Formulae and Statistical Tables for GCE Mathematics and ...

1 First Issued September 2004 For the new specifications for first teaching from September 2004 GCE Mathematics ADVANCED SUBSIDIARY Mathematics (5361) ADVANCED SUBSIDIARY PURE Mathematics (5366) ADVANCED SUBSIDIARY FURTHER Mathematics (5371) ADVANCED Mathematics (6361) ADVANCED PURE Mathematics (6366) ADVANCED FURTHER Mathematics (6371) GCE Statistics ADVANCED SUBSIDIARY STATISTICS (5381) ADVANCED STATISTICS (6381) Formulae and Statistical Tablesfor GCE Mathematics and GCE StatisticsGeneral Certificate of Education 19 MSPM Further copies of this booklet are available from: AQA Logistics Centre (Guildford), Deacon Field Office, Stag Hill House, Guildford, Surrey, GU2 7XJ Telephone: 0870 410 1036 Fax: 01483 452819 or download from the AQA website Copyright 2010 AQA and its licensors.

2 All rights reserved. COPYRIGHT AQA retains the copyright on all its publications, including the specimen units and mark schemes/ teachers guides. However, registered centres of AQA are permitted to copy material from this booklet for their own internal use, with the following important exception: AQA cannot give permission to centres to photocopy any material that is acknowledged to a third party even for internal use within the centre. Set and published by the Assessment and Qualifications Alliance. The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 3644723 and a registered charity number 1073334.

3 Registered address AQA, Devas Street, Manchester, M15 6EX. 3klj Contents Page 4 Pure Mathematics 9 Mechanics 10 Probability and Statistics Statistical Tables 15 Table 1 Cumulative Binomial Distribution Function 22 Table 2 Cumulative Poisson Distribution Function 24 Table 3 Normal Distribution Function 25 Table 4 Percentage Points of the Normal Distribution 26 Table 5 Percentage Points of the Student s t-Distribution 27 Table 6 Percentage Points of the 2 Distribution 28 Table 7

4 Percentage Points of the F-Distribution 30 Table 8 Critical Values of the Product Moment Correlation Coefficient 31 Table 9 Critical Values of Spearman s Rank Correlation Coefficient 32 Table 10 Critical Values of the Wilcoxon Signed Rank Statistic 33 Table 11 Critical Values of the Mann-Whitney Statistic 34 Table 12 Control Charts for Variability 35 Table 13 Random Numbers klm 4 PURE Mathematics Mensuration Surface area of sphere 2 4r= Area of curved surface of cone heightslant =r Arithmetic series []dnanlanSdnaunn)1(2)()1( =2121 +=+= + Geometric series 1 for 11)1( 1< = == rraSrraSraunnnn Summations ()1211+= =nnrnr )12)(1(6112++= =nnnrnr 224113)1(+= =nnrnr Trigonometry the Cosine rule Abccbacos2222 += Binomial Series ++ ++ + +=+ nbbarnbanbanabanrrnnnnn( 2 1)( ) where )!

5 (!!C rnrnrnrn == <++ ++ ++=+nxxrrnnnxnnnxxrn ,1( )1()1( )1(1)1( ) Logarithms and exponentials axxalne= Complex numbers )sini(cos)}sini(cos{ nnrrnn+=+ sinicosei+= The roots of 1=nz are given by nkzi 2e=, for 1 , ,2 ,1 ,0 = N R 5klj Maclaurin s series .. )0(f! )0(f!2)0(f)0f()f()(2+++ + +=rrrxxxx xrxxxxrx allfor ! !21)exp( +++++== xrxxxxxrr< + + + =++1( )1( 32)1ln( 1) xrxxxxxrr allfor )!12()1( !5! ++ + + =+ xrxxxxrr allfor )!2()1( !4! + + + = Hyperbolic functions 1sinhcosh22= xx xxxcoshsinh22sinh= xxx22sinhcosh2cosh+= xxxx( 1lncosh}{21 += )1 }{1lnsinh21++= xxx )1( 11lntanh211< += xxxx Conics Ellipse Parabola Hyperbola Rectangular hyperbola Standard form 12222=+byax axy42= 12222= byax 2cxy= Asymptotes none none byax = 0,0==yx Trigonometric identities BABABA sincoscossin)sin( = BABABA sinsincoscos)cos( = () )( tantan1tantan)

