List MF20. List of Formulae and Statistical Tables. Cambridge Pre-U Mathematics (9794) and Further Mathematics (9795)

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1 List MF0 List of Formulae and Statistical Tables Cambridge Pre-U Mathematics (979) and Further Mathematics (979) For use from 07 in all aers for the above syllabuses. CST7

2 Mensuration Surface area of shere = πr Area of curved surface of cone = πr slant height Trigonometry a = b + c bc cos A Arithmetic series u n = a + (n )d S n = n(a + l) = n{a + (n )d} Geometric series u n = ar n n a( r ) S n = r S = a r for r < PURE MATHEMATICS Summations n r= n r= r = nn ( + )(n + ) = ( + ) r n n Binomial series n n n+ + = r r+ r+ n n n n n n n n r r n ( a+ b) = a + a b+ a b a b b, (n N), where r n n( n ) n( n )...( n r+ ) r ( + ) = + n ( <, n R).... r n! = n n Cr = r r!( n r)! Logarithms and eonentials e ln a = a Comle numbers {r(cos θ + i sin θ)} n = r n (cos nθ + i sin nθ) e iθ = cos θ + i sin θ The roots of z n = are given by z = e π ki n, for k = 0,,,..., n

3 Maclaurin s series r f() = f(0) + f (0) + f (0) f (r) (0) +...! r! e r = e() = for all! r! ln( + ) = ( ) r + r +... ( < ) r sin = ( ) r r for all!! (r + )! cos = +! tan = sinh = +!... + ( ) r r +... for all! ( r)! + +!... + ( ) r r ( ) r + r for all (r + )! cosh = for all!! ( r)! tanh = + + Hyerbolic functions cosh sinh = sinh = sinh cosh cosh = cosh + sinh r r ( < < ) r + cosh = ln { + } ( ) sinh = ln { + + } tanh = ln + ( < ) Coordinate geometry The erendicular distance from (h, k) to a + by + c = 0 is ah + bk + c a + b The acute angle between lines with gradients m and m is tan m m + mm

4 Trigonometric identities sin(a ± B) = sin A cos B ± cos A sin B cos(a ± B) = cos A cos B sin A sin B tan(a ± B) = tan A± tan B (A ± B (k + tan Atan B )π) For t = tan A : sin A = t t, cos A = + t + t A+ B A B sin A + sin B = sin cos A+ B A B sin A sin B = cos sin A+ B A B cos A + cos B = cos cos A+ B A B cos A cos B = sin sin Vectors The resolved art of a in the direction of b is a.b b The oint dividing AB in the ratio λ : μ is µ a+ λ b λ + µ i a b ab ab Vector roduct: a b = a b sin θ ˆ n= j a b = ab ab k a b ab ab If A is the oint with osition vector a = a i + a j + a k and the direction vector b is given by b = b i + b j + b k, then the straight line through A with direction vector b has cartesian equation a y a z a = = (= λ) b b b The lane through A with normal vector n = n i + n j + n k has cartesian equation n + n y + n z + d = 0 where d = a.n The lane through non-collinear oints A, B and C has vector equation r = a + λ(b a) + μ(c a) = ( λ μ)a + λb + μc The lane through the oint with osition vector a and arallel to b and c has equation r = a + sb + tc The erendicular distance of (α, β, γ) from n + n y + n z + d = 0 is nα + n β + n γ + d + + n n n Matri transformations cosθ sinθ Anticlockwise rotation through θ about O: sinθ cosθ cos θ sin θ Reflection in the line y = (tan θ): sin θ cos θ

5 Differentiation f() f () tan k k sec k sin cos tan + sec sec tan cot cosec cosec cosec cot sinh cosh cosh sinh tanh sech sinh + cosh tanh Integration (+ constant; a > 0 where relevant) f() f( ) d sec k tan k k tan ln sec cot ln sin cosec ln cosec + cot = ln tan( ) sec ln sec + tan = ln tan( + π) sinh cosh cosh sinh tanh ln cosh sin ( < a) a a a + a tan a cosh a a or ln{ + a } ( > a) sinh a + a or ln{ + + a } a+ ln = tanh ( < a) a a a a a a ln a a + a dv du u d = uv v d d d

6 Area of a sector A = r dθ (olar coordinates) A = dy d y dt (arametric form) dt dt Arc length s = s = s = dy + d (cartesian coordinates) d d dy + dt dt dt (arametric form) dr r + dθ (olar form) dθ Surface area of revolution S = π yds S =π y ds Numerical solution of equations The Newton-Rahson iteration for solving f() = 0: n + = n f( n ) f( ) n

