List MF19
List of formulae and statistical tables
Cambridge International AS & A Level
Mathematics (9709) and Further Mathematics (9231)
For use from 2020 in all papers for the above syllabuses.
CST319
*2508709701*
PURE MATHEMATICS
Mensuration
Volume of sphere = 43 πr 3
Surface area of sphere = 4πr 2
Volume of cone or pyramid = 13 × base area × height
Area of curved surface of cone = πr × slant height
Arc length of circle = rθ
( θ in radians)
Area of sector of circle = 12 r 2θ
( θ in radians)
Algebra
For the quadratic equation ax 2 + bx + c = 0 :
x=
−b ± b 2 − 4ac
2a
For an arithmetic series:
un = a + (n − 1)d ,
S n = 12 n( a + l ) = 12 n{2a + (n − 1) d }
For a geometric series:
un = ar n −1 ,
Sn =
a(1 − r n )
1− r
(r ≠ 1) ,
S∞ =
a
1− r
( r <1 )
Binomial series:
n
 n
 n
(a + b) n = a n +   a n −1b +   a n − 2b 2 +   a n −3b3 + K + b n , where n is a positive integer
1
 2
 3
n
n!
and   =
 r  r!(n − r )!
(1 + x) n = 1 + nx +
n(n − 1) 2 n(n − 1)(n − 2) 3
x +
x + K , where n is rational and x < 1
2!
3!
2
Trigonometry
tan θ ≡
cos 2 θ + sin 2 θ ≡ 1 ,
sin θ
cos θ
1 + tan 2 θ ≡ sec 2 θ ,
cot 2 θ + 1 ≡ cosec 2 θ
sin( A ± B) ≡ sin A cos B ± cos A sin B
cos( A ± B) ≡ cos A cos B m sin A sin B
tan( A ± B ) ≡
tan A ± tan B
1 m tan A tan B
sin 2 A ≡ 2sin A cos A
cos 2 A ≡ cos 2 A − sin 2 A ≡ 2cos 2 A − 1 ≡ 1 − 2sin 2 A
tan 2 A ≡
2 tan A
1 − tan 2 A
Principal values:
− 12 π ⩽ sin −1 x ⩽ 12 π ,
0 ⩽ cos −1 x ⩽ π ,
Differentiation
f( x )
f ′( x )
xn
nx n −1
ln x
1
x
ex
ex
sin x
cos x
cos x
− sin x
tan x
sec 2 x
sec x
sec x tan x
cosec x
− cosec x cot x
cot x
− cosec 2 x
tan −1 x
1
1 + x2
uv
v
v
u
v
If x = f(t ) and y = g(t ) then
− 12 π < tan −1 x < 12 π
dy dy dx
=
÷
dx dt dt
3
du
dv
+u
dx
dx
du
dv
−u
dx
dx
v2
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f( x )
∫ f( x ) dx
xn
x n +1
n +1
1
x
ln x
ex
ex
sin x
− cos x
cos x
sin x
sec 2 x
tan x
1
x + a2
1
2
x − a2
1
x
tan −1  
a
a
1
x−a
ln
2a x + a
1
a − x2
1
a+x
ln
2a a − x
2
2
dv
du
∫ u dx dx = uv −∫ v dx dx
f ′( x)
∫ f ( x) dx = ln f ( x)
Vectors
If a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then
a.b = a1b1 + a2b2 + a3b3 = a b cos θ
4
(n ≠ −1)
( x > a)
( x < a)
FURTHER PURE MATHEMATICS
Algebra
Summations:
n
∑
r =1
n
∑
r = 12 n(n + 1) ,
r =1
n
∑ r = n (n + 1)
r 2 = 16 n(n + 1)(2n + 1) ,
3
r =1
1
4
2
2
Maclaurin’s series:
f( x) = f(0) + x f ′(0) +
x2
xr
f ′′(0) + K + f ( r ) (0) + K
r!
