Sixth Term Examination Papers MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES

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1 Sixth Term Examiatio Papers MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES

2 Pure Mathematics Mesuratio Surface area of sphere = 4πr Area of curved surface of coe = πr slat height Trigoometry a = b + c bc cos A Arithmetic Series u = a +( )d S = (a + l) = {a +( )d} Geometric Series u = ar S = a( r ) r S = a for r < r Summatios r= r= r = ( + )( + ) 6 r 3 = 4 ( + ) Biomial Series ( r ) + ( r + ) = ( + r + ) (a + b) = a + ( ) a b + ( ) a b ( r ) a r b r b ( ), where ( r ) = C r = ( + x) = + x +! r!( r)! ( ) x ( )... ( r + ) x r r ( x <, ) Logarithms ad expoetials e x l a = a x Complex Numbers {r(cos θ + isiθ)} = r (cos θ + isiθ) e iθ = cos θ + isiθ πki The roots of = aregiveby =e,fork = 0,,,...,

3 Maclauri s Series f(x) =f(0)+xf (0)+ x! f (0) xr r! f (r) (0)+... e x = exp(x) = + x + x xr ! r! forallx l( + x) =x x + x x r +( )r r ( < x ) si x = x x3 3! + x5 5!... x r+ +( )r +... (r + )! forallx cos x = x! + x4 4!... x r +( )r (r)! +... forallx ta x = x x3 3 + x x r+ +( )r +... ( x ) r + sih x = x + x3 3! + x5 xr forallx 5! (r + )! cosh x = + x! + x4 xr ! (r)! +... forallx tah x = x + x3 3 + x5 xr ( < x < ) 5 r + Hyperbolic Fuctios cosh x sih x = sih x = sihx cosh x cosh x = cosh x + sih x cosh x = l{x + (x )} (x ) sih x = l{x + (x + )} tah x = Coordiate Geometry + x l ( ) ( x <) x The perpedicular distace from (h, k) to ax + by + c = 0is ah + bk + c (a + b ) The acute agle betwee lies with gradiets m ad m is ta m m + m m Trigoometric Idetities si(a ± B) =si A cos B ± cos A si B cos(a ± B) =cos A cos B si A si B ta A ± ta B ta(a ± B) = ta A ta B (A ± B (k + )π) For t = ta A: sia = t si A + si B = si A + B si A si B = cos A + B cos A + cos B = cos A + B cos A cos B = si A + B t,cosa = + t + t cos A B si A B cos A B si A B 3

4 Vectors The resolved part of a i the directio of b is a.b b The poit dividig AB i the ratio λ : μ is μa + λb λ + μ a b 3 a 3 b = ( a 3 b a b 3 ) a b a b i a b Vector product: a b = a b si θ ˆ = j a b k a 3 b 3 If A is the poit with positio vector a = a i + a j + a 3 k ad the directio vector b is give by b = b i + b j + b 3 k, the the straight lie through A with directio vector b has cartesia equatio x a b = y a b = a 3 b 3 (= λ ) The plae through A with ormal vector = i + j + 3 k has cartesia equatio x + y + 3 +d = 0, where d = a. The plae through o-colliear poits A, B ad C has vector equatio r = a + λ (b a)+μ(c a) =( λ μ)a + λb + μc The plae through the poit with positio vector a ad parallel to b ad c has equatio r = a + sb + tc The perpedicular distace of (α, β, γ ) from x + y + 3 +d = 0is α + β + 3 γ + d ( ) Matrix trasformatios Aticlockwise rotatio through θ about O: ( cos θ si θ si θ cos θ ) cos θ si θ Reflectioitheliey =(ta θ)x: ( si θ cos θ ) Differetiatio f(x) ta kx si x cos x ta x sec x cot x cosec x sihx cosh x tah x sih x cosh x tah x f (x) k sec kx ( x ) ( x ) + x sec x ta x cosec x cosec x cot x cosh x sih x sech x ( + x ) (x ) x 4

