bodies:
Triangular lamina: 23 along median from vertex
Circular arc, radius r, angle at centre 2α: sinr αα
from centre
Sector of circle, radius r, angle at centre 2α: 2 sin
3r αα
from centre
Mechanics M3
Candidates sitting M3 may also require those formulae listed under Mechanics M2, and Core Mathematics C12 and C34.
Motion in a circle
Transverse velocity: v rθ= �
Transverse acceleration: v rθ= ���
Radial acceleration: 2
2 vrθr
− = −�
Centres of mass
For uniform bodies:
Solid hemisphere, radius r: 38 r from centre
Hemispherical shell, radius r: 12 r from centre
Solid cone or pyramid of height h: 14 h above the base on the line from centre of base to vertex
Conical shell of height h: 13 h above the base on the line from centre of base to vertex
Universal law of gravitation
Force = Gmmd
1 22
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 13
Statistics S1Probability
P( ) P( ) P( ) P( )A B A B A B∪ = + − ∩
P( ) P( ) P( )A B A B A∩ = |
P | || ( | (
( ) P( ) P( )P( ) P ) P( ) P )
A B B A AB A A B A A
=+ ′ ′
Discrete distributions
For a discrete random variable X taking values xi with probabilities P(X = xi)
16 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION
The values z in the table are those which a random variable Z ~ N(0, 1) exceeds with probability p; that is, P(Z > z) = 1 – Ф(z) = p.
p z p z0.5000 0.0000 0.0500 1.64490.4000 0.2533 0.0250 1.96000.3000 0.5244 0.0100 2.32630.2000 0.8416 0.0050 2.57580.1500 1.0364 0.0010 3.09020.1000 1.2816 0.0005 3.2905
Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013 17
Statistics S2
Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listed under Core Mathematics C12 and C34.
Discrete distributions
Standard discrete distributions:
Distribution of X P(X = x) Mean Variance
Binomial B(n, p)nxp px n x
− −( )1 np np(1 – p)
Poisson Po(λ) e!
xλ λx
− λ λ
Continuous distributions
For a continuous random variable X having probability density function f
Expectation (mean): E( ) f( )dX μ x x x= = ∫ Variance: 2 2 2 2Var( ) ( ) f( ) d f( )dX σ x μ x x x x x μ= = − = −∫ ∫ For a function g(X): E(g( )) g( ) f( )X x x x= ∫ d
Cumulative distribution function: F( ) P( ) f( )x X x t tx
0 00= ≤ =
−∞∫ d
Standard continuous distribution:
Distribution of X P.D.F. Mean Variance
Uniform (Rectangular) on [a, b]1
b a−12 ( )a b+ 1
122( )b a−
18 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION
The tabulated value is P(X ≤ x), where X has a binomial distribution with index n and parameter p.
26 Pearson Edexcel IAS/IAL in Mathematics Formulae List – Issue 1 – June 2013
CRITICAL VALUES FOR CORRELATION COEFFICIENTS
These tables concern tests of the hypothesis that a population correlation coefficient ρ is 0. The values in the tables are the minimum values which need to be reached by a sample correlation coefficient in order to be significant at the level shown, on a one-tailed test.