AQA Formulae Booklet - Wood Green Academy

35 Pages·2004·1.1 MB·English

abc Formulae and

Statistical Tables

for GCE Mathematics

and GCE Statistics

Issued September 2004

For the new specifications for first teaching from September 2004

GCE Mathematics

ADVANCED SUBSIDIARY MATHEMATICS (5361)
ADVANCED SUBSIDIARY PURE MATHEMATICS (5366)
ADVANCED SUBSIDIARY FURTHER MATHEMATICS (5371)

ADVANCED MATHEMATICS (6361)
ADVANCED PURE MATHEMATICS (6366)
ADVANCED FURTHER MATHEMATICS (6371)

GCE Statistics

ADVANCED SUBSIDIARY STATISTICS (5381)
ADVANCED STATISTICS (6381)

General Certificate of Education
19MSPM
Further copies of this booklet are available from:
Publications Department, Aldon House, 39 Heald Grove, Rusholme, Manchester, M14 4NA
Telephone: [phone] Fax: [phone]
or
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© Assessment and Qualifications Alliance 2004

COPYRIGHT
AQA retains the copyright on all its publications, including the specimen units and mark schemes/ teachers’
guides. However, registered centres of AQA are permitted to copy material from this booklet for their own
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any material that is acknowledged to a third party even for internal use within the centre.

Set and published by the Assessment and Qualifications Alliance.

The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 3644723 and a registered charity number 1073334.
Registered address AQA, Devas Street, Manchester M15 6EX. Dr Michael Cresswell Director General.
Contents

Page
4 Pure Mathematics

9 Mechanics

10 Probability and Statistics

Statistical Tables

15 Table 1 Cumulative Binomial Distribution Function

22 Table 2 Cumulative Poisson Distribution Function

24 Table 3 Normal Distribution Function

25 Table 4 Percentage Points of the Normal Distribution

26 Table 5 Percentage Points of the Student’s t-Distribution

27 Table 6 Percentage Points of the χ2 Distribution

28 Table 7 Percentage Points of the F-Distribution

30 Table 8 Critical Values of the Product Moment Correlation Coefficient

31 Table 9 Critical Values of Spearman’s Rank Correlation Coefficient

32 Table 10 Critical Values of the Wilcoxon Signed Rank Statistic

33 Table 11 Critical Values of the Mann-Whitney Statistic

34 Table 12 Control Charts for Variability

35 Table 13 Random Numbers
klj 3
PURE MATHEMATICS
Mensuration
Surface area of sphere =4πr2
Area of curved surface of cone = πr × slantheight
Arithmetic series
u = a+(n−1)d
n

[ ]
S = 1n(a+l)= 1n 2a+(n−1)d
n 2 2
Geometric series
u = arn−1
n
a(1−rn)
S =
n 1−r
a
S = for r <1
∞ 1−r
Summations
n
∑ ( )
r = 1n n +1
2
r=1
n
∑r2 = 1n(n+1)(2n+1)
6
r=1
n
∑r3 = 1n2(n+1)2
4
r=1

Trigonometry – the Cosine rule
a2 = b2 + c2 − 2bc cosA
Binomial Series
n n n
(a+b)n = an +  an−1b+  an−2b2 + K +  an−rbr + K +bn (n∈N )
1 2 r
n n!
where r= nCr = r!(n−r)!
n(n−1) n(n−1)K(n−r+1)
(1+ x)n =1+nx+ x2 + K + xr + K ( x <1, n∈ R )
1.2 1.2Kr
Logarithms and exponentials
ax = exlna
Complex numbers
{r(cosθ+isinθ)}n =rn(cosnθ+isinnθ)
eiθ=cosθ+isinθ
2πki
The roots of zn =1 are given by z =e n , for k =0, 1, 2, K , n−1
4 klm
Maclaurin’s series
x2 xr
f(x) =f(0)+ xf′(0)+ f′′(0)+ K + f(r)(0)+ K
2! r!
x2 xr
ex =exp(x)=1+ x+ + K + + K for all x
2! r!
x2 x3 xr
ln(1+ x) = x− + − K +(−1)r+1 + K (−1< x 1)
2 3 r
x3 x5 x2r+1
sinx=x− + − K +(−1)r + K for all x
3! 5! (2r +1)!
x2 x4 x2r
cosx=1− + − K +(−1)r + K for all x
2! 4! (2r)!
Hyperbolic functions
cosh2 x−sinh2 x =1
sinh2x=2sinhxcoshx
cosh2x = cosh2 x+sinh2 x
{ }
cosh−1 x =ln x+ x2 −1 (x 1)
{ }
sinh−1 x = ln x+ x2 +1
1+ x
tanh−1 x = 1ln  ( x <1)
2 1− x
Conics
Rectangular
Ellipse Parabola Hyperbola
hyperbola
Standard x2 y2 x2 y2
+ =1 y2 =4ax − =1 xy=c2
form a2 b2 a2 b2
x y
Asymptotes none none =± x=0,y=0
a b
Trigonometric identities
sin(A± B)=sin AcosB±cosAsinB
cos(A±B)=cosAcosBmsin AsinB
tanA±tanB ( )
tan(A±B)= A±B≠(k+ 1)π
1mtanAtanB 2
A+B A−B
sinA+sinB = 2sin cos
2 2
A+ B A− B
sin A−sinB=2cos sin
2 2
A+B A−B
cosA+cosB=2cos cos
2 2
A+B A−B
cosA−cosB=−2sin sin
2 2
5
klj
Vectors
a.b
The resolved part of a in the direction of b is
b

