GCE AS and A Level MATHEMATICS FORMULA BOOKLET

1 WJEC CBAC Ltd. GCE AS and A Level MATHEMATICS FORMULA BOOKLET From September 2017 Issued 2017 WJEC CBAC Ltd. 2 Pure MATHEMATICS Mensuration Surface area of sphere = 24r Area of curved surface of cone = heightslant r Arithmetic Series Sn = 21n(a + l) = 21n [2a + (n 1)d] Geometric Series Sn = rran 1)1( S = ra 1 for | r | < 1 Summations )12)(1(6112 nnnrnr 224113)1( nnrnr Binomial Series 111rnrnrn )( 2 1)(221NN nbbarnbanbanabanrrnnnnn where )!(!!C rnrnrnrn ) ,1( )1()1( )1(1)1(2RR nxxrrnnnxnnnxxrn Logarithms and exponentials xaxa lne Complex Numbers )sini(cos)}sini(cos{ nnrrnn The roots of 1 nz are given by nkzi2e , for 1 , ,2 ,1 ,0 nk WJEC CBAC Ltd.

2 3 Maclaurin s and Taylor s Series )0(! )0(!2)0()0()()(2 rrfrxfxfxfxf )(!)( )(!2)()()()()()(2 afraxafaxafaxafxfrr )(! )(!2)()()()(2 afrxafxafxafxafrr xrxxxxrx allfor ! !21)exp(e2 )11( )1( 32)1ln(132 xrxxxxxrr xrxxxxxrr allfor )!12()1( !5!3sin1253 xrxxxxrr allfor )!2()1( !4!21cos242 )11( 12)1( 53t an12531 xrxxxxxrr xrxxxxxr allfor )!12( !5!3sinh1253 xrxxxxr allfor )!2( !4!21cosh242 )11( 12 53t anh12531 xrxxxxxr Hyperbolic Functions 1sinhcosh22 xx xxxcoshsinh22sinh xxx22sinhcosh2cosh )1( 1lncosh}{21 xxxx }{1lnsinh21 xxx )1( 11lnt anh211 xxxx WJEC CBAC Ltd. 4 Trigonometric Identities BABABA sincoscossin)sin( BABABA sinsincoscos)cos( ))(( t ant an1t ant an)t an(21 kBABABABA For At21t an : 212sinttA , 2211costtA 2cos2sin2sinsinBABABA 2sin2cos2sinsinBABABA 2cos2cos2coscosBABABA 2sin2sin2coscosBABABA Vectors The resolute of a in the direction of b is The point dividing AB in the ratio : is ba The equation of a plane in Cartesian form is 123n x n y n z k The perpendicular distance between two skew lines is ().

3 D b a nn, where a and b are position vectors of points on each line and n is a mutual perpendicular to both lines. The perpendicular distance between a point and a line is 1122axbycDab , where the coordinates of the point are 11( , )xyand the equation of the line is given by ax by c . The perpendicular distance between a point and a plane is 123123222nnnkDnnn , where ,, are the coordinates of the point and 123n x n y n z k is the equation of the plane. Matrix transformations Anticlockwise rotation through about O: cos sinsincos Reflection in the line xy)(tan : 2cos2sin2sin 2cos WJEC CBAC Ltd. 5 Differentiation Function Derivative )()(xgxf 2))(()()()()(xgxgxfxgxf tan x sec2x xsec xxtansec xcot x2cosec xcosec xxcotcosec x1sin 211x x1cos 211x x1t an 211x xsinh xcosh xcosh xsinh xtanh x2sech x1sinh 211x x1cosh 112 x x1tanh 211x WJEC CBAC Ltd.

4 6 Integration (+ constant; 0 a where relevant) Function Integral xtan xsecln xcot xsinln xcosec )tan(lncotcosecln21xxx xsec )tan(lntansecln4121 xxx sec2 x tan x xsinh xcosh xcosh xsinh xtanh xcoshln 221xa )( sin1axax 221xa axa1t an1 221ax )( lncosh}{221axaxxax 221xa }{221lnsinhaxxax 221xa )( t anh1ln211axaxaxaxaa 221ax axaxa ln21 xxuvuvxxvudddddd Area of a sector d221rA (polar coordinates) WJEC CBAC Ltd. 7 Numerical MATHEMATICS Numerical integration The trapezium rule: }) (2){(d121021 bannyyyyyhxy , where nabh Numerical Solution of Equations The Newton-Raphson iteration for solving 0)( xf: )()(1nnnnxfxfxx Mechanics Motion in a circle Transverse velocity: rrv Radial acceleration: rrvr222 Centres of Mass of Uniform Bodies Triangular lamina: 23 along median from vertex Semi circle: 43r from straight edge along axis of symmetry Quarter circle: 43rx 43ry from vertex WJEC CBAC Ltd.

5 8 Probability & Statistics Probability )P()P()P()P(BABABA )|P()P()P(ABABA )P()|P()P()|P()P()|P()|P(AABAABAABBA Bayes Theorem: )|P()P()|P()P()|P(iijjjABAABABA Discrete distributions For a discrete random variable X taking values ix with probabilities ip Expectation (mean): iipxX )E( Variance: 2222)()Var( iiiipxpxX For a function )g(X: iipxX)g())E(g( Standard discrete distributions: Distribution of X )P(xX Mean Variance Binomial ),B(pn xnxppxn )1( np )1(pnp Poisson )Po( !exx Continuous distributions For a continuous random variable X having probability density function f Expectation (mean): xxxfXd)()E( Variance: 2222d)(d)f()()Var( xxfxxxxX For a function )g(X: xxfxXd)()g())E(g( Cumulative distribution function: xttfxXxd)()P()F( Standard continuous distributions: Distribution of X Mean Variance Uniform (Rectangular) on [a, b] U[a,b] ab 1 )(21ba 2121)(ab Normal ) ,N(2 221e21 x 2 Exponential Exp( ) ex 1 21 WJEC CBAC Ltd.

6 9 Expectation algebra For independent random variables X and Y )E()E()E(YXXY , )Var()Var()Var(22 YbXabYaX Sampling distributions For a random sample nXXX , , ,21 of n independent observations from a distribution having mean and variance 2 X is an unbiased estimator of , with nX2)Var( 2S is an unbiased estimator of 2 , where 1)(22 nXXSi For a random sample of n observations from ) ,N(2 )1 ,0N(~/nX )1(~/X ntnS If X is the observed number of successes in n independent Bernoulli trials in each of which the probability of success is p, and nXY , then pY )E( and nppY)1()Var( For a random sample of xn observations from ) ,N(2xx and, independently, a random sample of yn observations from ) ,N(2yy )1 ,0N(~)()(22yyxxyxnnYX WJEC CBAC Ltd.

7 10 Correlation and Regression For a sample of n pairs of observations ,iixy 222ixxiixSxxxn 222iyyiiySyyyn iixyiii ixySxxyyx yn A measure of linear association between two variables X and Y is given by the Pearson product -moment correlation coefficient r. For the sample (x1,y1), (x2,y2),.. , (xn,yn), it is given by xyxxyySrSS . Given data, the parameters and of the linear regression model may be estimated using the principle of least squares. The least squares estimate of the parameter is given by xyxxSS . The least squares estimate of the parameter is given by yx . The least squares regression line is given by yx.

8 Spearman's rank correlation coefficient is given by 22611isdrnn . SS/MLJ/W05(16)E