7 MECHANICS Motion in a circle Transverse velocity: v= rθ Transverse acceleration: v = rθ v Radial acceleration: r θ = r PROBABILITY 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 )P( B A ) j ΣP( A)P( B A) i j i Discrete distributions For a discrete random variable X taking values i with robabilities i Eectation (mean): E(X) = μ = Σ i i Variance: Var(X) = σ = Σ( i μ) i = Σ i i μ For a function g(x) : E(g(X)) = Σ g( i ) i The robability generating function (P.G.F.) of X is G X (t) = E(t X ), and E(X) = G X (), Var(X) = G X() + G X() {G X()} For Z = X + Y, where X and Y are indeendent: G Z (t) = G X (t) G Y (t) The moment generating function (M.G.F.) of X is M X (t) = E(e tx ), and E(X) = M X (0), E(X n ) = M ( n) X (0), Var(X) = M X (0) { M X (0)} For Z = X + Y, where X and Y are indeendent: M Z (t) = M X (t) M Y (t) Standard discrete distributions Distribution of X P(X = ) Mean Variance P.G.F. M.G.F. n Binomial B(n, ) ( ) n n n( ) ( + t) n ( + e t ) n Poisson Po(λ) e λ λ! λ λ e λ(t ) t λ(e ) e Geometric Geo() on,, ( ) t ( t ) e ( )e t t 7

8 Continuous distributions For a continuous random variable X having robability density function (P.D.F.) f Eectation (mean): E(X) = μ = f( )d Variance: Var(X) = σ = ( µ ) f( )d= f( )d µ For a function g(x) : E(g(X)) = g( )f( )d Cumulative distribution function: F() = P(X ) = f( t)dt The moment generating function (M.G.F.) of X is M X (t) = E(e tx ), and E(X) = M X (0), E(X n ) = M ( n) X (0), Var(X) = M X (0) { M X (0)} For Z = X + Y, where X and Y are indeendent: M Z (t) = M X (t) M Y (t) Standard continuous distributions Distribution of X P.D.F Mean Variance M.G.F. Uniform (Rectangular) on [a, b] b a + b a ( a b) bt at ( ) e e ( b at ) Eonential λ e λ λ λ λ λ t Normal N(μ, σ ) e σ π µ σ μ σ e µ t+ σ t Eectation algebra For indeendent random variables X and Y E(XY) = E(X) E(Y), Var (ax ± by) = a Var(X) + b Var (Y) 8

9 Samling distributions For a random samle X, X,, X n of n indeendent observations from a distribution having mean μ and variance σ σ X is an unbiased estimator of μ, with Var( X ) = n S is an unbiased estimator of σ, where S Σ( Xi X) = n For a random samle of n observations from N(μ, σ ) X µ ~ N(0,) σ / n X µ ~ t n (also valid in matched-airs situations) S / n If X is the observed number of successes in n indeendent Bernoulli trials, in each of which the robability of success is, and Y = X n, then E(Y) = and Var(Y) = ( ) n For a random samle of n observations from N(μ, σ ) and, indeendently, a random samle of n y observations from N(μ y, σ y ) ( X Y) ( µ µ y) ~ N(0,) σ σ y + n n If σ = y ( X Y) ( µ ) σ y = σ µ y (unknown) then tn+ ny, where S + n n y Correlation and Regression For a set of n airs of values ( i, y i ) S = Σ( i ) ( Σ ) = Σ i i n S yy = Σ(y i y ) ( Σy ) = Σ y i i n S y = Σ( i )(y i y ) = Σ i y i ( Σi )( Σyi ) n The roduct-moment correlation coefficient is ( Σi)( Σyi) S Σy ( )( ) i i y Σ i yi y r = = = n S Syy { Σ( i ) }{ Σ( yi y) } ( Σi) ( Σyi) Σi Σyi n n Sy Σ( i )( yi y) The regression coefficient of y on is b = = S Σ( i ) Least squares regression line of y on is y = a + b where a = y b + y y ( n ) S ( n ) S S = n + n y 9

10 CUMULATIVE BINOMIAL PROBABILITIES n n r r P( Xø ) = Cr ( ) r= 0 n = / / / / = n = / / / / = n = / / / / = n = / / / / =

11 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / / / / / n = 9 = n = 0 = n = =

12 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / n = = n = =

13 CUMULATIVE BINOMIAL PROBABILITIES / / / / / / / / n = 8 = n = 0 =

14 CUMULATIVE BINOMIAL PROBABILITIES / / / / n = =

15 CUMULATIVE BINOMIAL PROBABILITIES / / / / n = 0 =

16 CUMULATIVE POISSON PROBABILITIES P( Xø ) = e r= 0 r λ λ λ = λ = λ = λ = λ = r!

17 CUMULATIVE POISSON PROBABILITIES λ = λ =

18 CUMULATIVE POISSON PROBABILITIES λ =

19 If Z has a normal distribution with mean 0 and variance, then, for each value of z, the table gives the value of Φ(z), where Φ(z) = P(Z z). For negative values of z, use Φ( z) = Φ(z). THE NORMAL DISTRIBUTION FUNCTION z ADD If Z has a normal distribution with mean 0 and variance, then, for each value of, the table gives the value of z such that P(Z z) =. Critical values for the normal distribution z

20 CRITICAL VALUES FOR THE t-distribution If T has a t-distribution with ν degrees of freedom, then, for each air of values of and ν, the table gives the value of t such that: P(T t) = ν =