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,,,...,

+ ) (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 +.....3.

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

+ x4 xr +... + 4! (r)! +... forallx tah x = x + x3 3 + x5 xr+ +... + +.

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

+ 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

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

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

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

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

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

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

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

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 ))

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

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

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

056 0.0067 0.0033 0.00 0.0005 0.0000 0.9988 0.994 0.9734 0.9645 0.94 0.8965 0.8369 0.790 0.7648 0.686 0.593 0.5000 0.4069 0.374 0.35 0.099 0.63 0.035 0.0579 0.0355 0.066 0.0086 0.00 3.0000 0.9995 0.

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

000 0.0000 0.0000 0.0000 0.0000 0.0000 0.996 0.9470 0.859 0.87 0.738 0.6007 0.468 0.377 0.3373 0.38 0.495 0.0898 0.0498 0.050 0.0 0.0083 0.0043 0.003 0.0003 0.000 0.0000 0.0000 0.0000 3 0.9994 0.

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

846 0.6479 0.5795 0.448 0.8 0.608 0.053 0.0839 0.0398 0.070 0.0065 0.00 0.0006 0.000 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.9958 0.9559 0.8535 0.8063 0.698 0.53 0.355 0.6 0.05 0.

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

7338 0.4797 0.407 0.73 0.353 0.0600 0.036 0.036 0.008 0.005 0.0007 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.989 0.908 0.70 0.6479 0.500 0.3057 0.646 0.07 0.0783 0.

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

000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 3 0.9659 0.7636 0.47 0.386 0.340 0.096 0.033 0.049 0.0097 0.004 0.0005 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.

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

939 0.6474 0.37 0.396 0.7 0.0374 0.0093 0.0033 0.009 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 4 0.9844 0.845 0.545 0.443 0.55 0.0979 0.030 0.

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

9098 0.878 0.844 0.8088 0.775 0.9998 0.9989 0.9964 0.99 0.9856 0.9769 0.9659 0.956 0.937 3.0000 0.9999 0.9997 0.999 0.998 0.9966 0.994 0.9909 0.9865 4.0000.0000.0000 0.9999 0.9998 0.9996 0.999 0.9986 0.

17 CUMULATIVE POISSON PROBABILITIES λ x = λ x =

785 0.7693 0.753 0.7367 0.799 0.709 0.6858 0.6684 0.650 0.6335 6 0.8893 0.8786 0.8675 0.8558 0.8436 0.83 0.880 0.8046 0.7908 0.7767 7 0.9489 0.947 0.936 0.990 0.94 0.934 0.9049 0.8960 0.8867 0.

18 CUMULATIVE POISSON PROBABILITIES λ x =

0375 0.003 0.007 0.0055 0.008 0.004 0.0007 0.0003 0.000 6 0.30 0.0786 0.0458 0.059 0.04 0.0076 0.0040 0.00 0.000 0.0005 7 0.0 0.43 0.0895 0.0540 0.036 0.080 0.000 0.0054 0.009 0.005 8 0.338 0.30 0.550 0.

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

5596 0.5636 0.5675 0.574 0.5753 4 8 6 0 4 8 3 36 0. 0.5793 0.583 0.587 0.590 0.5948 0.5987 0.606 0.6064 0.603 0.64 4 8 5 9 3 7 3 35 0.3 0.679 0.67 0.655 0.693 0.633 0.6368 0.6406 0.6443 0.6480 0.

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 =

598 7.73 8.60 5 0.77.476.05.57 3.365 4.03 4.773 5.894 6.869 6 0.78.440.943.447 3.43 3.707 4.37 5.08 5.959 7 0.7.45.895.365.998 3.499 4.09 4.785 5.408 8 0.706.397.860.306.896 3.355 3.833 4.50 5.04 9 0.

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