Ocr Formula Booklet

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ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATIONADVANCED GENERAL CERTIFICATE OF EDUCATIONMATHEMATICSLIST OF FORMULAEANDSTATISTICAL TABLES(List MF1)MF1CST252January 2007Pure MathematicsMensurationSurface area of sphere = 4r2Area of curved surface of cone = r slant heightTrigonometrya2= b2+ c2 2bc cos AArithmetic Seriesun = a + (n 1)dSn = 12n(a + l) = 12n{2a + (n 1)d}Geometric Seriesun = arn1Sn = a(1 rn)1 rS = a1 r for |r| < 1Summationsnr=1r2= 16n(n + 1)(2n + 1)nr=1r3= 14n2(n + 1)2Binomial Series

nr + nr + 1 = n + 1r + 1

(a + b)n= an+ n1an1b + n2an2b2+ . . . + nr anrbr+ . . . + bnn ,where nr = nCr = n!r!(n r)!(1 + x)n= 1 + nx + n(n 1)1.2 x2+ . . . + n(n 1) . . . (n r + 1)1.2.3 . . . r xr+ . . . |x| < 1, n Logarithms and exponentialsex ln a= axComplex Numbers{r(cos + i sin )}n= rn(cos n + i sin n)ei= cos + i sin The roots of n= 1 are given by = e2kin, for k = 0, 1, 2, . . . , n 12Maclaurins Seriesf(x) = f(0) + xf

(0) + x22!f

(0) + . . . + xrr!f(r)(0) + . . .ex= exp(x) = 1 + x + x22! + . . . + xrr! + . . . for all xln(1 + x) = x x22 + x33 . . . + (1)r+1xrr + . . . (1 < x 1)sin x = x x33! + x55! . . . + (1)r x2r+1(2r + 1)! + . . . for all xcos x = 1 x22! + x44! . . . + (1)r x2r(2r)! + . . . for all xtan1x = x x33 + x55 . . . + (1)r x2r+12r + 1 + . . . (1 x 1)sinh x = x + x33! + x55! + . . . + x2r+1(2r + 1)! + . . . for all xcosh x = 1 + x22! + x44! + . . . + x2r(2r)! + . . . for all xtanh1x = x + x33 + x55 + . . . + x2r+12r + 1 + . . . (1 < x < 1)Hyperbolic Functionscosh2x sinh2x = 1sinh 2x = 2 sinh x cosh xcosh 2x = cosh2x + sinh2xcosh1x = ln{x +(x2 1)} (x 1)sinh1x = ln{x +(x2+ 1)}tanh1x = 12 ln 1 + x1 x |x| < 1Coordinate GeometryThe perpendicular distance from (h, k) to ax + by + c = 0 is |ah + bk + c|(a2+ b2)The acute angle between lines with gradients m1 and m2 is tan1

m1 m21 + m1m2

Trigonometric Identitiessin(A B) = sin Acos B cos Asin Bcos(A B) = cos Acos B sin Asin Btan(A B) = tan A tan B1 tan Atan B A B k + 12For t = tan 12A: sin A = 2t1 + t2, cos A = 1 t21 + t2sin A + sin B = 2 sin A + B2 cos A B2sin A sin B = 2 cos A + B2 sin A B2cos A + cos B = 2 cos A + B2 cos A B2cos A cos B = 2 sin A + B2 sin A B23VectorsThe resolved part of a in the direction of b is a.b|b|The point dividing AB in the ratio : is a + b + Vector product: a b = |a| |b| sin n =

i a1 b1j a2 b2k a3 b3

= a2b3 a3b2a3b1 a1b3a1b2 a2b1

If A is the point with position vector a = a1i + a2j + a3k and the direction vector b is given byb = b1i + b2j + b3k, then the straight line through A with direction vector b has cartesian equationx a1b1= y a2b2= a3b3(= )The plane through A with normal vector n = n1i + n2j + n3k has cartesian equationn1x + n2y + n3 + d = 0, where d = a.nThe 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 has equation r = a + sb + tcThe perpendicular distance of (, , ) from n1x + n2y + n3 + d = 0 is

n1 + n2 + n3 + d

(n21 + n22 + n23)Matrix transformationsAnticlockwise rotation through about O: cos sin sin cos

