OCR A Level Further Mathematics A H245 Formulae Booklet

A Level Further Mathematics A (H245)SpecimenFormulae Booklet OCR 2017QN 603/1325/0H245Turn over

2Pure MathematicsArithmetic seriesSn 12 n(a l ) 12 n{2a (n 1)d}Geometric seriesa(1 r n )1 raS for r 11 rSn Binomial series n Cr a n r b r n n!where n Cr r r !(n r )!n(n 1) 2(1 x)n 1 nx x 2!(n r 1) rx r! n(n 1)Seriesnn11r 1Maclaurin series 1, n f (0) 2f ( r ) (0) rx . x .2!r!e x exp( x) 1 x x2xr for all x . 2!r! 1, n Specf( x) f(0) f (0) x ln(1 x) x x(n ) ,im r 2 6 n(n 1)(2n 1) , r 3 4 n2 (n 1)2r 1 bnen(a b)n a n n C1 a n 1b n C2 a n 2b2 x 2 x3xr . ( 1) r 1 . ( 1 x 1)23rsin x x x3 x5x 2 r 1 . ( 1)r . for all x3! 5!(2r 1)!cos x 1 x2 x4x2r ( 1)r for all x2! 4!(2r )!(1 x)n 1 nx n(n 1) 2x 2! n(n 1)(n r 1) rx r!Matrix transformations 0 1 Reflection in the line y x : 1 0 cos Anticlockwise rotation through about O: sin sin cos OCR 2017H245 x

3Rotations through θ about the coordinate axes. The direction of positive rotation is taken to beanticlockwise when looking towards the origin from the positive side of the axis of rotation.00 1 R x 0 cos sin 0 sin cos 0 sin 10 0 cos cos sin R z sin cos 00001 en cos R y 0 sin Differentiationf ( x)f ( x)imtan kxecsec xcotxcosec xarcsin x or sin 1 xSparccosx or cos 1 xarctanx or tan 1 xk sec2 kxsec x tan x cosec2 x cosec x cot x1 1 x211 x211 x2dudvu dy v dx u dxQuotient rule y , v dxv2Differentiation from first principlesf ( x h) f ( x )f ( x) limh 0hIntegrationf ( x)dx ln f ( x) cf ( x) 1 f (x) f ( x) dx n 1 f ( x) nIntegration by parts OCR 2017 un 1 c dvdudx uv v dxdxdxH245Turn over

41 b f ( x)dxb a aThe mean value of f ( x) on the interval [a, b] isArea of sector enclosed by polar curve isa x12a x21a x1 r d 222x2 a2 x sin 1 ( x a) a 1 x tan 1 a a x sinh 1 or ln( x x 2 a 2 ) a x cosh 1 or ln( x x 2 a 2 ) a ( x a)en122 f( x)dxf( x)21Numerical methodsb a y dx 12 h{( y0 yn ) 2( y1 y2 yn 1 ) }, where h b animTrapezium rule:Complex numbersCircles: z a kSpHalf lines: arg z a f( xn )f ( xn )ecThe Newton-Raphson iteration for solving f( x) 0 : xn 1 xn Lines: z a z bDe Moivre’s theorem: {r cos isin }n r n cos n isin n 2 k i for k 0,1, 2, ., n 1Roots of unity: The roots of z n 1 are given by z exp n Vectors and 3-D coordinate geometryCartesian equation of the line through the point A with position vector a a1i a2 j a3k in directionu u1i u2 j u3k isx a1 y a2 z a3 u1u2u3Cartesian equation of a plane is n1x n2 y n3 z d 0 a1 b1 i a1 b1 a2b3 a3b2 Vector product: a b a2 b2 j a2 b2 a3b1 a1b3 a b k a b a b a b 332 1 1 2 3 3 OCR 2017H245