6 Tan(21+ = kBABABABA 2cos2sin2sinsinBABABA +=+ 2sin2cos2sinsinBABABA += 2cos2cos2coscosBABABA +=+ 2sin2sin2coscosBABABA + = klm 6 Vectors The resolved part of a in the direction of b is The position vector of the point dividing AB in the ratio : is ++ba Vector product: === 122131132332332211 sinbababababababababakjinbaba If A is the point with position vector kjia321aaa++= and the direction vector b is given by kjib321bbb++=, then the straight line through A with direction vector b has cartesian equation = = = 332211bazbaybax The plane through A with normal vector kjin321nnn++= has cartesian equation dznynxn=++321 where The plane through non-collinear points CBA and , has vector equation cbaacabar ++ = + +=)1()()

7 ( The plane through the point with position vector a and parallel to b and c has equation cbarts++= Matrix transformations Anticlockwise rotation through about O: cos sinsincos Reflection in the line :)(tanxy = 2cos2sin2sin 2cos The matrices for rotations (in three dimensions) through an angle about one of the axes are cossin0sincos0001 for the x-axis cos0sin010sin0cos for the y-axis 1000cossin0sincos for the z-axis 7klj Differentiation )f(x )(fx x1sin 211x x1cos 211x x1tan 211x+ kxtan kxk2sec xcosec xxcotcosec xsec xxtansec xcot x2cosec xsinh xcosh xcosh xsinh xtanh x2sech x1sinh 211x+ x1cosh 112 x x1tanh 211x )( g)( fxx 2))( g()( g )f()g( )

8 ( fxxxxx Integration (+ constant; 0>a where relevant) )f(x xxd)f( xtan xsecln xcot xsinln xcosec )tan(lncotcosecln21xxx=+ xsec ) tan(lntansecln4121+=+xxx kx2sec kxktan 1 xsinh xcosh xcosh xsinh xtanh xcoshln INTEGRATION Formulae CONTINUE OVER THE PAGE klm 8221xa )( sin1axax< 221xa+ axa1tan1 221ax )( lnor cosh}{221axaxxax> + 221xa+ }{221lnor sinhaxxax++ 221xa )( tanh1ln211axaxaxaxaa< = + 221ax axaxa+ ln21 =xxuvuvxxvudddddd Area of a sector = d212rA (polar coordinates))

9 Arc length +=xxysddd12 (cartesian coordinates) + =22ddddtytxstd (parametric form) Surface area of revolution xxyySxddd1 22 += (cartesian coordinates) + =ttytxySxddddd 222 (parametric form) Numerical integration The trapezium rule: +++++ bannyyyyyhxy 121021)} (2){(d .., where nabh = The mid-ordinate rule: ++++ bannyyyyhxy 21232321) (d .., where nabh = Simpson s rule: ()()( ){} +++++++++ bannnyyyyyyyyhxy where nabh = and n is even 9klj Numerical solution of differential equations For )(fddxxy= and small h, recurrence relations are: Euler s method: ).

10 (f1nnnxhyy+=+ hxxnn+=+1 For :) ,f(ddyxxy= Euler s method: ) ,f(1rrrryxhyy+=+ Improved Euler method: ) ,f( ), ,f( where),(12121211kyhxhkyxhkkkyyrrrrrr++== ++=+ Numerical solution of equations The Newton-Raphson iteration for solving :0)f(=x )(f)f(1nnnnxxxx =+ MECHANICS Motion in a circle Transverse velocity: rv= Transverse acceleration: rv= Radial acceleration: rvr22 = Centres of mass For uniform bodies Triangular lamina: 32 along median from vertex Solid hemisphere, radius r: r83 from centre Hemispherical shell, radius r: r21 from centre Circular arc, radius r, angle at centre :2 sinr from centre Sector of circle, radius r, angle at centre :2 3sin2r from centre Solid cone or pyramid of height h: h41 above the base on the line from centre of base to vertex Conical shell of height h: h31 above the base on the line from centre of base to vertex Moments of inertia For uniform bodies of mass m Thin rod, length 2l, about perpendicular axis through centre: 231ml Rectangular lamina about axis in plane bisecting edges of length 2l.)