Mathematical Formulae and Statistical Tables

1 GCEE dexcel GCE in Mathematics Mathematical Formulae and Statistical TablesFor use in Edexcel Advanced Subsidiary GCE and Advanced GCE examinationsCore Mathematics C1 - C4 Further Pure Mathematics FP1 - FP3 Mechanics M1 - M5 Statistics S1 - S4 Modified large print version produced by V I Resourcing LimitedFor use from June 2009 This copy is the property of Edexcel. It is not to be removed from the examination room or marked in any 2 The Formulae in this booklet have been arranged according to the unit in which they are first introduced.

2 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).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 3 TABLE OF CONTENTSPage 14 Core Mathematics C1 14 Mensuration14 Arithmetic series 15 Core Mathematics C2 15 Cosine rule16 Binomial series17 Logarithms and exponentials 17 Geometric series18 Numerical integrationPage 4 TABLE OF CONTENTS (continued)Page19 Core Mathematics C3 19 Logarithms and exponentials 20 -21 Trigonometric identities22 Differentiation23 Core Mathematics C4 24 Integration Page 5 TABLE OF CONTENTS (continued)

3 Page25 Further Pure Mathematics FP125 Summations 26 Numerical solution of equations 27 Conics28 Matrix transformations29 Further Pure Mathematics FP229 Area of sector29 Complex numbers30 31 Maclaurin s and Taylor s SeriesPage 6 TABLE OF CONTENTS (continued)Page32 Further Pure Mathematics FP3 32 -35 Vectors36 Hyperbolics37 Conics38 -39 Differentiation 40 -41 Integration42 Arc length43 Surface area of revolutionPage 7 TABLE OF CONTENTS (continued)Page45 Mechanics M2 45 Centres of mass 46 Mechanics M346 Motion in a circle47 Centres of mass48 Universal law of gravitation44 44 Mechanics M1 There are no Formulae given for M1 in addition to those candidates are expected to know.

4 Page 8 TABLE OF CONTENTS (continued)Page5 Mechanics M5 5 -5 Moments of inertia 5 Moments as vectors4 4 There are no Formulae given for M4 in addition to those candidates are expected to know. Mechanics M4 Page 9 TABLE OF CONTENTSPage54 Statistics S154 Probability55 Discrete distributions56 Continuous distributions 57 - 59 Correlation and regression60-64 The Normal distribution function65 Percentage points of the Normal distributionPage 10 TABLE OF CONTENTS (continued)Page66 Statistics S266 Discrete distributions 67 -68 Continuous distributions69-103 Binomial Cumulative Distribution Function104-111 Poisson Cumulative Distribution FunctionPage 11 TABLE OF CONTENTS (continued)

5 Page112 Statistics S3112 Expectation algebra113 -114 Sampling distributions 115 Correlation and regression115 Non parametric tests116-121 Percentage Points of the 2 Distribution 122-125 Critical Values for Correlation Coefficients126-127 Random NumbersPage 12 TABLE OF CONTENTS (continued)Page128 Statistics S4128-129 Sampling distributions130-132 Percentage Points of Student's t Distribution 133-140 Percentage Points of the F DistributionPage 13 There are no Formulae provided for Decision Mathematics units D1 and Mathematics C1 MensurationSurface area of sphere = 4 r2 Area of curved surface of cone = r slant heightArithmetic seriesun = a + (n 1) dSn = 12n( a + l ) = 12n[2a + (n 1) d]Page 14 Core Mathematics C2 Candidates sitting C2 may also require those Formulae listed under Core Mathematics C1 Cosine rulea2 = b2 + c2 2bc cos APage 15 Binomial series (n ) where = nCr = n!

6 R!(nr)!(1 + x)n = 1 + nx + x2 + + xr + ( I x I < 1, n )nrn2(a + b)n = an + an 1 b + an 2 b2 + + an r br + + bn n1 nr nnnr(1) (1)..1 2 .. rnn(1)12+ Page 16 Logarithms and exponentialslogx=logxlogababGeometric seriesun = ar n 1Sn = arrn()11S = for I r I < 1 ar1 Page 17 Numerical integrationThe trapezium rule: by dx h { (y0 + yn) + 2 (y1 + y2 + + yn 1) },where h = bana12 Page 18 Core Mathematics C3 Candidates sitting C3 may also require those Formulae listed under Core Mathematics C1 and C2 Logarithms and exponentialseaxaxln==Page 19 Trigonometric identitiessin(AB)=sinAcosBcosAsinB cos()coscossinsinABABAB= ) tan(AB)=tanAtanB1tanAtanB (AB(k)12 + Page 20 Trigonometric identities (continued)sinAsinB=2cossinAB2AB2+ cosAcosB=2coscosABAB22+ +coscosAB=2sinsinAB2AB2 +sinAsinB=2sincosAB2AB2+ +Page 21 Differentiationf(x) f (x)

