List MF19. List of formulae and statistical tables. Cambridge International AS & A Level Mathematics (9709) and Further Mathematics (9231)

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1 List MF9 List of fomulae ad statistical tables Cambidge Iteatioal AS & A Level Mathematics (9709) ad Futhe Mathematics (93) Fo use fom 00 i all papes fo the above syllabuses. CST39 * *

2 PURE MATHEMATICS Mesuatio Volume of sphee = 4 3 π 3 Suface aea of sphee = 4π Volume of coe o pyamid = base aea height 3 Aea of cuved suface of coe = π slat height Ac legth of cicle = θ (θ i adias) Aea of secto of cicle = (θ i adias) θ Algeba Fo the quadatic equatio Fo a aithmetic seies: Fo a geometic seies: a + b + c = 0 : b± b 4ac = a u = a+ ( ) d, S = ( a+ l) = { a+ ( ) d} u = a a( ), S = ( ) Biomial seies: ( a+ b) = a + a b+ a b + a b + K + b 3 3 3! ad =!( )! a, S = ( < ), whee is a positive itege ( ) ( )( ) 3 ( + ) = K, whee is atioal ad <! 3!

3 Tigoomety cos θ + si θ, siθ taθ cosθ + ta θ sec θ, si( A± B) si Acos B± cos Asi B cos( A± B) cos Acos Bm si Asi B cot θ + cosec θ ta A± ta B ta( A± B) m taatab si A si Acos A cos A cos A si A cos A si A taa ta A ta A Picipal values: π si π, 0 cos π, ta π < < π Diffeetiatio f( ) f( ) l e si cos e cos si ta sec sec sec ta cosec cosec cot cot ta uv If = f( t) ad y = g( t) the dy = dy d d dt dt u v cosec + du dv v + u d d du dv v u d d v 3

4 Itegatio (Abitay costats ae omitted; a deotes a positive costat.) f( ) e si cos sec a dv du u d= uv v d d d f( ) d = l f( ) f( ) + a a f( ) d + + l e cos si ta ta a a a l a + a a+ l a a ( ) ( > a) ( < a) Vectos a= ai+ a j+ a k ad b= bi+ bj+ b3k the If 3 ab. = ab + ab + ab = a b cosθ 3 3 4

5 FURTHER PURE MATHEMATICS Algeba Summatios: = ( + ), = = ( + )(+ ), 6 = = 3 = ( + ) 4 Maclaui s seies: ( ) f( ) = f(0) + f (0) + f (0) + K+ f (0) + K!! e = ep( ) = K+ + K (all )!! 3 + l( + ) = + K+ ( ) + K ( < ) si = + K+ ( ) + K (all ) 3! 5! ( + )! 4 cos = + K+ ( ) + K (all )! 4! ( )! ta = + K+ ( ) + K ( ) sih = K+ + K (all ) 3! 5! ( + )! 4 cosh = K+ + K (all )! 4! ( )! tah = K+ + K ( < < ) Tigoomety If t = ta the: t si = ad + t t cos = + t Hypebolic fuctios cosh sih, sih sih cosh, ( ) sih = l + + ( ) cosh = l + ( ) + tah = l ( ) < cosh cosh + sih 5

6 Diffeetiatio f( ) f( ) si cos sih cosh tah sih cosh tah cosh sih sech + Itegatio (Abitay costats ae omitted; a deotes a positive costat.) f( ) f( ) d sec l sec ta l ta( ) + = + π ( < π ) 4 cosec l cosec + cot = l ta ( ) (0 < <π ) sih cosh cosh sih sech a a a + si cosh sih tah a a a ( < a) ( > a) 6

