DRAFT. Formulae and Statistical Tables for A-level Mathematics SPECIMEN MATERIAL. First Issued September 2017

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1 Fist Issued Septembe 07 Fo the ew specifictios fo fist techig fom Septembe 07 SPECIMEN MATERIAL Fomule d Sttisticl Tbles fo A-level Mthemtics AS MATHEMATICS (7356) A-LEVEL MATHEMATICS (7357) AS FURTHER MATHEMATICS (7366) A-LEVEL FURTHER MATHEMATICS (7367)

2 Futhe copies of this booklet e vilble fom: Telephoe: F: o dowlod fom the AQA website Copyight 06 AQA d its licesos. All ights eseved. COPYRIGHT AQA etis the copyight o ll its publictios, icludig the specime uits d mk schemes/ teches guides. Howeve, egisteed cetes of AQA e pemitted to copy mteil fom this booklet fo thei ow itel use, with the followig impott eceptio: AQA cot give pemissio to cetes to photocopy y mteil tht is ckowledged to thid pty eve fo itel use withi the cete. Set d published by the Assessmet d Qulifictios Allice. The Assessmet d Qulifictios Allice (AQA) is compy limited by gutee egisteed i Egld d Wles d egisteed chity umbe Registeed ddess AQA, Devs Steet, Mcheste, M5 6EX.

3 Cotets Pge 4 Pue Mthemtics 0 Mechics 0 Pobbility d Sttistics Sttisticl Tbles Tble Cumultive Poisso Distibutio Fuctio 4 Tble Pecetge Poits of the Studet s t-distibutio 5 Tble 3 Pecetge Poits of the χ Distibutio 6 Tble 4 Citicl Vlues of the Poduct Momet Coeltio Coefficiet 3

4 PURE MATHEMATICS Biomil seies ( + b) = + b+ b + + b + + b ( )! whee = C =!( )! ( ) ( ) ( + ) ( + ) = ( <, ) Aithmetic seies S= ( + l) = + ( ) d Geometic seies ( ) S = S = fo < Tigoomety: smll gles Fo smll gle θ, siθ θ cos θ θ tθ θ [ ] Tigoometic idetities si( A± B) = si Acos B± cos Asi B cos( A± B) = cos Acos B si Asi B t( A± B) = t A± t B ± ( + ) t At B si A+ si B = si A+ B cos A B si A si B = cos A+ B si A B cos A+ cos B = cos A+ B cos A B cos A cos B = si A+ B si A B ( A B k π) 4

5 Diffeetitio f( ) f ( ) t k k sec k cosec cosec cot sec sec t cot f ( ) g ( ) si cos t cosec f ( ) g ( ) f() g () ( g ( )) th sih cosh th + sech + 5

6 Itegtio u dv d = uv v du d d d (+ costt; > 0 whee elevt) f( ) f( ) d t cot l sec l si cosec l cosec + cot = l t ( ) sec l sec + t = l t ( + π) sec k th + + t k k l cosh si ( < ) t Numeicl solutio of equtios 4 { } cosh o l + ( > ) sih o l{ } l th = ( < ) l + The Newto-Rphso itetio fo solvig f( ) = 0 : Numeicl itegtio b f( ) = + f( ) The tpezium ule: y d h {( y 0 + y ) + ( y + y y + + )}, whee Comple umbes [ ( cos θ + i si θ)] = ( cos θ + i si θ) The oots of z = e give by z e π = ki, fo k = 0,,,, b h = 6

7 Mti tsfomtios Aticlockwise ottio though θ bout O: Reflectio i the lie y = (t θ ) : cos θ siθ cos θ si θ si θ cos θ siθ cos θ The mtices fo ottios (i thee dimesios) though gle θ bout oe of the es e Summtios = = cos θ siθ fo the -is 0 siθ cos θ cos θ 0 siθ 0 0 fo the y-is siθ 0 cos θ cos θ siθ 0 siθ cos θ 0 fo the z-is 0 0 = ( + )( + ) 6 4 = ( + ) 3 Mclui s seies = !! ( ) f( ) f( 0) f( 0) f ( 0) f ( 0) e = ep( ) = fo ll!! 3 + l( + ) = + + ( ) + ( < ) si = + + ( ) + fo ll 3! 5! ( + )! 4 cos = + + ( ) + fo ll! 4! ( )! 7