7 Tan kx k sec2 kxsec x sec x tan xcot x cosec2 xcosec x cosec x cot x f(x)g(x)f(x)g(x)(g(x))2f(x)g(x) Page 22 Core Mathematics C4 Candidates sitting C4 may also require those Formulae listed under Core Mathematics C1, C2 and C3 The Formulae start on the next 23 Integration (+ constant)f (x) f (x) dxsec2 kx tan kxtan x lnIsec xIcot x lnIsin xIcosec x lnIcosec x + cot xI ,lnIItan 12( x )sec x u dx = uv v dx 1k214 dvdxdudx Page 24lnIsec x + tan xI , lnItan( 1 x + )IFurther Pure Mathematics FP1 Candidates sitting FP1 may also require those Formulae listed under Core Mathematics C1 and C2 Summations n r 2 = n (n + 1)(2n + 1)r = 1 16 n r3 = n2 (n + 1)2r = 1 14 Page 25 Numerical solution of equationsThe Newton Raphson iteration for solving f(x) = 0 : x = xn fxfxnn()()n + 1 Page 26 ConicsParabolaRectangular Hyperbola Standard Formy2 = 4axxy = c2 Parametric Form(at2, 2at) ct, Foci(a, 0)Not required Directricesx = aNot requiredctPage 27 Matrix transformationsAnticlockwise rotation through about O : Reflection in the line y = (tan ) x.

8 In FP1, will be a multiple of 45 cos sin sin cos cos 2 sin 2 sin 2 cos 2 Page 28 Further Pure Mathematics FP2 Candidates sitting FP2 may also require those Formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1 C4 Area of a sectorA = 12 r 2 d (polar coordinates)Complex numbersei = cos + i sin {r(cos + i sin )}n = rn (cos n + i sin n )The roots of zn = 1 are given by z = ekin2, for k = 0, 1, 2, , n 1 Page 29 Maclaurin s and Taylor s Seriesf(x) = f(0) + x f (0) + x22!f (0) + +xrr!f (r) (0) + f(x) = f(a) + (x a) f ( a ) +()!xa22f (a) + + ()!xarrf (r) (a) + f(a + x) = f(a) + x f (a) + x22!

9 F (a) + + xrr!f (r) (a) + ex = exp(x) = 1 + x + x22!+ +xrr!+ for all xln (1 + x) = x x22 + x33 + ( 1)r + 1 xrr+ ( 1 < x 1) Page 30 Maclaurin s and Taylor s Series (continued)sin x = x x33!+ x55! + ( 1)r xrr2121()!+ for all xcos x = 1 x22!+ x44! + ( 1)r xrr22()! + for all xarctan x = x x33 + x55 + ( 1)r xrr2121 + ( 1 x 1 ) ++++Page 31 Further Pure Mathematics FP3 Candidates sitting FP3 may also require those Formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1 C4 VectorsThe resolved part of a in the direction of b is point dividing AB in the ratio : is I I +ab + Page 32 Vectors (continued)Vector product: a b = IaIIbI sin n = = a.

10 (b c) = = b.(c a) = c.(a b) --i j ka1 a2 a3b1 b2 b3 _____a2b3 a3b2a3b1 a1b3a1b2 a2b1 a1 a2 a3b1 b2 b3c1 c2 c3_____ _____ Page 33 Vectors (continued)If A is the point with position vector a = a1 i + a2 j + a3 k and thedirection vector b is given byb = b1 i + b2 j + b3 k , then the straight line through A withdirection vector b has cartesian equationxab11= yab22= zab33( = )The plane through A with normal vector n = n1 i + n2 j + n3 k has cartesian equation n1 x + n2 y + n3 z + d = 0 where d = a .n Page 34 Vectors (continued)The plane through non collinear points A, B and C has vector equationr = a + (b a) + (c a) = (1 )a + b + cThe plane through the point with position vector a and parallel to b and c hasequation r = a + sb + tcThe perpendicular distance of ( , , ) from n1x + n2y + n3z + d = 0is + +In1 + n2 + n3 + dInnn122232 Page 35 Hyperbolic functionscosh2 x sinh2 x = 1sinh 2x = 2 sinh x cosh xcosh 2x = cosh2 x + sinh2 xarcosh x = ln { x + x2 1 } (x 1)arsinh x = ln { x + x2 + 1 }artanh x = 12 ln ( I x I < 1)