Edexcel GCE in Mathematics - mathsnetalevel.com

1 GCE Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables For use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinations Core Mathematics C1 C4 Further Pure Mathematics FP1 FP3 Mechanics M1 M5 Statistics S1 S4 For use from January 2005 UA015572 TABLE OF CONTENTS Page 4 Core Mathematics C1 4 Mensuration 4 Arithmetic series 5 Core Mathematics C2 5 Cosine rule 5 Binomial series 5 Logarithms and exponentials 5 Geometric series 5 Numerical integration 6 Core Mathematics C3 6 Logarithms and exponentials 6 Trigonometric identities 6 Differentiation 7 Core Mathematics C4 7 Integration 8 Further Pure Mathematics FP1 8 Summations 8 Area of a sector 8 Numerical solution of equations 9 Further Pure Mathematics FP2 9 Hyperbolic functions 9 Coordinate geometry 9 Conics 10 Differentiation 10 Integration 11 Curvature 11 Arc length 11 Surface area of revolution 12 Further Pure Mathematics FP3 12 Complex numbers 12 Maclaurin s and Taylor s Series 13 Vectors 14 Matrix transformations 14 Taylor polynomials 14 Numerical differentiation

2 UA015572 Edexcel AS/A level Mathematics Formulae List: C1 C4, FP1 FP3 Contents Page Issue 1 November 2004 1 16 Mechanics M1 16 There are no formulae given for M1 in addition to those candidates are expected to know. 16 Mechanics M2 16 Centres of mass 16 Mechanics M3 16 Motion in a circle 16 Centres of mass 16 Universal law of gravitation 17 Mechanics M4 17 There are no formulae given for M4 in addition to those candidates are expected to know. 17 Mechanics M5 17 Moments of inertia 17 Moments as vectors 18 Statistics S1 18 Probability 18 Discrete distributions 18 Continuous distributions 19 Correlation and regression 20 The Normal distribution function 21 Percentage points of the Normal distribution 22 Statistics S2 22 Discrete distributions 22 Continuous distributions 23 Binomial cumulative distribution function 28 Poisson cumulative distribution function 29 Statistics S3 29 Expectation algebra 29 Sampling distributions 29 Correlation and regression 29 Non-parametric

3 Tests 30 Percentage points of the 2 distribution 31 Critical values for correlation coefficients 32 Random numbers 33 Statistics S4 33 Sampling distributions 34 Percentage points of Student s t distribution 35 Percentage points of the F distribution There are no formulae provided for Decision Mathematics units D1 and UA015572 Edexcel AS/A level Mathematics Formulae List: M1 M5, S1 S4 Contents Page Issue 1 November 2004 The formulae in this booklet have been arranged according to the unit in which they are first introduced. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit ( candidates sitting C3 might be expected to use formulae first introduced in C1 or C2).

4 It may also be the case that candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification. UA015572 Edexcel AS/A level Mathematics Formulae List Issue 1 November 2004 3 Core Mathematics C1 Mensuration Surface area of sphere = 4 r 2 Area of curved surface of cone = r slant height Arithmetic series un = a + (n 1)d Sn = 21n(a + l) = 21n[2a + (n 1)d] 4 UA015572 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 Issue 1 November 2004 Core Mathematics C2 Candidates sitting C2 may also require those formulae listed under Core Mathematics C1. Cosine rule a2 = b2 + c2 2bc cos A Binomial series 2 1)(221nrrnnnnnab=+bbarnbanbana++ ++ + + KK (n ) )!

5 (!!C rnrnrnrn == where <+ + ++ ++=+nnnxxn21)1(1)1(nxxrrnnnxr ,1( 21)1()1( 2 KKKK ) Logarithms and exponentials axxbaloglog=blog eometric series un = ar 1 Sn = G nr ran 1)1( S = r 1 for r < 1 a Numerical integration The trapezium rule: baxyd 21h{(y0 + yn) + 2(y1 + y2 + .. + yn 1)}, where nabh = UA015572 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 Issue 1 November 2004 5 Core Mathematics C3 Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2. Logarithms and exponentials xaxa=lne Trigonometric identities BABABA sincoscossin)(sin = BABABA sinsincoscos)(cosm= ))(( tantan1tantan)(tan21 + = kBABABABAm 2cos2sin2sinsinBABABA +=+ 2sin2cos2sinsinBABABA += 2cos2cos2coscosBABABA +=+ coscosA 2sin2sin2 BABAB + = Differentiation sec x sec x tan x x sec2 x cosec x cosec x cot x f(x) f (x) tan kx k sec2 kx cot co)g()f(xx ))(g()(g)f( )g()(f2xxxxx 6 UA015572 Edexcel AS/A level Mathematics Formulae List.)