8 Vectos The esolved pt of i the diectio of b is.b b i b b 3 b 3 Vecto poduct: b = b si θ ˆ = j b = b 3 b 3 k 3 b 3 b b If A is the poit with positio vecto = i+ j+ 3k, the the stight lie though A with diectio vecto b = bi+ bj+ b3k hs equtio y z 3 = = = λ (Ctesi fom) b b b3 o ( ) b = 0 (vecto poduct fom) the ple though A d pllel to b d c hs vecto equtio = + sb+ tc Ae of secto A= dθ (pol coodites) Hypebolic fuctios Coics sih = sih cosh cosh sih = cosh cosh sih = + { } { } cosh = l + ( ) sih = l th = l ( ) < Stdd fom Pmetic fom Asymptotes Ecceticity Ellipse Pbol Hypebol y y + = y = 4 = b b = cos θ = t = sec θ y = bsiθ y = t y = btθ oe oe y = ± b b b 8

9 Futhe umeicl itegtio b The mid-odite ule: y d hy ( + y y + y 3 ) whee b h = b Simpso s ule: d {( 0 + ) + 4 ( ) + ( ) } y h y y y y y y y y 3 Numeicl solutio of diffeetil equtios Fo d y = f( ) d smll h, ecuece eltios e: d Eule s method: y = y + + hf ( ); = + + h dy Fo f(, y) : d = Eule s method: y = y + + hf(, y ) whee b h = d is eve Impoved Eule method: y = y + ( k + k ), whee k = hf, y ), k = + hf( + h, y + k ) Ac legth s= dy + d d (Ctesi coodites) s = d dy + dt dt dt (pmetic fom) Sufce e of evolutio dy S = π y + d d (Ctesi coodites) d d π y S = y + dt (pmetic fom) dt dt Tigoomety: t substitutio Witig t = t θ, si θ = t + t cos θ t t = + ( 9

10 MECHANICS Costt cceletio s = ut + t s= ut+ t s = vt t s= vt t v = u + t v= u+ t = ( + ) s= ( u+ v ) s u vt v = u + s Cetes of mss Fo uifom bodies Tigul lmi: 3 t log medi fom vete Solid hemisphee, dius : 3 fom cete 8 Hemispheicl shell, dius : fom cete Cicul c, dius, gle t cete α : si α fom cete α Secto of cicle, dius, gle t cete α : si α fom cete 3α Solid coe o pymid of height h: h bove the bse o the lie fom cete of bse to vete 4 Coicl shell of height h: h bove the bse o the lie fom cete of bse to vete 3 PROBABILITY d STATISTICS Pobbility P( A B) = P( A) + P( B) P( A B) P( A B) = P( A) P( B A) Discete distibutios Stdd discete distibutios: Distibutio of X P( X = ) Me Vice Biomil B(, p ) p ( p ) p p( p) Poisso Po( λ ) λ λ e λ λ! 0

11 Smplig distibutios Fo dom smple X, X,, X of idepedet obsevtios fom distibutio hvig me µ d vice σ : X is ubised estimto of µ, with V( X ) = σ S is ubised estimto of σ, whee S Fo dom smple of obsevtios fom N( µ, σ ) X µ ~ N( 0, ) σ X µ S t ~ Distibutio-fee (o-pmetic) tests = ( Xi X) ( Oi Ei) Cotigecy tbles: is ppoimtely distibuted s χ Ei

12 TABLE CUMULATIVE POISSON DISTRIBUTION FUNCTION The tbulted vlue is P( X ), whee X hs Poisso distibutio with me λ. λ λ λ λ

13 λ λ

14 TABLE PERCENTAGE POINTS OF THE STUDENT'S t-distribution The tble gives the vlues of stisfyig P( X ) = p, whee X is dom vible hvig the studet's t-distibutio with v degees of feedom. 0 p p v v

15 TABLE 6 PERCENTAGE POINTS OF THE χ DISTRIBUTION The tble gives the vlues of stisfyig P( X ) = p, whee X is dom vible hvig the χ distibutio with v degees of feedom. p p v v

16 TABLE 3 CRITICAL VALUES OF THE PRODUCT MOMENT CORRELATION COEFFICIENT The tble gives the citicl vlues, fo diffeet sigificce levels, of the poduct momet coeltio coefficiet,, fo vyig smple sizes,. Oe til 0% 5%.5% % 0.5% Oe til Two til 0% 0% 5% % % Two til