Reection in the line y = (tan )x: cos 2 sin 2sin 2 cos 2

Differentiationf(x) f

(x)tan kx k sec2kxsin1x 1(1 x2)cos1x 1(1 x2)tan1x 11 + x2sec x sec x tan xcot x cosec2xcosec x cosec x cot xsinh x cosh xcosh x sinh xtanh x sech2xsinh1x 1(1 + x2)cosh1x 1(x2 1)tanh1x 11 x2If y = f(x)g(x) then dydx = f

(x)g(x) f(x)g

(x){g(x)}24Integration ( + constant; a > 0 where relevant)f(x) f(x) dxsec2kx 1k tan kxtan x ln |sec x|cot x ln |sin x|cosec x ln |cosec x + cot x| = ln

tan 12x

sec x ln |sec x + tan x| = ln

tan(12x + 14)

sinh x cosh xcosh x sinh xtanh x ln cosh x1(a2 x2) sin1

xa |x| < a1a2+ x21a tan1

xa

1(x2 a2) cosh1

xa or ln{x +(x2 a2)} (x > a)1(a2+ x2) sinh1

xa or ln{x +(x2+ a2)}1a2 x212a ln

a + xa x

= 1a tanh1

xa |x| < a1x2 a212a ln

x ax + a

udvdx dx = uv

vdudx dxArea of a sectorA = 12 r2d (polar coordinates)A = 12

xdydt ydxdt dt (parametric form)Numerical MathematicsNumerical integrationThe trapezium rule: bay dx 12h{(y0 + yn) + 2(y1 + y2 + . . . + yn1)}, where h = b anSimpsons Rule: bay dx 13h{(y0 + yn) + 4(y1 + y3 + . . . + yn1) + 2(y2 + y4 + . . . + yn2)},where h = b an and n is evenNumerical Solution of EquationsThe Newton-Raphson iteration for solving f(x) = 0: xn+1 = xn f(xn)f

(xn)5MechanicsMotion in a circleTransverse velocity: v = rTransverse acceleration: v = rRadial acceleration: r2= v2rCentres of Mass (for uniform bodies)Triangular lamina: 23 along median from vertexSolid hemisphere, radius r: 38r from centreHemispherical shell, radius r: 12r from centreCircular arc, radius r, angle at centre 2: r sin from centreSector of circle, radius r, angle at centre 2: 2r sin 3 from centreSolid cone or pyramid of height h: 14h above the base on the line from centre of base to vertexConical shell of height h: 13h above the base on the line from centre of base to vertexMoments of Inertia (for uniform bodies of mass m)Thin rod, length 2l, about perpendicular axis through centre: 13ml2Rectangular lamina about axis in plane bisecting edges of length 2l: 13ml2Thin rod, length 2l, about perpendicular axis through end: 43ml2Rectangular lamina about edge perpendicular to edges of length 2l: 43ml2Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: 13m(a2+ b2)Hoop or cylindrical shell of radius r about axis: mr2Hoop of radius r about a diameter: 12mr2Disc or solid cylinder of radius r about axis: 12mr2Disc of radius r about a diameter: 14mr2Solid sphere, radius r, about diameter: 25mr2Spherical shell of radius r about a diameter: 23mr2Parallel axes theorem: IA = IG + m(AG)2Perpendicular axes theorem: I

= Ix + Iy (for a lamina in the x-y plane)6Probability & StatisticsProbabilityP(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

)Bayes Theorem: P(Aj | B) = P(Aj)P(B | Aj)P(Ai)P(B | Ai)Discrete distributionsFor a discrete random variable X taking values xi with probabilities piExpectation (mean): E(X) = = xipiVariance: Var(X) = 2= (xi )2pi = x2ipi 2For a function g(X): E(g(X)) = g(xi)piThe probability generating function of X is GX(t) = E(tX), andE(X) = G

