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10.1 Formulae and tables
Content
Mathematical notation
Mathematical formulae and definitions
formulae & tables
10.1.1
10.1.2
FORMULAE
section
10
formulae & tables
10-1
10.1.1 Mathematical notation
The list which follows summarises the notation used in the Syndicate’s
Mathematics examinations. Although primarily directed towards A Level,
the list also applies, where relevant, to examinations at all other levels and
other boards.
1. Set notation


is an element of
is not an element of
x1 , x2 , ... 
the set with elements x1 , x2 , ...
x : ... 
the set of all x such that
n  A
the number of elements in set A

the empty set

universal set
A'
the complement of the set A
the set of integers, 0,  1,  2,  3 , ... 
the set of positive integers, 1, 2, 3, ... 

the set of rational numbers
the set of positive rational numbers, x 


0
:x0
the set of positive rational numbers and zero,
x  : x  0 
the set of real numbers
the set of positive real numbers, x 


:x0
0
the set of positive real numbers and zero,
x  : x  0 
n
the real n tuples
the set of complex numbers

is a subset of


is a proper subset of

is not a proper subset of

union

intersection
a, b
the closed interval x 
a, b 
the interval x 
:a  x  b 
a, b
a, b
the interval x 
:a  x  b 
AL eGuide Mathematics
is not a subset of
the open interval x 
:a  x  b 
:a  x  b 
 cosmic
2. Miscellaneous Symbols

is equal to

is not equal to

is identical to or is congruent to

is approximately equal to

;
is proportional to
 ; 
is less than or equal to; is not greater than
;
is greater than; is much greater than
 ; 
is greater than or equal to; is not less than

infinity
is less than; is much less than
3. Operations
ab
a plus b
ab
a minus b
a  b , ab , a  b
a multiplied by b
ab,
a:b
n
a
i 1
a
i
a
, ab
b
a divided by b
the ratio of a to b
a1  a2  ...  an
the positive square root of the real number a
a
the modulus of the real number a
n!
n factorial for n 
n
 
r 
the binomial coefficient
n!
, for n , r 
r !  n  r !

 0,  0!  1
n  n  1 ...  n  r  1
r!

 0, 0  r  n
, for n 
, r

 0
10
formulae & tables
10-3
4. Functions
f
function f
f x
the value of the function f at x
f: A  B
f is a function under which each element of
set A has an image in set B
f: x
the function f maps the element x to the element
y
y
f -1
the inverse of the function f
g f , gf
the composite function of f and g which is
defined by  g f  x  or gf  x   g  f  x  
lim f  x 
x a
the limit of f  x  as x tends to a
x ;  x
an increment of x
dy
dx
the derivative of y with respect to x
dn y
dx n
the n th derivative of y with respect to x
n
f '  x  , f "  x  , …, f    x 
the first, second, …, n th derivatives of f  x 
with respect to x
 y dx

b
a
indefinite integral of y with respect to x
the definite integral of y with respect to x for
y dx
values of x between a and b
y
x
the partial derivative of y with respect to x
x, x,…
the first, second, … derivatives of x with respect
to time
5. Exponential and Logarithmic Functions
e
base of natural logarithms
x
e , exp x
exponential function of x
loga x
logarithm to the base a of x
ln x
natural logarithm of x
lg x
logarithm of x to base 10
6. Circular Functions and Relations



sin, cos, tan
cosec, sec, cot



cosec , sec , cot 

sin1, cos1, tan1
1
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1
1
the circular functions
the inverse circular functions
 cosmic
7. Complex Numbers
i
square root of –1
z
a complex number, z  x  iy
 r  cos  isin  , r 
i
 re , r 

0

0
Re z
the real part of z , Re  x  iy   x
Im z
the imaginary part of z , Im  x  iy   y
z
the modulus of z , x  iy  x 2  y 2 , r  cos   i sin   r
arg z
the argument of z , arg  r  cos   i sin     ,     
z
the complex conjugate of z ,  x  iy   x  iy

8. Matrices
M
M
a matrix M
1
the inverse of the square matrix M
MT
the transpose of the matrix M
det M
the determinant of the square matrix M
9. Vectors
a
the vector a
AB
the vector represented in magnitude and direction by the
directed line segment AB
â
a unit vector in the direction of the vector a
i, j, k
unit vectors in the directions of the cartesian coordinate
axes
a
the magnitude of a
AB
the magnitude of AB
a b
the scalar product of a and b
a b
the vector product of a and b
10
formulae & tables
10-5
10. Probability and Statistics
A, B, C, etc.
events
AB
union of events A and B
AB
intersection of the events A and B
P  A
probability of the event A
A'
complement of the event A , the event ‘not A ’
P A B
probability of the event A given the event B
X , Y , R, etc.
random variables
x, y, r , etc.
value of the random variables X , Y , R, etc.
x1, x2 , ...
observations
f1, f2 , ...
frequencies with which the observations, x1, x2 , ... occur
px
the value of the probability function P  X  x  of the
discrete random variable X
p1, p2 , ...
probabilities of the values x1, x2 , ... of the discrete
random variable X
f  x , g x  , …
the value of the probability density function of the
continuous random variable X
Fx , Gx , …
the value of the (cumulative) distribution function
P  X  x  of the random variable X
E X 
expectation of the random variable X
E g  X  
expectation of g  X 
Var  X 
variance of the random variable X
B  n, p 
binominal distribution, parameters n and p

