List MF9

UNIVERSITY OF CAMBruMGE INTERNATflONAL EXAMINATIONS

General Certificate of Education Advanced l-,evelGenenal Gertificate of Education Advanced Subsidiiary Level

Advanced International Certificate of Educa*ion

MATHHMATTffiS {ffi?ffiS, S7S$}HTGHHR futA"["fr*ffi ffifr&T"tc$ {ffi r,fi $}

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UNI\rER$ITY of CAMER xIIGEInternationa I Hxa rninations

PUR,M MAT}IEN4A'fYC5

Algebra

For the quadratic equaticn a-rz +bx + c = 0:

, = &*j(b:--{{t}2a

For an arithmetic series:

ua=f,t+(n*1)d, .!n *ln{a+I)=[n{fu:t+(n*l}dl

For a geometric series:

un=a{-t, s"=*3 (r*t)" s*=#; (lrlcr)

Binomial expansion:

(a+b)n = an +[T)"-'r" (;)"-u'.[!),,'*r3 +"..+t,4, where n is a positive inreger

. (n\ n!u"o

f, )=,11"-a

(l + "r)" = | + nx *n(n=-l\ ,z r nfu- lxn - 2) 1

Zl L x' 4. ..., where n is rational and lxl < I

Trtgonornetry

Arc length of circle = r9 {9 in radians)

Area of sectclr of circle = $r20 ( 6 in radians)

tane = lintcos0

cos2g +sin20 = 1, I +tanx6 * sec?8, cot2d * I =6ssss2gsin(A t 8) = sin Acos^8"1-*;osAsin Icos(,A t 8) * eos A cos .E -T sin .4 sin $

tan(A * a; = 3l3ll"** t' l+tan Atanll

sin 24 x 2 sin ,4 cos d

cos24 = cosz A -sinz A s llcosz A-- i c 1-* 2sin?.4

wnLA*J1ry"$"i-- tana A

Principal values:

$rSsin-rx{*lw0Scss-ixSn

j.;"r'< tan-lx"r $n

Dffirentiation

trf r = f(r)

Integration

and y = g(r)

f(x)

lnx

v_

"dvdvdvtnen --i- = -'- -- --dir rir dr

er

sin -r

eos #

f&a x

uv

! \-dl

nf*l

I.X

F;U

cosJ* sin "r

sec2 xdw r-lv

l';-*ff;-fi-T CLT

drr civ1/;- - ld

=-&r drv2

r^, ,

J r(.{J

'r"4

,n+l-*--:"i- {: ttL'#,?+_[

lnlxl+"r

e.r{- ,r

'*co$.{ + c

sir: "r {- c

tfrn.r * r.'

f(x)

-n

IEJ'

sin xcos.r

3sec-;g

( dv, ( du-J"a**-rr- J

v6crx

I f'(x) ,

J tG)*=lnlf(r)l+r

Vectors

If a = af + a2g+ ark and b = hi +W+fuX< r.fiem

a.b * rllfir + azlh*c1b3 * faflblcos#

Numerical integration

Trapeziurn rule:

I:rk)& * *r,{yo + z(yr * 9z *. .."*."r,u-r} .t. yntr ,where h = !=n

MECT{,A,NICS

It niformly ac c ele rat e d motia n

v=u*at, .r={(u+v)/' s=wt+iatz, u2=Lt2+"'hs

Motion of a projectile

Equation of trajectory is: ^ Exz

y = rranu _.ffiffi1

Elastic strings and sPrings

Motian in a circle

For uniform circular rnotion, the acceleration is directed towa.rds the centre and has magnitude

-u2r0r;

Centres of mass af unifarm bodies

Triangular lamina: J atong median frorn vertex

Solid hemisphere of radius r: frr frorn centte

Hemispherical shell of radius r: $r frorn centre

Circular arc of rarlius r and angle * ' l'.I{ from centre

&

Circular sector of radius r and angle fu, T#g from centre

Solid cone or pyramid cf height n: |h frorn vertex

2'xt =T'

PROBASILTT'Y AND STATISTI CS

Summary statistics

For ungrouped data:

standard deviation =

For grouped data:

standard cleviation =

Discrete random variables

E(X) =rro

Var(X) :T'xz P- {E(X)}2

For the binomial distribution E{n, p) :

/n\o. =

L; )n'tt- P)'-n " lt = nP ,

For the Poisson distribution Po(a) :

f*X=-

n

Y,xfx=F

F=E '

oz = np{l- p)

6'2 =arrt

p.'= E-a *-:

,r!'

C ont i nua us random v ar iab I e s

E(X) = J.rf(x)dx

Var(X) = f rz r1x; cx * {E(X)}?

Sampling and testing

Unbiased estimators:

f*T =4,

n

Central l-imit Theorem:

^2 _ ! {,, _e (E")2 )S'- = --";l ,"^.tt- -"- |**rt n )

x - nrfr, $l\l

Approximate distrj bution o f sample propnrtion :

w{'u, ek'$l\&)

TI{E NORKf Ag, $ISTRTBTIT'XffF{ S'T.]NCTTSN

If Z has a normal distribution with m*rn 0 alrd

variance I then, foreach value ofs, the table gives

the value of @(z), where

O(z) = p{Z S r.) "

For negative values of z use S(-e) = 1* <F(e) .

