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Pearson Edexcel InternationalAdvanced Subsidiary/AdvancedLevel in Mathematics,Further Mathematicsand Pure MathematicsMathematical Formulae and Statistical TablesFor use in Pearson Edexcel International Advanced Subsidiaryand Advanced Level examinationsPure Mathematics P1 – P4Further Pure Mathematics FP1 – FP3Mechanics M1 – M3Statistics S1 – S3First examination from January 2019This copy is the property of Pearson. It is not to be removed from theexamination room or marked in any way.P59773A 2019 Pearson Education Ltd.1/1/1/1/1/

Edexcel, BTEC and LCCI qualificationsEdexcel, BTEC and LCCI qualifications are awarded by Pearson, the UK’s largest awarding body offeringacademic and vocational qualifications that are globally recognised and benchmarked. For furtherinformation, please visit our qualification website at qualifications.pearson.com. Alternatively, you canget in touch with us using the details on our contact us page at qualifications.pearson.com/contactusAbout PearsonPearson is the world’s leading learning company, with 35,000 employees in more than 70 countriesworking to help people of all ages to make measurable progress in their lives through learning. Weput the learner at the centre of everything we do, because wherever learning flourishes, so do people.Find out more about how we can help you and your learners at qualifications.pearson.comReferences to third party material made in this Mathematical Formulae and Statistical Tables documentare made in good faith. Pearson does not endorse, approve or accept responsibility for the content ofmaterials, which may be subject to change, or any opinions expressed therein. (Material may includetextbooks, journals, magazines and other publications and websites.)All information in this document is correct at time of publication.ISBN 978 1 4469 4983 2All the material in this publication is copyright Pearson Education Limited 2017

ContentsIntroduction1Pure Mathematics P13Mensuration3Cosine rule3Pure Mathematics P23Arithmetic series3Geometric series3Logarithms and exponentials3Binomial series3Numerical integration3Pure Mathematics P34Logarithms and exponentials4Trigonometric identities4Differentiation4Integration5Pure Mathematics P45Binomial series5Integration5Further Pure Mathematics FP16Summations6Numerical solution of equations6Conics6Matrix transformations6Further Pure Mathematics FP27Area of a sector7Complex numbers7Maclaurin’s and Taylor’s Series7Further Pure Mathematics FP38Vectors8Hyperbolic functions9Conics9

Differentiation10Integration10Arc length11Surface area of revolution11Mechanics M111There are no formulae given for M1 in addition to those candidates are expected to know. 11Mechanics M211Centres of mass11Mechanics M312Motion in a circle12Centres of mass12Universal law of gravitation12Statistics S112Probability12Discrete distributions12Continuous distributions13Correlation and regression13The Normal Distribution Function14Percentage Points Of The Normal Distribution15Statistics S216Discrete distributions16Continuous distributions16Binomial Cumulative Distribution Function17Poisson Cumulative Distribution Function22Statistics S323Expectation algebra23Sampling distributions23Correlation and regression23Non-parametric tests23Percentage Points Of The χ 2 Distribution Function24Critical Values For Correlation Coefficients25Random Numbers26

IntroductionThe formulae in this booklet have been arranged by unit. A student sitting a unit may be required to useformulae that were introduced in a preceding unit (e.g. students sitting units P3 and P4 might be expected touse formulae first introduced in units P1 and P2).It may also be the case that students sitting Mechanics and Statistics units need to use formulae introduced inappropriate Pure Mathematics units, as outlined in the specification.No formulae are required for the unit Decision Mathematics D1.Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 20171

2Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Pure Mathematics P1MensurationSurface area of sphere 4 πr 2Area of curved surface of cone πr slant heightCosine rulea2 b2 c2 2bc cos APure Mathematics P2Arithmetic seriesun a (n 1)dSn 11n(a l) n[2a (n 1)d ]22Geometric seriesun ar n 1a (1 r n )1 raS for r 11 rSn Logarithms and exponentialsloga x log b xlog b aBinomial series n n n (a b) n a n a n 1b a n 2b 2 a n rb r b n (n ) 1 2 r n n!where n Cr r r !(n r )!(1 x) n 1 nx n(n 1) 2n(n 1) (n r 1) rx x ( x 1, n )1 21 2 rNumerical integrationThe trapezium rule: bay dx 1b ah{(y0 yn ) 2( y1 y2 . yn 1)}, where h 2nPearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 20173

Pure Mathematics P3Candidates sitting Pure Mathematics P3 may also require those formulae listed underPure Mathematics P1 and P2.Logarithms and exponentialse x ln a a xTrigonometric identitiessin (A B) sin A cos B cos A sin Btan(A B) cos (A B) cos A cos Bsin A sin Btan A tan B1(A B (k ) π)1 tan A tan B2sin A sin B 2 sinA BA Bcos22sin A sin B 2 cosA BA Bsin22cos A cos B 2 cosA BA Bcos22cos A cos B 2 sinA BA Bsin22Differentiationf (x)f ′(x)tan kxk sec2 kxsec xsec x tan xcot x cosec2 xcosec x cosec x cot xf ( x)g( x)f ' ( x) g ( x) f ( x) g' ( x)(g ( x)) 24Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Integration( constant)f (x) f (x) dxsec2 kx1tan kxktan xln sec x cot xln sin x Pure Mathematics P4Candidates sitting Pure Mathematics P4 may also require those formulae listed underPure Mathematics P1, P2 and P3.Binomial series(1 x) n 1 nx n(n 1) 2n(n 1) (n r 1) rx x ( x 1, n )1 21 2 rIntegration( constant) f (x) dxf (x)cosec xsec xdv u dx dx uv 1 ln cosec x cot x , ln tan ( x) 211ln sec x tan x , ln tan ( x π) 24duvdxdxPearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 20175

Further Pure Mathematics FP1Candidates sitting Further Pure Mathematics FP1 may also require those formulae listed underPure Mathematics P1 and P2.Summationsn r2 1n(n 1)(2n 1)63 1 2n (n 1)24r 1n rr 1Numerical solution of equationsThe Newton-Raphson iteration for solving f ( x ) 0 : xn 1 xn f ( xn )f ' ( xn )ConicsParabolaRectangularHyperbolaStandardFormy2 4axxy c2ParametricForm(at 2, 2at) c ct , tFoci(a, 0)Not requiredDirectricesx aNot requiredMatrix transformations cos θAnticlockwise rotation through θ about O: sin θ cos 2θReflection in the line y (tan θ)x: sin 2θ sin θ cos θ sin 2θ cos 2θ In FP1, θ will be a multiple of 45 .6Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Further Pure Mathematics FP2Candidates sitting Further Pure Mathematics FP2 may also require those formulae listed underFurther Pure Mathematics FP1, and Pure Mathematics P1, P2, P3 and P4.Area of a sectorA 1 2r dθ (polar coordinates)2 Complex numberseiθ cos θ i sin θ{r(cos θ i sin θ)}n r n (cos n θ i sin n θ)The roots of z 1 are given by z en2 πk in, for k 0, 1, 2, , n 1Maclaurin’s and Taylor’s Seriesf (x) f (0) x f ′(0) x2x r (r)f ″(0) f (0) r!2!f (x) f (a) (x a) f ′(a) f (a x) f (a) x f ′(a) ex exp (x) 1 x ln(1 x) x ( x a)2( x a ) r (r)f ″(a) f (a) r!2!x2x r (r)f ″(a) f (a) r!2!x2xr for all xr!2!x2x3xr ( 1)r 1 ( 1 x 1)r23sin x x x3x5x 2 r 1 for all x ( 1)r3!5!(2r 1)!cos x 1 x2x4x2r for all x ( 1)r2!4!(2r )!arctan x x x3x5x 2 r 1 ( 1)r ( 1 x 1)352r 1Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 20177