2!
e x = exp( x) = 1 + x +
(all x)
x 2 x3
xr
+ − K + (−1) r +1 + K
2
3
r
(–1 < x ⩽ 1)
x3 x5
x 2 r +1
+ − K + (−1) r
+K
3! 5!
(2r + 1)!
(all x)
x2 x4
x2r
+
− K + (−1) r
+K
2! 4!
(2r )!
(all x)
x3 x5
x 2 r +1
+ − K + (−1) r
+K
3
5
2r + 1
(–1 ⩽ x ⩽ 1)
ln(1 + x) = x −
sin x = x −
x2
xr
+K + +K
r!
2!
cos x = 1 −
tan −1 x = x −
sinh x = x +
x3 x5
x 2 r +1
+ +K +
+K
3! 5!
(2r + 1)!
(all x)
x2 x4
x2r
+
+K+
+K
2! 4!
(2r )!
(all x)
x3 x5
x 2 r +1
+ +K +
+K
3
5
2r + 1
(–1 < x < 1)
cosh x = 1 +
tanh −1 x = x +
Trigonometry
If t = tan 12 x then:
sin x =
Hyperbolic functions
cosh 2 x − sinh 2 x ≡ 1 ,
2t
1+ t2
cos x =
and
1− t2
1+ t2
cosh 2 x ≡ cosh 2 x + sinh 2 x
sinh 2 x ≡ 2sinh x cosh x ,
sinh −1 x = ln( x + x 2 + 1)
cosh −1 x = ln( x + x 2 − 1)
1+ x 
tanh −1 x = 12 ln 

1− x 
5
(x ⩾ 1)
(| x | < 1)
Differentiation
f( x )
f ′( x )
sin −1 x
1
1 − x2
cos −1 x
−
1
1 − x2
sinh x
cosh x
cosh x
sinh x
tanh x
sech 2 x
sinh −1 x
1
1 + x2
1
cosh −1 x
x2 − 1
1
1 − x2
tanh −1 x
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f( x )
∫ f( x ) dx
sec x
ln| sec x + tan x | = ln| tan( 12 x + 14 π) |
( x < 12 π )
cosec x
− ln| cosec x + cot x | = ln| tan( 12 x) |
(0 < x < π)
sinh x
cosh x
cosh x
sinh x
sech 2 x
tanh x
1
x
sin −1  
a
( x < a)
 x
cosh −1  
a
( x > a)
2
a −x
2
1
2
x −a
2
 x
sinh −1  
a
1
2
a +x
2
6
MECHANICS
Uniformly accelerated motion
v = u + at ,
v 2 = u 2 + 2as
s = ut + 12 at 2 ,
s = 12 (u + v)t ,
FURTHER MECHANICS
Motion of a projectile
Equation of trajectory is:
y = x tan θ −
gx 2
2V 2 cos 2 θ
Elastic strings and springs
T=
λx
l
E=
,
λ x2
2l
Motion in a circle
For uniform circular motion, the acceleration is directed towards the centre and has magnitude
ω 2r
v2
r
or
Centres of mass of uniform bodies
Triangular lamina: 23 along median from vertex
Solid hemisphere of radius r: 83 r from centre
Hemispherical shell of radius r: 12 r from centre
Circular arc of radius r and angle 2α:
r sin α
Circular sector of radius r and angle 2α:
α
from centre
2r sin α
from centre
3α
Solid cone or pyramid of height h: 34 h from vertex
7
PROBABILITY & STATISTICS
Summary statistics
For ungrouped data:
x=
Σx
,
n
standard deviation =
Σ( x − x ) 2
Σx 2
=
− x2
n
n
x=
Σxf
,
Σf
standard deviation =
Σ( x − x ) 2 f
Σx 2 f
=
− x2
Σf
Σf
For grouped data:
Discrete random variables
Var( X ) = Σx 2 p − {E( X )}2
E( X ) = Σxp ,
For the binomial distribution B(n, p) :
n
pr =   p r (1 − p) n − r ,
r
µ = np ,
σ 2 = np(1 − p )
For the geometric distribution Geo(p):
pr = p(1 − p) r −1 ,
µ=
1
p
For the Poisson distribution Po(λ )
pr = e − λ
λr
r!