6 Core Mathematics C3 Issue 1 November 2004 Core Mathematics C4 Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3. Integration (+ constant) f(x) xxd)f( sec2 k k1 tan kx xxsecln xtanxsinln xcotxcosec )tan(lncotcosecln21xxx=+ xsec )tan(lntansecln4121 +=+xxx =xxuvuvxxvudddddd UA015572 Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 Issue 1 November 2004 7 Further Pure Mathematics FP1 Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1, C2, C3 and C4. Summations )12)(1(6112++= =nnnrnr 224113)1(+= =nnrnr Area of a sector A = d22r (polar coo 1rdinates) umerical solution of equations The Newton-Raphson iteration for solving N0)f(=x: )(f)f(1nnnnxxxx =+ 8 UA015572 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 Issue 1 November 2004 Further Pure Mathematics FP2 Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1 C4.

7 Hyperbolic functions 1sinhcosh22= xx xxxcoshsinh22sinh= xxx22sinhcosh2cosh+= )1( 1lnarcosh}{2 +=xxxx }{1lnarsinh2++=xxx )1( 1lnartanh2 =x11< +xxx Coordinate geometry The perpendicular distance from (h, k) to 0=++cbyax is 22bacbkah+++ m1 and is m221211arctanmmmm+ The acute angle between lines with gradients Conics Ellipse Parabola Hyperbola Rectangular Hyperbola Standard Form 12222=+byax axy42= 122= yx 22ba2cxy= Parametric Form )sin ,cos( ba (a sec , b tan ( a cosh , b sinh ) )2 ,(2atat ) ) ,(tcct Eccentricity 1<e )1(222eab = 1=e 1>e )1(222 =eab e = 2 Foci )0 ,(ae )0 ,(a )0 ,(ae ( 2c, 2c) Directrices ea =x ax = eax = x + y = 2c byax = 0 ,0==yx Asymptotes none none UA015572 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 Issue 1 November 2004 9 Differentiation f(x) f (x) xarcsin 211x xarccos 211x 211x+xatarcn xsinh xcosh xxsinh cosh xtanh x2sech xarsinh 21x+ 1xarcosh 112 x h x 211x artan Integration(+ constant.)

8 A where relevant) f(x) >0 xxd)f( xsinh xcosh xcosh xsinh xtanh xcoshln 221xa )( arcsinaxax< 221xa+ axaarctan1 )( lnarcosh}{22axaxxax> += 221ax 221xa+ }{22lnarsinhaxxax++= 221xa )( artanh1ln21axaxaxaxaa< = + 221ax axaxa+ ln21 10 UA015572 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 Issue 1 November 2004 Curvature Radius of curvature: = 1 = ds =d 22232dddd1xyxy = +yxyxyx&&&&&&&& +2322)( Arc length xxysddd12 += (cartesian coordinates) ttyxdd22 tddd + = (parametric form) Surface area of revolution s 22dd2d2 dddxxySysyttt ==+ UA015572 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 Issue 1 November 2004 11 Further Pure Mathematics FP3 Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1 and FP2, and Core Mathematics C1 C4.

9 Complex numbers sinicosei+= )sini(cos)}sini(cos{ nnrrnn+=+ The roots of 1=n are givenz by nki2e , for z=1 , ,2 ,1 ,0 =nkK Maclaurin s and Taylor s Series KK )0(f! )0(f!2)0(f)0f()f()(2+++ + +=rrrxxxx KK )(f!)( )(f!2)()(f)()f()f()(2+ ++ + +=araxaaxaaxaxrr KK )(f! )(f!2)(f)f()f()(2+++ + +=+arxaxaxaxarr xrxxxxrx allfor ! !21)exp(e2KK+++++== )11( )1( 32)1(ln132 < + + + =++xrxxxxxrrKK xrxxxxxrr allfor )!12()1( !5!3sin1253KK++ + + =+ xrxxxxrr allfor )!2()1( !4!21cos242KK+ + + = )11( 12)1( 53arctan1253 ++ + + =+xrxxxxxrrKK xrxxxxxr allfor )!12( !5!3sinh1253KK++++++=+ xrxxxxr allfor )!2( !4!21cosh242KK+++++= )11( 12 53artanh1253<< ++++++=+xrxxxxxrKK 12 UA015572 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 Issue 1 November 2004 Vectors The resolved part of a in the direction of b is : is ++ba The point dividing AB in the ratio Vector product.

10 === 122131132332321321 sinbababababababbbaaakjinbaba )()()( = == cccbbb )()()( = If A is the point with position vector aakji++= and given by b321bbb++=, then the straight line through A with direction vector b has cartesian quation the direction vector b isikje)( 321 = = = azayax 321bbb The plane through A with normal vector kjin321nnn++= has cartesian equation ==+++ddznynxn where0321 r points The plane through non-collineaA, B and C has vector equation cbaacabar ++ = + +=)1()()( The plane through the point with position vector a and parallel to b and c has equation The perpendicular distance of cbarts++= ) , ,( from 0321=+++dznynxn is 232221321nnndnnn+++++ . UA015572 Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 Issue 1 November 2004 13 Matrix transformations Anticlockwise rotation through about O: co cos sinsins Reflection in the line xy)(tan =: 2cos2sin 2sin 2cosThe mtions (in 3D) through an angle atrices for rota about one of the axes are cos010 for the y-axis for the z-axis aylor polynomials cossin0sincos0001 for the x-axis 0sin sin0cos 1000cossin0sincos Terror)(f!