Further Pure Mathematics FP3Candidates sitting Further Pure Mathematics FP3 may also require those formulae listed underFurther Pure Mathematics FP1, and Pure Mathematics P1, P2, P3 and P4.VectorsThe resolved part of a in the direction of b isThe point dividing AB in the ratio λ : μ isa.bbμa λbλ μijVector product: a b a b sin θ n̂ a1 a2b1 b2a1 a2a.(b c) b1 b2c1 c2 a2b3 a3b2 ka3 a3b1 a1b3 a b a b b31 22 1a3b3 b.(c a) c.(a b)c3If A is the point with position vector a a1i a2 j a3k and the direction vector b is given byb b1i b2 j b3k, then the straight line through A with direction vector b has cartesian equationz a3x a1y a2 ( λ)b1b2b3The plane through A with normal vector n n1i n2 j n3k has cartesian equationn1 x n2 y n3 z d 0 where d a.nThe plane through non-collinear points A, B and C has vector equationr a λ(b a) μ(c a) (1 λ μ)a λb μcThe plane through the point with position vector a and parallel to b and c has equationr a sb tcThe perpendicular distance of (α, β, γ) from n1 x n2 y n3 z d 0 is8n1α n2 β n3γ dn12 n22 n32.Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Hyperbolic functionscosh2 x sinh2 x 1sinh 2 x 2 sinh x cosh xcosh 2 x cosh2 x sinh2 xarcosh x ln{x x 2 1} (x 1)arsinh x ln{x x 2 1}artanh x 1 1 ln 2 1 x ( x 1)x Standard Formx2y2 1a2a2y2 4axx2y2 1a2b2xy c2ParametricForm(a cos θ, b sin θ)(at 2, 2at)(a sec θ, b tan θ)( a cosh θ, b sinh θ) c ct , tEccentricitye 1b a2(1 e2)e 1e 1b a2(e2 1)e 2Foci( ae, 0)(a, 0)( ae, 0)( 2c, 2c)DirectricesAsymptotes2x aenonex anone2aexy abx Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017x y 2cx 0, y 09

Differentiationf (x)arcsin xf ′(x)11 x211 x2arccos x arctan x11 x2sinh xcosh xcosh xsinh xtanh xsech2 xarsinh xarcosh xartanh x11 x21x2 111 x2Integration( constant; a 0 where relevant)f (x) f (x) dxsinh xcosh xcosh xsinh xtanh xln cosh x1a x221a2 x21x a22 x arcsin ( x a) a 1 x arctan a a x arcosh , ln{x a x 2 a 2 } (x a)1a2 x21a x1 x artanh ( x a)ln a 2a a xa1x2 a21x aln2a x a101a x22Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Arc lengths s 2 dy 1 d x (cartesian coordinates) dx dx dy dt (parametric form)dtdt22Surface area of revolutionSx 2π y ds 2π 2π y 1 dx dy y dt dt dt 2 dy d xdx 22Mechanics M1There are no formulae given for M1 in addition to those candidates are expected to know.Candidates sitting M1 may also require those formulae listed under Pure Mathematics P1.Mechanics M2Candidates sitting M2 may also require those formulae listed under Pure Mathematics P1, P2, P3 and P4.Centres of massFor uniform bodies:Triangular lamina:2along median from vertex3Circular arc, radius r, angle at centre 2α :r sin αfrom centreαSector of circle, radius r, angle at centre 2α :2r sin αfrom centre3αPearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 201711

