417318 list of formulae and statistical tables PDF

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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 =

4 3

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 ax2 + 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 ) ( r ≠ 1) , 1−r

S∞ =

a 1− r

(

r a )

(

x < a)

FURTHER PURE MATHEMATICS Algebra Summations: n

∑r =

n

1 2 n (n

∑r

+ 1) ,

r =1

n

2

∑r

= 16 n( n + 1)(2n + 1) ,

r =1

3

= 14 n 2 (n + 1)2

r =1

Maclaurin’s series:

f( x ) = f(0) + x f ′(0) +

x2 x r (r) f′′ (0) + K + f (0) + K r! 2!

e x = exp( x) = 1 + x +

(all x)

x2 x3 xr + − K + ( −1) r+1 + K r 2 3

(–1 < x ⩽ 1)

x3 x5 x2 r+1 + − K + (− 1)r +K 3! 5! (2 r + 1)!

(all x)

x2 x4 x2r + − K + (−1)r +K 2! 4! (2r )!

(all x)

ln(1 + x) = x − sin x = x −

x2 xr +K + +K r! 2!

cos x = 1 −

tan −1 x = x −

x3 x5 x2 r +1 + − K + ( −1)r +K 3 5 2 r +1

sinh x = x +

(–1 ⩽ x ⩽ 1)

x 3 x5 x2 r + 1 + +K+ +K 3! 5! (2 r + 1)!

(all x)

x2 x4 x 2r + + K+ +K 2! 4! (2r )!

(all x)

cosh x = 1 +

tanh −1 x = x +

x3 x5 x2 r+1 + + K+ +K 3 5 2r + 1

(–1 < x < 1)

Trigonometry If t = tan 12 x then:

sin x =

Hyperbolic functions cosh2 x − sinh2 x ≡ 1,

2t 1+t2

and

cos x =

sinh 2 x ≡ 2sinh xcosh x , sinh cosh

−1

−1

1− t 2 1+ t 2

cosh 2 x ≡ cosh 2 x + sinh 2 x

x = ln ( x + x2 +1 )

x = ln( x + x2 −1)

 1+ x  − tanh 1 x = 21 ln    1− x 

5

(x ⩾ 1) (| x | < 1)

Differentiation

f( x )

f ′( x )

sin −1 x

1 1− x 2

−

cos 1 x

−

1 1 − x2

sinh x

cosh x

coshx

sinh x

tanh x

sech 2 x

sinh −1 x

1 1+ x 2 1

cosh −1 x

x 2 −1 1 1 −x 2

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) |

(

cosecx

− ln| cosec x + cot x | = ln| tan( 12 x ) |

(0 < x < π)

sinh x

coshx

cosh x

sinh x

sech2 x

tanh x

1

x  sin −1    a

2

a −x

2

x cosh −1    a

1 2

x −a

2

x sinh −1    a

1 2

a +x

2

6

x < 12 π )

(

x < a)

(x > a )

MECHANICS Uniformly accelerated motion v = u + at ,

s = ut + 12 at 2 ,

s = 12 ( u + v) t ,

v 2 = u 2 + 2as

FURTHER MECHANICS Motion of a projectile Equation of trajectory is:

y = x tan θ −

gx 2 2V cos2 θ 2

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: 38 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: Solid cone or pyramid of height h:

3 4

α

from centre

2r sin α from centre 3α

h from vertex

7

PROBABILITY & STATISTICS Summary statistics For ungrouped data: x=

Σx , n

Σ( x − x ) 2 Σx 2 = −x2 n n

standard deviation =

For grouped data: x=

Σxf , Σf

standard deviation =

Σ ( x − x )2 f Σ x2 f = − x2 Σf Σf

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 

σ 2 = np(1 − p)

µ = np ,

For the geometric distribution Geo(p): − p r = p (1 − p ) r 1 ,

µ=

1 p

For the Poisson distribution Po( λ) −λ

pr = e

λr r!