,
Continuous random variables
E( X ) = x f( x) dx ,
∫
σ2 =λ
µ =λ ,
∫
Var( X ) = x 2 f( x) dx − {E( X )}2
Sampling and testing
Unbiased estimators:
x=
Σx
,
n
s2 =
Σ( x − x ) 2
1  2 ( Σx ) 2 
=
 Σx −

n −1
n −1
n 
Central Limit Theorem:

σ2 
X ~ N  µ,

n 

Approximate distribution of sample proportion:
p (1 − p) 

N  p,

n


8
FURTHER PROBABILITY & STATISTICS
Sampling and testing
Two-sample estimate of a common variance:
s2 =
Probability generating functions
G X (t ) = E(t X ) ,
Σ( x1 − x1 ) 2 + Σ( x2 − x2 ) 2
n1 + n 2 − 2
E( X ) = G ′X (1) ,
9
Var( X ) = G ′′X (1) + G ′X (1) − {G ′X (1)}2
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 Φ(z), where
Φ(z) = P(Z ⩽ z).
For negative values of z, use Φ(–z) = 1 – Φ(z).
1
2
3
4
6
7
8
9
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
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
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
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 11
8 9
7 8
6 6
5 5
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
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
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
0.5040
0.5438
0.5832
0.6217
0.6591
0.5080
0.5478
0.5871
0.6255
0.6628
0.5120
0.5517
0.5910
0.6293
0.6664
0.5160
0.5557
0.5948
0.6331
0.6700
0.5199
0.5596
0.5987
0.6368
0.6736
0.5239
0.5636
0.6026
0.6406
0.6772
0.5279
0.5675
0.6064
0.6443
0.6808
0.5319
0.5714
0.6103
0.6480
0.6844
0.5359
0.5753
0.6141
0.6517
0.6879
4
4
4
4
4
8
8
8
7
7
12
12
12
11
11
16
16
15
15
14
0.5
0.6
0.7
0.8
0.9
0.6915
0.7257
0.7580
0.7881
0.8159
0.6950
0.7291
0.7611
0.7910
0.8186
0.6985
0.7324
0.7642
0.7939
0.8212
0.7019
0.7357
0.7673
0.7967
0.8238
0.7054
0.7389
0.7704
0.7995
0.8264
0.7088
0.7422
0.7734
0.8023
0.8289
0.7123
0.7454
0.7764
0.8051
0.8315
0.7157
0.7486
0.7794
0.8078
0.8340
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
3
3
3
3
3
7
7
6
5
5
10
10
9
8
8
1.0
1.1
1.2
1.3
1.4
0.8413
0.8643
0.8849
0.9032
0.9192
0.8438
0.8665
0.8869
0.9049
0.9207
0.8461
0.8686
0.8888
0.9066
0.9222
0.8485
0.8708
0.8907
0.9082
0.9236
0.8508
0.8729
0.8925
0.9099
0.9251
0.8531
0.8749
0.8944
0.9115
0.9265
0.8554
0.8770
0.8962
0.9131
0.9279
0.8577
0.8790
0.8980
0.9147
0.9292
0.8599
0.8810
0.8997
0.9162
0.9306
0.8621
0.8830
0.9015
0.9177
0.9319
2
2
2
2
1
5
4
4
3
3
1.5
1.6
1.7
1.8
1.9
0.9332
0.9452
0.9554
0.9641
0.9713
0.9345
0.9463
0.9564
0.9649
0.9719
0.9357
0.9474
0.9573
0.9656
0.9726
0.9370
0.9484
0.9582
0.9664
0.9732
0.9382
0.9495
0.9591
0.9671