µa+λb
The position vector of the point dividing AB in the ratio λ:µ is
λ+µ

i a b a b −a b 
1 1 2 3 3 2
 
Vector product: a×b = a b sinθ nˆ = j a b = a b −a b
2 2  3 1 1 3 
k a b a b −a b 
 
3 3 1 2 2 1

If A is the point with position vector a=a i+a j+a k and the direction vector b is given by
1 2 3
b=bi+b j+b k , then the straight line through A with direction vector b has cartesian equation
1 2 3
x−a y−a z−a
1 = 2 = 3 =λ
b b b
1 2 3

The plane through A with normal vector n=n i+n j+n k has cartesian equation
1 2 3
n x+n y+n z = d where d =a.n
1 2 3

The plane through non-collinear points A, B and C has vector equation
r=a+λ(b−a)+µ(c−a)=(1−λ−µ)a+λb+µc

The plane through the point with position vector a and parallel to b and c has equation
r=a+sb+tc

Matrix transformations
cosθ −sinθ
Anticlockwise rotation through θ about O:  
sinθ cosθ
cos2θ sin2θ
Reflection in the line y =(tanθ)x:  
sin2θ −cos2θ
The matrices for rotations (in three dimensions) through an angle θ about one of the axes are

1 0 0 
 
0 cosθ −sinθ for the x-axis
 
0 sinθ cosθ
 
 cosθ 0 sinθ
 
0 1 0 for the y-axis
 
−sinθ 0 cosθ
 
cosθ −sinθ 0
 
sinθ cosθ 0 for the z-axis
 
 0 0 1
 
6 klm
Differentiation
f(x) f′(x)
1
sin−1 x
1−x2
1
cos−1 x −
1−x2
1
tan−1 x
1+ x2
tankx ksec2kx
cosecx −cosecxcotx
secx secxtanx
cotx −cosec2 x
sinhx coshx
coshx sinhx
tanhx sech2 x
1
sinh−1x
1+ x2
1
cosh−1 x
x2 −1
1
tanh−1 x
1− x2
f (x) f′ (x) g(x)−f(x) g′ (x)

g (x) (g (x))2

Integration
(+ constant; a>0 where relevant)
f(x) ∫f(x)dx
tanx lnsecx
cotx lnsinx
cosecx −lncosecx+cotx =lntan(1x)
2
secx lnsecx+tanx =lntan(1x+ 1π)
2 4
1
sec2kx tan kx
k
sinhx coshx
coshx sinhx
tanhx lncoshx

INTEGRATION FORMULAE CONTINUE OVER THE PAGE
7
klj
1 x
sin−1  (x <a)
a2 −x2 a
1 1 x
tan−1 
a2 + x2 a a
1 x { }
cosh−1  or ln x+ x2 −a2 (x>a)
x2 −a2 a
1 x { }
sinh−1  or ln x+ x2 +a2
a
a2 +x2  
1 1 a+ x 1 x
ln = tanh−1  (x <a)
a2 −x2 2a a− x a a
1 1 x−a
ln
x2 −a2 2a x+a
∫ dv ∫ du
u dx=uv− v dx
dx dx