X(1)Var(X) = G

X(1) + G

X(1) {G

X(1)}2For Z = X + Y, where X and Y are independent: GZ(t) = GX(t)GY(t)Standard discrete distributionsDistribution of X P(X = x) Mean Variance P.G.F.Binomial B(n, p) nxpx(1 p)nxnp np(1 p) (1 p + pt)nPoisson Po() e xx! e(t1)Geometric Geo(p) on 1, 2, p(1 p)x1 1p1 pp2pt1 (1 p)tContinuous distributionsFor a continuous random variable X having probability density function fExpectation (mean): E(X) = = xf(x) dxVariance: Var(X) = 2= (x )2f(x) dx = x2f(x) dx 2For a function g(X): E(g(X)) = g(x)f(x) dxCumulative distribution function: F(x) = P(X x) = xf(t) dtThe moment generating function of X is MX(t) = E(etX) andE(X) = M

X(0)E(Xn) = M(n)X (0)Var(X) = M

X(0) {M

X(0)}2For Z = X + Y, where X and Y are independent: MZ(t) = MX(t)MY(t)7Standard continuous distributionsDistribution of X P.D.F. Mean Variance M.G.F.Uniform (Rectangular) on [a, b] 1b a12(a + b) 112(b a)2 ebt eat(b a)tExponential ex 112 tNormal N(, 2) 1 (2)e12

x 2 2et+122t2Expectation algebraCovariance: Cov(X, Y) = E(X X)(Y Y) = E(XY) XYVar(aX bY) = a2Var(X) + b2Var(Y) 2abCov(X, Y)Product moment correlation coefcient: = Cov(X, Y)XYIf X = aX

+ b and Y = cY

+ d, then Cov(X, Y) = ac Cov(X

, Y

)For independent random variables X and YE(XY) = E(X)E(Y)Var(aX bY) = a2Var(X) + b2Var(Y)Sampling distributionsFor a random sample X1, X2, . . . , Xn of n independent observations from a distribution having mean and variance 2X is an unbiased estimator of , with Var(X) = 2nS2is an unbiased estimator of 2, where S2= (Xi X)2n 1For a random sample of n observations from N(, 2)X /n N(0, 1)X S/n tn1 (also valid in matched-pairs situations)If X is the observed number of successes in n independent Bernoulli trials in each of which theprobability of success is p, and Y = Xn, thenE(Y) = p and Var(Y) = p(1 p)nFor a random sample of nx observations from N(x, 2x) and, independently, a random sample ofny observations from N(y, 2y)(X Y) (x y)

2xnx+ 2yny

N(0, 1)If 2x = 2y = 2(unknown) then(X Y) (x y)

S2p

1nx+ 1ny

tnx+ny2,where S2p = (nx 1)S2x + (ny 1)S2ynx + ny 28Correlation and regressionFor a set of n pairs of values (xi, yi)Sxx = (xi x)2= x2i ( xi)2nSyy = (yi y)2= y2i (yi)2nSxy = (xi x)(yi y) = xiyi (xi)(yi)nThe product moment correlation coefcient isr = Sxy(SxxSyy) = (xi x)(yi y)_{[(xi x)2}[(yi y)2}| = xiyi (xi)( yi)n___x2i ( xi)2n __y2i (yi)2n __Spearmans rank correlation coefcient is rs = 1 6d2n(n2 1)The regression coefcient of y on x is b = SxySxx= (xi x)(yi y)(xi x)2Least squares regression line of y on x is y = a + bx where a = y bxDistribution-free (non-parametric) testsGoodness-of-t test and contingency tables:(Oi Ei)2Ei 2Approximate distributions for large samplesWilcoxon Signed Rank test: T N[14n(n + 1), 124n(n + 1)(2n + 1)}Wilcoxon Rank Sum test (samples of sizes m and n, with m n):W N[12m(m+ n + 1), 112mn(m+ n + 1)}9CUMULATIVEBINOMIALPROBABILITIESn=5p0.050.10.151/60.20.250.31/30.350.40.450.50.550.60.652/30.70.750.85/60.