5The distance between skew lines is D b a n, where a and b are position vectors of points on eachnline and n is a mutual perpendicular to both linesThe distance between a point and a line is D ax1 by1 ca 2 b2( x1 , y1 ) and the equation of the line is given by ax by cThe distance between a point and a plane is D b n pn, where the coordinates of the point are, where b is the position vector of the point andthe equation of the plane is given by r n pSmall angle approximationsTrigonometric identitiessin( A B) sin A cos B cos Asin Bcos( A B) cos A cos B sin Asin Btan A tan B1 tan A tan B A B k 12imtan( A B) Hyperbolic functionseccosh 2 x sinh 2 x 1sinh 1 x ln[ x ( x 2 1)]ensin , cos 1 12 2 , tan where θ is small and measured in radiansSpcosh 1 x ln[ x ( x 2 1)] , x 11 1 x tanh 1 x ln , 1 x 12 1 x Simple harmonic motionx A cos t B sin t x R sin t StatisticsProbabilityP( A B) P( A) P( B) P( A B)P( A B) P( A)P( B A) P( B)P( A B ) OCR 2017orP( A B) H245P( A B)P (B )Turn over

6Standard deviation x x 2n f x x x2 x 2 orn 2 fx 2 x2 f fSampling distributionsFor any variable X , E( X ) , Var( X ) 2and X is approximately normally distributed when n isnlarge enough (approximately n 25 ) X 2 N(0, 1)If X N , 2 then X N , andn / n Unbiased estimates of the population mean and variance are given by2 xn x 2 x and nn 1 n n Expectation algebraUse the following results, including the cases where a b 1 and/or c 0 :en1. E aX bY c aE X bE Y c ,2. if X and Y are independent then Var aX bY c a 2 Var X b2 Var Y .Expectation: E( X ) xi piimDiscrete distributionsX is a random variable taking values xi in a discrete distribution with P( X xi ) piecVariance: 2 Var( X ) ( xi )2 pi xi2 pi 2P( X x ) n xn x p (1 p)x 1nSpBinomial B(n, p)Uniform distribution over 1, 2, , n, U (n)Geometric distribution Geo( p)Poisson Po( ) 1 p x 1 pe xx!E( X )Var( X )npnp(1 – p)n 121p1 2n 1121 pp2 Continuous distributionsX is a continuous random variable with probability density function (p.d.f.) f( x)Expectation: E( X ) x f( x)d xVariance: 2 Var( X ) ( x )2 f( x)d x x 2 f( x)d x 2xCumulative distribution function F( x) P( X x) f (t )dt OCR 2017H245

7Continuous uniform distribution over a, b p.d.f.1b aExponential e x Normal N , 2x 1 1 e 2 2 E( X )Var( X )1 a b 21 1 b a 2121 2 22Percentage points of the normal distributionIf Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives thevalue of z such that P(Z z ) p etric tests (Oi Ei )2 v2EienGoodness-of-fit test and contingency tables:Approximate distributions for large samples1n(n 1)(2n 1)24imWilcoxon Signed Rank test: T N 14 n(n 1), Wilcoxon Rank Sum test (samples of sizes m and n, with m n ) :ecW N 12 m(m n 1), 121 mn(m n 1) Correlation and regressionFor a sample of n pairs of observations ( xi , yi ) xi 2, S yy ( yi y ) SpS xx ( xi x ) 2xi2nS xy ( xi x )( yi y ) xi yi 2yi2 yi n2, xi yinProduct moment correlation coefficient: r The regression coefficient of y on x is b S xyS xxS xyS xx S yy xy xi yi in i 22 xi yi 22 x y i i nn ( xi x )( yi y ) ( xi x )2Least squares regression line of y on x is y a bx where a y bxSpearman’s rank correlation coefficient: rs 1 OCR 20176 di2 n n2 1H245Turn over

8Critical values for Spearman’s rank correlation coefficient, rsCritical values for the product moment correlation coefficient, 520.49580.48690.47850.47050.4629 OCR .33700.33420.3314

9Critical values for the 2 distributionpIf X has a 2 distribution with v degrees of freedomthen, for each pair of values of p and v, the tablegives the value of x such thatP X x p .xOp0.010.0250.0533v 1 0.0 1571 0.0 9821 0.0239322 0.02010 0.050640.10263 0.11480.21580.35184 .2112.3124.8137.2149.4 OCR 2017Specimen56789H245