CI.s160 CI.51!)9 il"523s

0.555? 0.5"59S 0.5e360.5948 0.5e87 0.5*260.6331 0.636S CI.ti4.CI{s

0.6700 CI.6735 fi,6'j',??

0.7054 0.7088 fi"?i?30.7389 0,?42? 0.?4540.7?04 f,.7734 0.??&4

0.7995 0.e0x 0.8fi510"82&r- s.8?89 0"s315

0.85$8 CI.8531 0"E554

0.8'7?9 0,8?49 $.877i)0.89?5 0"&944 0.S962

0.9099 0.9115 {!.9i310.9251 0.9265 0.92?9

0.9382 0.9394 0.94060.9495 CI.9505 0.95150.9591 0.9599 {:}"9t5011

0"9671 0.96?8 0.96860.9?38 0.974i4 CI.9?5$

0.9793 0"9?9ll $"9,903

0.9838 0.q84? fr.9&46

s.9875 0.98?8 0.!)SSi

0"99M 0.990fi 0.q9$?0.9927 0.9929 0.993 t

0.9945 0,9946 0.99480.9959 0.99i60 0.9961

0.9959 il.99?0 0"yr7l{}.9s77 $.{}9'18 0.99?9

0.9984 0^9984 0.998.5

{i"5279 CI"531S 0.53590.5fr75 0.5?14 0.57530,6tXr4 0.61CI3 0.6141

0.644:1 Li.6480 0.r5_517

s.6808 0"6844 $.6879

0.?15? i3.7190 0.7224s.T486 0.75t? 0"?549CI.7794 $"7823 0.785?0.80?8 $.8t06 f}.8133{,,s34$ 0.8365 0"8:i89

0.85?7 0.8599 0.8S21

r).8?90 *.8E1{i 0,8830{1.8s}80 0.,$$9? $.90150"9x4? 0.9r62. 0.91]1CI.9?92 {i.9306 0,93 t9

&.9418 0.t4?9 0.9441

CI.9525 0,9*53-5 0.9545(J.s616 CI.9625 0.9633$"9$93 0.969e 0.970i6

8.9?55 0.9?51 0.9?6?

0.9808 0.981? 0.98t?0.9s5CI 0,9854 $.98570.9884. 0.S88? $.9S900.99 r { 0.9913 {!"991 60.9932 S.9S34 0.99-?6

CI.gE4S CI"995t fJ.9952

0.996? 0.9963 0.99640"99?2 0.9973 0.99?40"99?9 0"9980 0,'.;s81

s.9985 r).9986 S.9986

Critical valqres f;'or the nonrm$l distributimm

If Zhas a noffnal distnibution with rnea.n 0 and

variance I then, for each valrte of p" the ta'ble

gives the value ofe such that

l2 3 4 5 6 7 8IADD

4 8124 812't 81247u4 7 ti3 ?!03 71036 935I35 8

?5 724 624 t6

?3 5t3 4

tzl

12 3

r2,3frztl20110t I0l I011ti i.,.I I

00 000 000 000 000 0

0.5040 0.5080 0.-5120

0.5438 0.5478 0.55170.5832 0.58?l 0.5910o,62t7 0.6255 0.6?930.6591 0.5528 0.6664

0.6950 0.6985 0.7019a;t291 A.7324 0.7357

0.7611 0.?542 t.76730.?910 0"7939 4"79670.8186 0.8212 0.8238

0,M38 0"8461 0.84850.8665 0.8686 0.8?080.8869 0.8888 0.89070.9049 0.9066 0.90820.9207 0.9222 $.9236

0.934-5 0"9357 0.93?00.9463 0.9474 0.94840.9564 0.95?3 0.95820.9649 0.9656 0.966l'0"97t9 4.9726 4.9732

0.9778 0,9?83 0.97880.9826 0,9830 0.9834

0.9864 0.9868 0.91i71

0.9896 0.9898 0.99010.9920 0.9922 0.9925

0.9940 0.9941 0.99430.9955 0.9956 0.995?0.9965 0.996? 0.99680"99?5 0.99?6 0.99770.9982 0,9982 0.9983

16 2t 24

t6 2A 24t5 19 ?3t5 t9 27t4 t8 22

t4 t7 20l3 l5 19

l? l5 l8lr 14 16

l0 13 15

91214810127 9 ll6 81067I

28 32 36

28 32 36?7 3t 3526 3A 3425 29 32

24 27 3l23 76 29

21 24 Z7L9 22 2518 20 23

16 19 21

14 t6 l8l3 t5 l7ll 13 t4r0 ll i3

8l0ll78967 8

56645534433 4233222t22llrllllll0ll000

0.00.1

0.20.30.4

0.5

0.60.?

0,80.9

l"ol.lt.21.3

1.4

1.5

1.6

t.t1.8

1.9

2.4z.l2.22.32.4

2.5

2.62.72"8

2.9

0.50000.53984.57910.6r790.6554

0.6915o.7257o.75800.78810.8159

0.84130.86430.88490.90320.9192

o.9332a.94520.95540.96410.9713

o.97720.98210.98610.98930.9918

0.99380.99s30"9965o.99740.9981

P(Z<z)=p,

0.75 0.90 0.95

0.614 1.282 1.&5