5 Itegratio ( + costat; a > 0 where relevat) f(x) f(x) dx sec kx ta kx k ta x l sec x cot x l si x cosec x l cosec x + cot x =l ta x sec x l sec x + ta x =l ta(x + π) 4 sih x cosh x cosh x sih x tah x l cosh x (a x ) si ( x ) a ( x < a) a + x a ta ( x a ) (x a ) cosh ( x a ) or l{x + (x a )} (x > a) (a + x ) sih ( x a ) or l{x + (x + a )} a x a l a + x a x = a tah ( x ) ( x < a) a x a a l x a x + a u dv dx dx = uv v du dx dx Area of a sector A = r dθ (polar coordiates) A = (x dy dt y dx dt ) dt (parametric form) Numerical Mathematics Numerical itegratio b The trapezium rule: y dx h{(y 0 + y )+(y + y y b a )}, whereh = a b Simpso s Rule: y dx h{(y y )+4(y + y y )+(y + y y )}, a where h = b a ad is eve Numerical Solutio of Equatios The Newto-Raphso iteratio for solvig f(x) =0: x + = x f(x ) f (x ) 5

6 Mechaics Motio i a circle Trasverse velocity: v = r θ Trasverse acceleratio: v = r θ Radial acceleratio: r θ = v r Cetres of Mass (for uiform bodies) Triagular lamia: 3 alogmediafromvertex Solid hemisphere, radius r: 3 r from cetre 8 Hemispherical shell, radius r: r from cetre Circular arc, radius r, agleatcetreα: r si α α Sector of circle, radius r, agleatcetreα: r si α 3α from cetre from cetre Solid coe or pyramid of height h: h above the base o the lie from cetre of base to vertex 4 Coical shell of height h: h above the base o the lie from cetre of base to vertex 3 Momets of Iertia (for uiform bodies of mass m) Thi rod, legth l, about perpedicular axis through cetre: 3 ml Rectagular lamia about axis i plae bisectig edges of legth l: 3 ml Thi rod, legth l, about perpedicular axis through ed: 4 3 ml Rectagular lamia about edge perpedicular to edges of legth l: 4 3 ml Rectagular lamia, sides a ad b, about perpedicular axis through cetre: 3 m(a + b ) Hoop or cylidrical shell of radius r about axis: mr Hoop of radius r about a diameter: mr Disc or solid cylider of radius r about axis: mr Disc of radius r about a diameter: 4 mr Solid sphere, radius r, about diameter: 5 mr Spherical shell of radius r about a diameter: 3 mr Parallel axes theorem: I A = I G + m(ag) Perpedicular axes theorem: I = I x + I y (foralamiaithex-y plae) 6

7 Probability & Statistics Probability P(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(A j B) = P(A j )P(B A j ) ΣP(A i )P(B A i ) Discrete distributios For a discrete radom variable X takig values x i with probabilities p i Expectatio (mea): E(X) =μ = Σ x i p i Variace: Var(X) =σ = Σ(x i μ) p i = Σ x i p i μ For a fuctio g(x): E(g(X)) = Σ g(x i )p i The probability geeratig fuctio of X is G X (t) =E(t X ),ad E(X) =G X () Var(X) =G X ()+G X () {G X ()} For Z = X + Y,whereX ad Y are idepedet: G Z (t) =G X (t)g Y (t) Stadard discrete distributios Distributio of X P(X = x) Mea Variace P.G.F. Biomial B(, p) ( x ) px ( p) x p p( p) ( p + pt) Poisso Po(λ ) e λ λ x x! Geometric Geo(p) o,, p( p) x p λ λ e λ(t ) p p pt ( p)t Cotiuous distributios For a cotiuous radom variable X havig probability desity fuctio f Expectatio (mea): E(X) =μ = xf(x) dx Variace: Var(X) =σ = (x μ) f(x) dx = x f(x) dx μ For a fuctio g(x): E(g(X)) = g(x)f(x) dx x Cumulative distributio fuctio: F(x) =P(X x) = f(t) dt The momet geeratig fuctio of X is M X (t) =E(e tx ) ad E(X) =M X (0) E(X )=M () X (0) Var(X) =M X (0) {M X (0)} For Z = X + Y,whereX ad Y are idepedet: M Z (t) =M X (t)m Y (t) 7