,

Continuous random variables E( X ) = x f(x ) dx ,

∫

µ = λ,

σ 2 =λ

∫

Var( X ) = x2 f( x) d x − {E( X )}2

Sampling and testing Unbiased estimators:

x=

Σx , n

s2 =

1  2 (Σx )2  Σ (x − x )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

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

20 20 19 19 18

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

14 13 12 11 10

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

7 6 6 5 4

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 2 2 1 1

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

0 0 0 0 0

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

0 0 0 0 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

0.5 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

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

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

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

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

5 6 ADD

7

8

9

24 24 23 22 22

28 28 27 26 25

32 32 31 30 29

36 36 35 34 32

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

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

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

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

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 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

0.975

0.99

0.995

0.9975

0.999

0.9995

ν=1

1.000

3.078

6.314

12.71

31.82

63.66

127.3

318.3

636.6

2 3 4

0.816 0.765 0.741

1.886 1.638 1.533

2.920 2.353 2.132

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

5

0.727

1.476

2.015

2.571

3.365

4.032

4.773

5.894

6.869

6 7 8 9

0.718 0.711 0.706 0.703

1.440 1.415 1.397 1.383

1.943 1.895 1.860 1.833

2.447 2.365 2.306 2.262

3.143 2.998 2.896 2.821

3.707 3.499 3.355 3.250

4.317 4.029 3.833 3.690

5.208 4.785 4.501 4.297

5.959 5.408 5.041 4.781

10

0.700

1.372

1.812

2.228

2.764

3.169

3.581

4.144

4.587

11 12 13 14

0.697 0.695 0.694 0.692

1.363 1.356 1.350 1.345

1.796 1.782 1.771 1.761

2.201 2.179 2.160 2.145

2.718 2.681 2.650 2.624

3.106 3.055 3.012 2.977

3.497 3.428 3.372 3.326

4.025 3.930 3.852 3.787

4.437 4.318 4.221 4.140

15

0.691

1.341

1.753

2.131

2.602

2.947

3.286

3.733

4.073

16 17 18 19

0.690 0.689 0.688 0.688

1.337 1.333 1.330 1.328

1.746 1.740 1.734 1.729

2.120 2.110 2.101 2.093

2.583 2.567 2.552 2.539

2.921 2.898 2.878 2.861

3.252 3.222 3.197 3.174

3.686 3.646 3.610 3.579

4.015 3.965 3.922 3.883

20

0.687

1.325

1.725

2.086

2.528

2.845

3.153

3.552

3.850

21 22 23 24

0.686 0.686 0.685 0.685

1.323 1.321 1.319 1.318

1.721 1.717 1.714 1.711

2.080 2.074 2.069 2.064

2.518 2.508 2.500 2.492

2.831 2.819 2.807 2.797

3.135 3.119 3.104 3.091

3.527 3.505 3.485 3.467

3.819 3.792 3.768 3.745

25

0.684

1.316

1.708

2.060

2.485

2.787

3.078

3.450

3.725

26 27 28 29

0.684 0.684 0.683 0.683

1.315 1.314 1.313 1.311

1.706 1.703 1.701 1.699

2.056 2.052 2.048 2.045

2.479 2.473 2.467 2.462

2.779 2.771 2.763 2.756

3.067 3.057 3.047 3.038

3.435 3.421 3.408 3.396

3.707 3.689 3.674 3.660

30

0.683

1.310

1.697

2.042

2.457

2.750

3.030

3.385

3.646

40 60 120

0.681 0.679 0.677 0.674

1.303 1.296 1.289 1.282

1.684 1.671 1.658 1.645

2.021 2.000 1.980 1.960

2.423 2.390 2.358 2.326

2.704 2.660 2.617 2.576

2.971 2.915 2.860 2.807

3.307 3.232 3.160 3.090

3.551 3.460 3.373 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 ...