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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 $}
$TATE$T$C$ {ffi3$0}
LI$T CIF FORru$UtAE
AF{ffi
TABLH$ OF TFIE NORMA$- fi}I$TR|ffiLTf;ffif{
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