0.9738
0.9394
0.9505
0.9599
0.9678
0.9744
0.9406
0.9515
0.9608
0.9686
0.9750
0.9418
0.9525
0.9616
0.9693
0.9756
0.9429
0.9535
0.9625
0.9699
0.9761
0.9441
0.9545
0.9633
0.9706
0.9767
1
1
1
1
1
2.0
2.1
2.2
2.3
2.4
0.9772
0.9821
0.9861
0.9893
0.9918
0.9778
0.9826
0.9864
0.9896
0.9920
0.9783
0.9830
0.9868
0.9898
0.9922
0.9788
0.9834
0.9871
0.9901
0.9925
0.9793
0.9838
0.9875
0.9904
0.9927
0.9798
0.9842
0.9878
0.9906
0.9929
0.9803
0.9846
0.9881
0.9909
0.9931
0.9808
0.9850
0.9884
0.9911
0.9932
0.9812
0.9854
0.9887
0.9913
0.9934
0.9817
0.9857
0.9890
0.9916
0.9936
2.5
2.6
2.7
2.8
2.9
0.9938
0.9953
0.9965
0.9974
0.9981
0.9940
0.9955
0.9966
0.9975
0.9982
0.9941
0.9956
0.9967
0.9976
0.9982
0.9943
0.9957
0.9968
0.9977
0.9983
0.9945
0.9959
0.9969
0.9977
0.9984
0.9946
0.9960
0.9970
0.9978
0.9984
0.9948
0.9961
0.9971
0.9979
0.9985
0.9949
0.9962
0.9972
0.9979
0.9985
0.9951
0.9963
0.9973
0.9980
0.9986
0.9952
0.9964
0.9974
0.9981
0.9986
5
ADD
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 such that
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
10
CRITICAL VALUES FOR THE t-DISTRIBUTION
If T has a t-distribution with ν degrees of freedom, then,
for each pair of values of p and ν, the table gives the value
of t such that:
P(T ⩽ t) = p.
p
0.75
0.90
0.95
ν=1
1.000
3.078
6.314
2
3
4
0.816
0.765
0.741
1.886
1.638
1.533
2.920
2.353
2.132
5
6
7
8
9
0.727
0.718
0.711
0.706
0.703
1.476
1.440
1.415
1.397
1.383
2.015
1.943
1.895
1.860
1.833
10
11
12
13
14
0.700
0.697
0.695
0.694
0.692
1.372
1.363
1.356
1.350
1.345
15
16
17
18
19
0.691
0.690
0.689
0.688
0.688
20
21
22
23
24
0.975
0.99
0.995
0.9975
0.999
0.9995
12.71
31.82
63.66
127.3
318.3
636.6
4.303
3.182
2.776
6.965
4.541
3.747
9.925
5.841
4.604
14.09
7.453
5.598
22.33
10.21
7.173
31.60
12.92
8.610
2.571
2.447
2.365
2.306
2.262
3.365
3.143
2.998
2.896
2.821
4.032
3.707
3.499
3.355
3.250
4.773
4.317
4.029
3.833
3.690
5.894
5.208
4.785
4.501
4.297
6.869
5.959
5.408
5.041
4.781
1.812
1.796
1.782
1.771
1.761
2.228
2.201
2.179
2.160
2.145
2.764
2.718
2.681
2.650
2.624
3.169
3.106
3.055
3.012
2.977
3.581
3.497
3.428
3.372
3.326
4.144
4.025
3.930
3.852
3.787
4.587
4.437
4.318
4.221
4.140
1.341
1.337
1.333
1.330
1.328
1.753
1.746
1.740
1.734
1.729
2.131
2.120
2.110
2.101
2.093
2.602
2.583
2.567
2.552
2.539
2.947
2.921
2.898
2.878
2.861
3.286
3.252
3.222
3.197
3.174
3.733
3.686
3.646
3.610
3.579
4.073
4.015
3.965
3.922
3.883
0.687
0.686
0.686
0.685
0.685
1.325
1.323
1.321
1.319
1.318
1.725
1.721