8 Stadard cotiuous distributios Distributio of X P.D.F. Mea Variace M.G.F. Uiform (Rectagular) o [a, b] b a Expoetial λe λx λ Normal N(μ, σ ) (a + b) (b e bt e at a) (b a)t λ λ λ t σ ( x μ (π) e σ ) μ σ e μt+ σ t Expectatio algebra Covariace: Cov(X, Y) =E((X μ X )(Y μ Y )) = E(XY) μ X μ Y Var(aX ± by) =a Var(X)+b Var(Y)±ab Cov(X, Y) Product momet correlatio coefficiet: ρ = Cov(X, Y) σ X σ Y If X = ax + b ad Y = cy + d, the Cov(X, Y) =ac Cov(X, Y ) For idepedet radom variables X ad Y E(XY) =E(X)E(Y) Var(aX ± by) =a Var(X)+b Var(Y) Samplig distributios For a radom sample X, X,..., X of idepedet observatios from a distributio havig mea μ ad variace σ X is a ubiased estimator of μ, with Var(X) = σ S is a ubiased estimator of σ,wheres = Σ(X i X) For a radom sample of observatios from N(μ, σ ) X μ σ/ N(0, ) X μ S/ t (also valid i matched-pairs situatios) If X is the observed umber of successes i idepedet Beroulli trials i each of which the probability of success is p, ady = X,the p( p) E(Y) =p ad Var(Y) = For a radom sample of x observatios from N(μ x, σ ) ad, idepedetly, a radom sample of x y observatios from N(μ y, σ y ) (X Y) (μ x μ y ) ( σ x + σ N(0, ) y ) x y If σ x = σ y = σ (ukow) the (X Y) (μ x μ y ) {Sp ( + )} x y t x + y, where S p = ( x )S x +( y )S y x + y 8

9 Correlatio ad regressio For a set of pairs of values (x i, y i ) S xx = Σ(x i x) = Σ x i (Σ x i ) S yy = Σ(y i y) = Σ y i (Σ y i ) S xy = Σ(x i x)(y i y) =Σ x i y i (Σ x i )(Σ y i ) The product momet correlatio coefficiet is r = S xy (Sxx S yy ) = Σ(x i x)(y i y) {(Σ(xi x) )(Σ(y i y) )} = Spearma s rak correlatio coefficiet is r s = (Σ x Σ x i y i i )(Σ y i ) {(Σ x i (Σ x i ) 6Σ d ( ) The regressio coefficiet of y o x is b = S xy S xx = Σ(x i x)(y i y) Σ(x i x) Least squares regressio lie of y o x is y = a + bx where a = y bx )(Σ y i (Σ y i ) )} Distributio-free (o-parametric) tests (O Goodess-of-fit test ad cotigecy tables: i E i ) χ ν Approximate distributios for large samples Wilcoxo Siged Rak test: T N( ( + ), ( + )( + )) 4 4 Wilcoxo Rak Sum test (samples of sizes m ad, withm ): W N( m(m + + ), m(m + + )) E i 9

10 CUMULATIVE BINOMIAL PROBABILITIES = 5 p / / / / x = = 6 p / / / / x = = 7 p / / / / x = = 8 p / / / / x =

11 CUMULATIVE BINOMIAL PROBABILITIES = 9 p / / / / x = = 0 p / / / / x = = p / / / / x =

12 CUMULATIVE BINOMIAL PROBABILITIES = 4 p / / / / x = = 6 p / / / / x =

13 CUMULATIVE BINOMIAL PROBABILITIES = 8 p / / / / x = = 0 p / / / / x =

14 CUMULATIVE BINOMIAL PROBABILITIES = 5 p / / / / x =

15 CUMULATIVE BINOMIAL PROBABILITIES = 30 p / / / / x =

16 CUMULATIVE POISSON PROBABILITIES λ x = λ x = λ x = λ x = λ x =

17 CUMULATIVE POISSON PROBABILITIES λ x = λ x =

18 CUMULATIVE POISSON PROBABILITIES λ x =

19 THE NORMAL DISTRIBUTION FUNCTION If Z has a ormal distributio with mea 0 ad variace the, for each value of, the table gives the value of Φ( ), where Φ( ) = P(Z ). For egative values of use Φ( ) = Φ( ) ADD If Z has a ormal distributio with mea 0 ad variace the, for each value of p,thetablegives the value of such that P(Z )=p. Critical values for the ormal distributio p

20 CRITICAL VALUES FOR THE t DISTRIBUTION If T has a t distributio with v degrees of freedom the, for each pair of values of p ad v, the table gives the value of t such that P(T t) =p. p v =

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