Area of a sector
A= 1 ∫r2dθ (polar coordinates)
2

Arc length
dy2
s = ∫ 1+  dx (cartesian coordinates)
dx
dx2 dy2
s=∫   +  dt (parametric form)
dt  dt 
Surface area of revolution
dy2
S = 2π∫ y 1+  dx (cartesian coordinates)
x dx
dx2 dy2
S =2π∫ y   +  dt (parametric form)
x dt   dt 
Numerical integration
The trapezium rule: ∫ by dx ≈ 1h{(y + y ) +2(y + y + K + y )}, where h= b−a
a 2 0 n 1 2 n−1 n
The mid-ordinate rule: ∫ by dx ≈ h(y + y + K +y + y ), where h= b−a
a 1 3 n−3 n−1 n
2 2 2 2
Simpson’s rule: ∫ by dx ≈ 1h{(y + y )+4(y + y +...+ y )+2(y + y +...+ y )}
a 3 0 n 1 3 n−1 2 4 n−2
b−a
where h= and n is even
n
8 klm
Numerical solution of differential equations
dy
For = f(x) and small h, recurrence relations are:
dx
Euler’s method: y = y +hf(x ); x = x +h
n+1 n n n+1 n

dy
For =f(x, y):
dx
Euler’s method: y = y +hf(x , y )
r+1 r r r
Improved Euler method: y = y + 1(k +k ), where k =hf(x , y ), k =hf(x + h, y + k )
r+1 r 2 1 2 1 r r 2 r r 1
Numerical solution of equations
f(x )
The Newton-Raphson iteration for solving f(x)=0: x = x − n
n+1 n f′(x )
n

MECHANICS
Motion in a circle
&
Transverse velocity: v=rθ
Transverse acceleration: v&=rθ&&
& v2
Radial acceleration: −rθ2 =−
r
Centres of mass
For uniform bodies
Triangular lamina: 2 along median from vertex
3
Solid hemisphere, radius r: 3r from centre
8
Hemispherical shell, radius r: 1r from centre
2
rsinα
Circular arc, radius r, angle at centre 2α: from centre
α
2rsinα
Sector of circle, radius r, angle at centre 2α: from centre
3α
Solid cone or pyramid of height h: 1h above the base on the line from centre of base to vertex
4
Conical shell of height h: 1h above the base on the line from centre of base to vertex
3
Moments of inertia
For uniform bodies of mass m
Thin rod, length 2l, about perpendicular axis through centre: 1ml2
3
Rectangular lamina about axis in plane bisecting edges of length 2l: 1ml2
3
Thin rod, length 2l, about perpendicular axis through end: 4ml2
3
Rectangular lamina about edge perpendicular to edges of length 2l: 4ml2
3
Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: 1m(a2 +b2)
3

MOMENTS OF INERTIA FORMULAE CONTINUE OVER THE PAGE
9
klj
Hoop or cylindrical shell of radius r about axis: mr2
Hoop of radius r about a diameter: 1mr2
2
Disc or solid cylinder of radius r about axis: 1mr2
2
Disc of radius r about a diameter: 1mr2
4
Solid sphere, radius r, about diameter: 2mr2
5
Spherical shell of radius r about a diameter: 2mr2
3
Parallel axes theorem: I =I +m(AG)2
A G
Perpendicular axes theorem: I =I +I (for a lamina in the x-y plane)
z x y
General motion in two dimensions
Radial velocity r&
&
Transverse velocity rθ
Radial acceleration &r&−rθ&2
Transverse acceleration rθ&&+2r&θ&= 1 d (r2θ&)
r dt
Moments as vectors
The moment about O of F acting through the point with position vector r is r×F
Universal law of gravitation
Gm m
Force= 1 2
d2

PROBABILITY and STATISTICS
Probability
P(A∪B)=P(A)+P(B)−P(A∩B)
P(A∩B)=P(A)×P(B | A)
( ) ( )
P A ×P B A
( )
j j
P A B =
j ∑n ( ) ( )
P A ×P B A
i i
i=1
Expectation algebra
Covariance: Cov(X, Y)=E((X −µ )(Y −µ ))=E(XY)−µ µ
X Y X Y
Var(aX ±bY)=a2 Var(X)+b2 Var(Y)±2abCov(X, Y)
Cov(X, Y)
Product moment correlation coefficient: ρ=
σ σ
X Y
For independent random variables X and Y
E(XY) = E(X)E(Y)
Var(aX ±bY)=a2 Var(X)+b2 Var(Y)
10 klm