1.717
1.714
1.711
2.086
2.080
2.074
2.069
2.064
2.528
2.518
2.508
2.500
2.492
2.845
2.831
2.819
2.807
2.797
3.153
3.135
3.119
3.104
3.091
3.552
3.527
3.505
3.485
3.467
3.850
3.819
3.792
3.768
3.745
25
26
27
28
29
0.684
0.684
0.684
0.683
0.683
1.316
1.315
1.314
1.313
1.311
1.708
1.706
1.703
1.701
1.699
2.060
2.056
2.052
2.048
2.045
2.485
2.479
2.473
2.467
2.462
2.787
2.779
2.771
2.763
2.756
3.078
3.067
3.057
3.047
3.038
3.450
3.435
3.421
3.408
3.396
3.725
3.707
3.689
3.674
3.660
30
40
60
120
0.683
0.681
0.679
0.677
1.310
1.303
1.296
1.289
1.697
1.684
1.671
1.658
2.042
2.021
2.000
1.980
2.457
2.423
2.390
2.358
2.750
2.704
2.660
2.617
3.030
2.971
2.915
2.860
3.385
3.307
3.232
3.160
3.646
3.551
3.460
3.373
∞
0.674
1.282
1.645
1.960
2.326
2.576
2.807
3.090
3.291
11
CRITICAL VALUES FOR THE χ 2 -DISTRIBUTION
If X has a χ 2 -distribution with ν degrees of
freedom then, for each pair of values of p and ν,
the table gives the value of x such that
P(X ⩽ x) = p.
p
0.01
0.025
0.05
0.9
0.95
0.975
0.99
0.995
0.999
ν=1
2
3
4
0.031571
0.02010
0.1148
0.2971
0.039821
0.05064
0.2158
0.4844
0.023932
0.1026
0.3518
0.7107
2.706
4.605
6.251
7.779
3.841
5.991
7.815
9.488
5.024
7.378
9.348
11.14
6.635
9.210
11.34
13.28
7.879
10.60
12.84
14.86
10.83
13.82
16.27
18.47
5
6
7
8
9
0.5543
0.8721
1.239
1.647
2.088
0.8312
1.237
1.690
2.180
2.700
1.145
1.635
2.167
2.733
3.325
9.236
10.64
12.02
13.36
14.68
11.07
12.59
14.07
15.51
16.92
12.83
14.45
16.01
17.53
19.02
15.09
16.81
18.48
20.09
21.67
16.75
18.55
20.28
21.95
23.59
20.51
22.46
24.32
26.12
27.88
10
11
12
13
14
2.558
3.053
3.571
4.107
4.660
3.247
3.816
4.404
5.009
5.629
3.940
4.575
5.226
5.892
6.571
15.99
17.28
18.55
19.81
21.06
18.31
19.68
21.03
22.36
23.68
20.48
21.92
23.34
24.74
26.12
23.21
24.73
26.22
27.69
29.14
25.19
26.76
28.30
29.82
31.32
29.59
31.26
32.91
34.53
36.12
15
16
17
18
19
5.229
5.812
6.408
7.015
7.633
6.262
6.908
7.564
8.231
8.907
7.261
7.962
8.672
9.390
10.12
22.31
23.54
24.77
25.99
27.20
25.00
26.30
27.59
28.87
30.14
27.49
28.85
30.19
31.53
32.85
30.58
32.00
33.41
34.81
36.19
32.80
34.27
35.72
37.16
38.58
37.70
39.25
40.79
42.31
43.82
20
21
22
23
24
8.260
8.897
9.542
10.20
10.86
9.591
10.28
10.98
11.69
12.40
10.85
11.59
12.34
13.09
13.85
28.41
29.62
30.81
32.01
33.20
31.41
32.67
33.92
35.17
36.42
34.17
35.48
36.78
38.08
39.36
37.57
38.93
40.29
41.64
42.98
40.00
41.40
42.80
44.18
45.56
45.31
46.80
48.27
49.73
51.18
25
30
40
50
60
11.52
14.95
22.16
29.71
37.48
13.12
16.79
24.43
32.36