N ,  2




normal distribution, mean  and variance  2
population mean
2
population variance
population standard deviation
x
sample mean
s2
unbiased estimate of population variance from a
2
1
xx
sample, s 2 

n 1

probability density function of the standardised normal
variable with distribution N  0, 1


corresponding cumulative distribution function
r
Cov  X , Y 
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

linear product-moment correlation coefficient for a
population
linear product-moment correlation coefficient for a
sample
covariance of X and Y
 cosmic
10.1.2 Mathematical formulae and definitions
PURE MATHEMATICS
Algebraic series
n
r 
r 1
1
2
n
n  n  1 ,
r
2
r 1
n
r
 61 n  n  1 2n  1 ,
 41 n2  n  1
3
r 1
2
Binomial expansion:
a  b
 n  n1  n  n2 2  n  n3 3
n
n
 a    a b    a b    a b  ...  b ,
1
2
3
 
 
 
n
 n
n!
where n is a positive integer and   
 r   n  r ! r !
Maclaurin's expansion:
1  x 
n
 1  nx 
ex  1 x 
sin x  x 
cos x  1 
x
x
 n  n  1

2!

2
2!
2!
x2
3
3!

f  x   f 0  f ' 0 x 
x
5
3!
 ... 
xr
r!
4
n
0
n!
x r  ...
2 r 1
 2r  1 !
x
x
 1
 all x 
 ...
2r
 2r  !
ln 1  x   x  21 x 2  31 x 3  ...  (1)r 1 
x n  ...
 all x 
x
r
4!
r!
f
 ...
 ...   1 
 ...   1 
2!
x 2  ... 
 n  n  1 ...  n  r  1
x 2  ... 
r
5!
x
x3
f ''  0 
xr
r
 ...
 all x 
 ...
 1  x  1
Trigonometry
sin  A  B   sin A cos B  cos A sin B
cos  A  B   cos A cos B sin A sin B
tan  A  B  
tan A  tan B
1 tan A tan B
sin3A  3 sin A  4 sin3 A
cos 3A  4 cos3 A  3 cos A
sin P  sinQ  2sin 21  P  Q  cos 21  P  Q 
sin P  sinQ  2cos 21  P  Q  sin 21  P  Q 
cos P  cos Q  2cos 21  P  Q  cos 21  P  Q 
cos P  cos Q  2sin 21  P  Q  sin 21  P  Q 
If t  tan 21 x , then:
sin x 
2t
1 t 2
and
cos x 
1 t 2
1 t 2
10
formulae & tables
10-7
Principal values:
 21   sin1 x  21 
x
 1
0  cos1 x  
x
 1
 21   tan1 x  21 
Integrals
(Arbitrary constants are omitted; a denotes a positive constant)
f x
 f  x  dx
1
1
x a
2
2
a
1
a x
2
2
x a
 x a

2a  x  a 
2
1
1
a x
2
x
sin1  
a
1
1
2
1  x 
tan  
a
2
2a
ln 
a x

a x
ln 
ln  sec x  tan x 
sec x
x
 a
 x  a
x
x
 a
 21  
Numerical Methods
Trapezium Rule:
 f  x  dx 
b
a
1
2
h y 0  2  y 1  y 2  ...  y n1   y n  ,
where h 
ba
n
The Newton-Raphson iteration for approximating a root of f  x   0 :
xr 1  xr 
f( xr )
f '( xr )
Vectors
The point dividing AB in the ratio  :  has position vector  a  b

AL eGuide Mathematics
 cosmic
PROBABILITY AND STATISTICS
Sampling and testing
Unbiased variance estimate from a single sample:
s2 
2
1  2  x  
1
2

 x 
 x  x 
n  1
n  n 1


  x1  x1     x2  x2 
2
Two-sample estimate of a common variance:
s2 
2
n1  n2  2
Regression and correlation
Estimated product moment correlation coefficient:
r
  x  x  y  y 
  x  x    y  y  
2
2
xy
n
 2  x  2   2  y  2 
  y 

 x 

n 
n 



xy 

Estimated regression line of y on x :
y  y  b  x2  x2  ,
where b 
  x  x  y  y 
x  x 
2
The Greek Alphabet