7 MECHANICS Uifomly acceleated motio v= u+ at, s = ( u+ v) t, s = ut + at, v = u + as Motio of a pojectile Equatio of tajectoy is: FURTHER MECHANICS g y = taθ V cos θ Elastic stigs ad spigs λ T =, l λ E = l Motio i a cicle Fo uifom cicula motio, the acceleatio is diected towads the cete ad has magitude ω o v Cetes of mass of uifom bodies Tiagula lamia: alog media fom vete 3 Solid hemisphee of adius : 3 fom cete 8 Hemispheical shell of adius : fom cete Cicula ac of adius ad agle α: siα fom cete α Cicula secto of adius ad agle α: si α fom cete 3α Solid coe o pyamid of height h: 3 h fom vete 4 7

8 PROBABILITY & STATISTICS Summay statistics Fo ugouped data: Fo gouped data: Σ =, stadad deviatio Σ( ) Σ = = Σf =, stadad deviatio Σ f Σ( ) f Σ f = = Σf Σf Discete adom vaiables E( X ) = Σ p, Va( X) = Σ p {E( X)} Fo the biomial distibutio B(, p ) : p = p ( p), µ = p, Fo the geometic distibutio Geo(p): p = p( p), σ = p( p) µ = p Fo the Poisso distibutio Po( λ ) p λ λ = e, µ = λ,! σ = λ Cotiuous adom vaiables E( X ) = f( ) d, Va( X) = f( ) d {E( X)} Samplig ad testig Ubiased estimatos: Σ =, s = = Σ Σ( ) ( Σ ) Cetal Limit Theoem: σ X ~N µ, Appoimate distibutio of sample popotio: p( p) N p, 8

9 Samplig ad testig Two-sample estimate of a commo vaiace: FURTHER PROBABILITY & STATISTICS s Σ( ) + Σ( ) = + Pobability geeatig fuctios X G X ( t ) = E( t ), E( X ) = G (), X Va( X ) = G () + G () {G ()} X X X 9

10 If Z has a omal distibutio with mea 0 ad vaiace, the, fo each value of z, the table gives the value of Φ(z), whee Φ(z) = P(Z z). Fo egative values of z, use Φ( z) = Φ(z). THE NORMAL DISTRIBUTION FUNCTION z ADD If Z has a omal distibutio with mea 0 ad vaiace, the, fo each value of p, the table gives the value of z such that P(Z z) = p. Citical values fo the omal distibutio p z

11 CRITICAL VALUES FOR THE t-distribution If T has a t-distibutio with ν degees of feedom, the, fo each pai of values of p ad ν, the table gives the value of t such that: P(T t) = p. p ν =

12 If X has a χ -distibutio with ν degees of feedom the, fo each pai of values of p ad ν, the table gives the value of such that P(X ) = p. CRITICAL VALUES FOR THE χ -DISTRIBUTION p ν =

13 WILCOXON SIGNED-RANK TEST The sample has size. P is the sum of the aks coespodig to the positive diffeeces. Q is the sum of the aks coespodig to the egative diffeeces. T is the smalle of P ad Q. Fo each value of the table gives the lagest value of T which will lead to ejectio of the ull hypothesis at the level of sigificace idicated. Citical values of T Level of sigificace Oe-tailed Two-tailed = Fo lage values of, each of P ad Q ca be appoimated by the omal distibutio with mea + ( ) 4 ad vaiace ( + )(+ ). 4 3

14 The two samples have sizes m ad, whee m. WILCOXON RANK-SUM TEST R m is the sum of the aks of the items i the sample of size m. W is the smalle of R m ad m( + m + ) R m. Fo each pai of values of m ad, the table gives the lagest value of W which will lead to ejectio of the ull hypothesis at the level of sigificace idicated. Citical values of W Level of sigificace Oe-tailed Two-tailed m = 3 m = 4 m = 5 m = Level of sigificace Oe-tailed Two-tailed m = 7 m = 8 m = 9 m = Fo lage values of m ad, the omal distibutio with mea mm ( + + ) ad vaiace m( m + + ) should be used as a appoimatio to the distibutio of R m. 4

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