40.48
14.61
18.49
26.51
34.76
43.19
34.38
40.26
51.81
63.17
74.40
37.65
43.77
55.76
67.50
79.08
40.65
46.98
59.34
71.42
83.30
44.31
50.89
63.69
76.15
88.38
46.93
53.67
66.77
79.49
91.95
52.62
59.70
73.40
86.66
99.61
70
80
90
100
45.44
53.54
61.75
70.06
48.76
57.15
65.65
74.22
51.74
60.39
69.13
77.93
85.53
96.58
107.6
118.5
90.53
101.9
113.1
124.3
95.02
106.6
118.1
129.6
100.4
112.3
124.1
135.8
104.2
116.3
128.3
140.2
112.3
124.8
137.2
149.4
12
WILCOXON SIGNED-RANK TEST
The sample has size n.
P is the sum of the ranks corresponding to the positive differences.
Q is the sum of the ranks corresponding to the negative differences.
T is the smaller of P and Q.
For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at
the level of significance indicated.
Critical values of T
One-tailed
Two-tailed
n=6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.05
0.1
2
3
5
8
10
13
17
21
25
30
35
41
47
53
60
Level of significance
0.025
0.01
0.05
0.02
0
2
0
3
1
5
3
8
5
10
7
13
9
17
12
21
15
25
19
29
23
34
27
40
32
46
37
52
43
0.005
0.01
0
1
3
5
7
9
12
15
19
23
27
32
37
For larger values of n, each of P and Q can be approximated by the normal distribution with mean 14 n(n + 1)
and variance 241 n(n + 1)(2n + 1) .
13
WILCOXON RANK-SUM TEST
The two samples have sizes m and n, where m ⩽ n.
Rm is the sum of the ranks of the items in the sample of size m.
W is the smaller of Rm and m(n + m + 1) – Rm.
For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of the
null hypothesis at the level of significance indicated.
Critical values of W
One-tailed
Two-tailed
n
3
4
5
6
7
8
9
10
One-tailed
Two-tailed
n
7
8
9
10
0.05
0.1
6
6
7
8
8
9
10
10
0.05
0.1
39
41
43
45
0.025
0.05
m=3
–
–
6
7
7
8
8
9
0.025
0.05
m=7
36
38
40
42
0.01
0.02
0.05
0.1
Level of significance
0.025 0.01
0.05 0.025
0.05
0.02
0.1
0.05
m=4
m=5
–
–
–
–
6
6
7
7
11
12
13
14
15
16
17
10
11
12
13
14
14
15
0.01
0.02
0.05
0.1
Level of significance
0.025 0.01
0.05 0.025
0.05
0.02
0.1
0.05
m=8
m=9
34
35
37
39
51
54
56
49
51
53
–
10
11
11
12
13
13
45
47
49
19
20
21
23
24
26
66
69
17
18
20
21
22
23
62
65
0.01
0.02
0.05
0.1
0.025
0.05
m=6
0.01
0.02
16
17
18
19
20
21
28
29
31
33
35
26
27
29
31
32
24
25
27
28
29
0.01
0.02
0.05
0.1
0.025
0.05
m = 10
0.01
0.02
59
61
82
78
74
For larger values of m and n, the normal distribution with mean 12 m(m + n + 1) and variance 121 mn(m + n + 1)
should be used as an approximation to the distribution of Rm.
14
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16