alpha


iota


rho


beta


kappa


sigma


gamma


lambda


tau


delta


mu


upsilon


epsilon


nu


phi


zeta


xi


chi


eta


omicron


psi


theta


pi


omega
10
formulae & tables
10-9
THE NORMAL DISTRIBUTION FUNCTION
If Z has a normal distribution with mean 0 and
variance 1 then, for each value of z , the table
gives the value of   z  , where
pdf = (z)
Φ(z)
 z  PZ  z .
For negative values of z use   z   1   z  .
z
z
z
0
1 2 3 4 5 6 7 8 9
ADD
z
0
1
2
3
4
5
6
7
8
9
0.0
0.1
0.2
0.3
0.4
0.5000
0.5398
0.5793
0.6179
0.6554
.5040
.5438
.5832
.6217
.6591
.5080
.5478
.5871
.6255
.6628
.5120
.5517
.5910
.6293
.6664
.5160
.5557
.5948
.6331
.6700
.5199
.5596
.5987
.6368
.6736
.5239
.5636
.6026
.6406
.6772
.5279
.5675
.6064
.6443
.6808
.5319
.5714
.6103
.6480
.6844
.5359
.5753
.6141
.6517
.6879
4
4
4
4
4
8
8
8
7
7
12
12
12
11
11
16
16
15
15
14
20
20
19
19
18
24
24
23
22
22
28
28
27
26
25
32
32
31
30
29
36
36
35
34
32
0.5
0.6
0.7
0.8
0.9
0.6915
0.7257
0.7580
0.7881
0.8159
.6950
.7291
.7611
.7910
.8186
.6985
.7324
.7642
.7939
.8212
.7019
.7357
.7673
.7967
.8238
.7054
.7389
.7704
.7995
.8264
.7088
.7422
.7734
.8023
.8289
.7123
.7454
.7764
.8051
.8315
.7157
.7486
.7794
.8078
.8340
.7190
.7517
.7823
.8106
.8365
.7224
.7549
.7852
.8133
.8389
3
3
3
3
3
7
7
6
5
5
10
10
9
8
8
14
13
12
11
10
17
16
15
14
13
20
19
18
16
15
24
23
21
19
18
27
26
24
22
20
31
29
27
25
23
1.0
1.1
1.2
1.3
1.4
0.8413
0.8643
0.8849
0.9032
0.9192
.8438
.8665
.8869
.9049
.9207
.8461
.8686
.8888
.9066
.9222
.8485
.8708
.8907
.9082
.9236
.8508
.8729
.8925
.9099
.9251
.8531
.8749
.8944
.9115
.9265
.8554
.8770
.8962
.9131
.9279
.8577
.8790
.8980
.9147
.9292
.8599
.8810
.8997
.9162
.9306
.8621
.8830
.9015
.9177
.9319
2
2
2
2
1
5
4
4
3
3
7
6
6
5
4
9
8
7
6
6
12
10
9
8
7
14
12
11
10
8
16
14
13
11
10
19
16
15
13
11
21
18
17
14
13
1.5
1.6
1.7
1.8
1.9
0.9332
0.9452
0.9554
0.9641
0.9713
.9345
.9463
.9564
.9649
.9719
.9357
.9474
.9573
.9656
.9726
.9370
.9484
.9582
.9664
.9732
.9382
.9495
.9591
.9671
.9738
.9394
.9505
.9599
.9678
.9744
.9406
.9515
.9608
.9686
.9750
.9418
.9525
.9616
.9693
.9756
.9429
.9535
.9625
.9699
.9761
.9441
.9545
.9633
.9706
.9767
1
1
1
1
1
2
2
2
1
1
4
3
3
2
2
5
4
4
3
2
6
5
4
4
3
7
6
5
4
4
8
7
6
5
4
10
8
7
6
5
11
9
8
6
5
2.0
2.1
2.2
2.3
2.4
0.9772
0.9821
0.9861
0.9893
0.9918
.9778
.9826
.9864
.9896
.9920
.9783
.9830
.9868
.9898
.9922
.9788
.9834
.9871
.9901
.9925
.9793
.9838
.9875
.9904
.9927
.9798
.9842
.9878
.9906
.9929
.9803
.9846
.9881
.9909
.9931
.9808
.9850
.9884
.9911
.9932
.9812
.9854
.9887
.9913
.9934
.9817
.9857
.9890
.9916
.9936
0
0
0
0
0
1
1
1
1
0
1
1
1
1
1
2
2
1
1
1
2
2
2
1
1
3
2
2
2
1
3
3
2
2
1
4
3
3
2
2
4
4
3
2
2
2.5
2.6
2.7
2.8
2.9
0.9938
0.9953
0.9965
0.9974
0.9981
.9940
.9955
.9966
.9975
.9982
.9941
.9956
.9967
.9976
.9982
.9943
.9957
.9968
.9977
.9983
.9945
.9959
.9969
.9977
.9984
.9946
.9960
.9970
.9978
.9984
.9948
.9961
.9971
.9979
.9985
.9949
.9962
.9972
.9979
.9985
.9951
.9963
.9973
.9980
.9986
.9952
.9964
.9974
.9981
.9986
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
1
0
1
1
1
1
0
Critical values for the normal distribution
If Z has a normal distribution with mean 0 and
variance 1 then, for each value of p , the table
gives the value of z , where
PZ  z  p .
p
0.75
0.90
0.95
0.975
0.99
0.995
0.9975
0.999
0.9995
z
0.674
1.282
1.645
1.960
2.326
2.576
2.807
3.090
3.291
AL eGuide Mathematics
 cosmic