10Wilcoxon signed rank testW is the sum of the ranks corresponding to the positive differences,W is the sum of the ranks corresponding to the negative differences,T is the smaller of W and W .For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesisat the level of significance indicated.Critical values of 172125303541475360SpecOne TailTwo Tailn 67891011121314151617181920Level of 15251929233427403246375243For larger values of n, each of W and W can be approximated by the normal distribution with mean1n4 n 1 and variance OCR 20171n24 n 1 2n 1 .H245

11Wilcoxon rank sum testThe two samples have sizes m and n, where m n .Rm is the sum of the ranks of the items in the sample of size m.W is the smaller of Rm and m m n 1 Rm .For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of thenull hypothesis at the level of significance indicated.Critical values of W0.050.1394143450.0250.05m 64547496669For larger values of m and n, the normal distribution with mean1mn12 m n 1 OCR 2017171820212223Level of significance0.025 0.010.05 0.0250.050.020.10.05m 8m .05m 3677889ecOne TailTwo Tailn789100.050.1SpOne TailTwo Tailn345678910Level of significance0.025 0.010.05 0.0250.050.020.10.05m 4m 28290.010.020.050.10.0250.05m 100.010.0259618278746265 m n 1 should be used as an approximation to the distribution of Rm .H2450.0250.05m 6and variance

12MechanicsKinematicsMotion in a straight linev u ats ut 12 at 2Motion in two dimensionsv u ats ut 12 at 2v2 u 2 2ass vt 12 at 2v v u u 2a ss 12 u v ts 12 u v ts vt 12 at 2Newton’s experimental lawBetween two smooth spheres v1 v2 e u1 u2 enBetween a smooth sphere with a fixed plane surface v euMotion in a circleRadial acceleration isimTangential velocity is v r v2or r 2 towards the centrerecTangential acceleration is v r SpCentres of massTriangular lamina:23along median from vertexSolid hemisphere, radius r:3r8Hemispherical shell, radius r:from centre12r from centreCircular arc, radius r, angle at centre 2α:r sin Sector of circle, radius r, angle at centre 2α:Solid cone or pyramid of height h:14from centre2r sin from centre3 h above the base on the line from centre of base to vertexConical shell of height h: 13 h above the base on the line from centre of base to vertex OCR 2017H245

13DiscreteInclusion-exclusion principleFor sets A, B and C:n A B C n A n B n C n A B n A C n B C n A B C The hierarchy of orders O 1 O log n O n O n log n O n2 O n3 . O a n O n! Sorting algorithmsBubble sort:Start at the left hand end of the list unless specified otherwise.enCompare the first and second values and swap if necessary. Then compare the (new) second valuewith the third value and swap if necessary. Continue in this way until all values have beenconsidered.Shuttle sort:imFix the last value then repeat with the reduced list until either there is a pass in which no swapsoccur or the list is reduced to length 1, then stop.ecStart at the left hand end of the list unless specified otherwise.SpCompare the second value with the first and swap if necessary, this completes the first pass. Nextcompare the third value with the second and swap if necessary, if a swap happened shuttle back tocompare the (new) second with the first as in the first pass, this completes the second pass.Next compare the fourth value with the third and swap if necessary, if a swap happened shuttleback to compare the (new) third value with the second as in the second pass (so if a swap happensshuttle back again). Continue in this way for n 1 passes, where n is the length of the list.Quick sort:The first value in any sublist will be the pivot, unless specified otherwise.Working from left to right, write down each value that is smaller than the pivot, then the pivot,then work along the list and write down each value that is not smaller than the pivot. This producestwo sublists (one of which may be empty) with the pivot between them and completes the pass.Next apply this procedure to each of the sublists from the previous pass, unless they consist of asingle entry, to produce further sublists. Continue in this way until no sublist has more than oneentry. OCR 2017H245