Mechanics M3Candidates sitting M3 may also require those formulae listed under Mechanics M2, andPure Mathematics P1, P2, P3 and P4.Motion in a circleTransverse velocity: v rθ̇Transverse acceleration: v̇ rθ̈v2Radial acceleration: rθ̇ 2 rCentres of massFor uniform bodies:3r from centre81Hemispherical shell, radius r: r from centre21Solid cone or pyramid of height h: h above the base on the line from centre of base to vertex41Conical shell of height h: h above the base on the line from centre of base to vertex3Solid hemisphere, radius r:Universal law of gravitationForce Gm1m2d2Statistics S1ProbabilityP(A B) P(A) P(B) P(A B)P(A B) P(A)P(B A)P(A B) P( B A)P( A)P( B A)P( A) P( B A' )P( A' )Discrete distributionsFor a discrete random variable X taking values xi with probabilities P(X xi)Expectation (mean): E(X ) μ xi P(X xi)Variance: Var(X ) σ 2 (xi μ)2 P(X xi) xi2 P(X xi) μ 2For a function g(X ) : E(g(X )) g(xi) P(X xi)12Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Continuous distributionsStandard continuous distribution:Distribution of XP.D.F.Normal N(μ, σ )1 x μ 1 e 2 σ σ 2π2MeanVarianceμσ22Correlation and regressionFor a set of n pairs of values (xi , yi)Sxx (xi x)2 xi2 ( xi ) 2nSyy ( yi y)2 yi2 ( yi ) 2nSxy (xi x)(yi y) xi yi ( xi )( yi )nThe product moment correlation coefficient isr S xyS xx S yy( xi )( yi )n2( xi ) ( yi ) 2 22 xy iin n ( xi x )( yi y ) { ( xi x ) 2 }{ ( yi y ) 2 } The regression coefficient of y on x is b S xyS xx xi yi ( xi x )( yi y ) ( xi x )2Least squares regression line of y on x is y a bx where a y bxPearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 201713

The Normal Distribution FunctionThe function tabulated below is Φ(z), defined as Φ(z) 9332 z e1 t22dt .99990.99991.00001.0000Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Percentage Points Of The Normal DistributionThe values z in the table are those which a random variable Z N(0, 1) exceeds with probability p;that is, P(Z z) 1 Φ(z) Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 201715

Statistics S2Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listedunder Pure Mathematics P1, P2, P3 and P4.Discrete distributionsStandard discrete distributions:Distribution of XP(X x)MeanVarianceBinomial B(n, p) n xn x x p (1 p)npnp(1 p)λxex!λλPoisson Po(λ) λContinuous distributionsFor a continuous random variable X having probability density function fExpectation (mean): E(X ) μ x f (x) d xVariance: Var(X ) σ 2 (x μ)2 f (x) d x x 2 f (x) d x μ 2For a function g(X ): E(g(X )) g(x) f (x) d xCumulative distribution function: F(x0) P(X x0) x0f (t) dt Standard continuous distribution:Distribution of XP.D.F.MeanVarianceUniform (Rectangular) on [a, b]1b a1(a b)21(b a)21216Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

Binomial Cumulative Distribution FunctionThe tabulated value is P(X x), where X has a binomial distribution with index n and parameter p.p n 5, x 01234n 6, x 012345n 7, x 0123456n 8, x 01234567n 9, x 012345678n 10, x 90510.97400.99520.99951.0000Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 300.82810.94530.98930.999017

p n 12, x 01234567891011n 15, x 01234567891011121314n 20, x 0.97930.99410.99870.99981.0000Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 2017

p n 25, x 012345678910111213141516171819202122n 30, x 00001.00001.00001.00001.0000Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics,Further Mathematics and Pure Mathematics – Mathematical Formulae and Statistical Tables– Issue 1 – November 2017 Pearson Education Limited 0.89980.95060.97860.99190.99740.99930.99981.000019

p n 40, x 00001.00001.00000.2

Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and Pure Mathematics Mathematical Formulae and Statistical Tables For use in Pearson Edexcel International Advanced Subsidiary and Advanced Level examinations Pure Mathematics P1 - P4 Further Pure Mathematics FP1 - FP3 Mechanics M1 - M3

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