Mathematical Formulae and Statistical Tables

1 List MF20. List of Formulae and Statistical Tables Cambridge Pre-U Mathematics (9794) and Further Mathematics (9795). For use from 2017 in all papers for the above syllabuses. CST317.. PURE MATHEMATICS. Mensuration Surface area of sphere = 4 r2. Area of curved surface of cone = r slant height Trigonometry a2 = b2 + c2 2bc cos A. Arithmetic series un = a + (n 1)d Sn = 12 n(a + l) = 12 n{2a + (n 1)d}. Geometric series un = arn 1. a (1 r n ). Sn =. 1 r a S = for r < 1. 1 r Summations n r r =1. 2. = 16 n(n + 1)(2n + 1). n.

2 = r r =1. 3 1. 4. n 2 (n + 1) 2. Binomial series n n n + 1 . + = . r r + 1 r + 1 . n n 1 n n 2 2 n n r r n n! (a + b) n= a n+ a b + a b + .. + a b + .. + b , (n ), where =. n =. n Cr 1 2 r r r !(n r )! n ( n 1) 2 n ( n 1) .. ( n r + 1) r (1 + x ) n =1 + nx + x + .. + x + .. ( x < 1, n ). r Logarithms and exponentials ex ln a = ax Complex numbers {r(cos + i sin )}n = rn(cos n + i sin n ). ei = cos + i sin . 2 ki The roots of zn = 1 are given by z = e n , for k = 0, 1, 2, .., n 1. 2. Maclaurin's series x2 x r (r). f(x) = f(0) + x f (0) + f (0) +.

3 + f (0) + .. 2! r! x2 xr ex = exp(x) = 1 + x + + .. + + .. for all x 2! r! x2 x3 xr ln(1 + x) = x + .. + ( 1)r + 1 + .. ( 1 < x 1). 2 3 r x3 x5 x 2 r +1. sin x = x + .. + ( 1)r + .. for all x 3! 5! (2r + 1)! x2 x4 x2r cos x = 1 + .. + ( 1)r + .. for all x 2! 4! (2r )! x3 x5 x 2 r +1. tan 1 x = x + .. + ( 1)r + .. ( 1 x 1). 3 5 2r + 1. x3 x5 x 2 r +1. sinh x = x + + + .. + + .. for all x 3! 5! (2r + 1)! x2 x4 x2r cosh x = 1 + + + .. + + .. for all x 2! 4! (2r )! x3 x5 x 2 r +1. tanh 1 x = x + + + .. + + .. ( 1 < x < 1).

4 3 5 2r + 1. Hyperbolic functions cosh2 x sinh2 x = 1. sinh 2x = 2 sinh x cosh x cosh 2x = cosh2x + sinh2x cosh 1 x = ln {x + x2 1 } (x 1). sinh 1 x = ln {x + x2 + 1 }. 1+ x . tanh 1 x = 1. ln (|x| < 1). 2. 1 x . Coordinate geometry ah + bk + c The perpendicular distance from (h, k) to ax + by + c = 0 is a 2 + b2. m1 m2. The acute angle between lines with gradients m1 and m2 is tan 1. 1 + m1m2. 3. 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 tan B. tan(A B) = (A B (k + 12 ) ).

5 1 tan A tan B. 2t 1 t2. For t = tan 12 A : sin A = , cos A =. 1+ t2 1+ t2. A+ B A B. sin A + sin B = 2 sin cos 2 2. A+ B A B. sin A sin B = 2 cos sin 2 2. A+ B A B. cos A + cos B = 2 cos cos 2 2. A+ B A B. cos A cos B = 2 sin sin 2 2. Vectors The resolved part of a in the direction of b is b a + b The point dividing AB in the ratio : is + . i a1 b1 a2b3 a3b2 .. n j Vector product: a b = |a||b| sin= a2 =. b2 a3b1 a1b3 . ab a b . k a3 b3 1 2 2 1 . If A is the point with position vector a = a1i + a2j + a3k and the direction vector b is given by b = b1i + b2j + b3k, then the straight line through A with direction vector b has cartesian equation x a1 y a2 z a3.

6 = = (= ). b1 b2 b3. The plane through A with normal vector n = n1i + n2j + n3k has cartesian equation n1x + n2y + n3z + d = 0. where d = The plane through non-collinear points A, B and C has vector equation r = a + (b a) + (c a) = (1 )a + b + c The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc n1 + n2 + n3 + d The perpendicular distance of ( , , ) from n1x + n2y + n3z + d = 0 is n12 + n22 + n32. Matrix transformations cos sin . Anticlockwise rotation through about O.

7 Sin cos . cos 2 sin 2 . Reflection in the line y = (tan )x: . sin 2 cos 2 . 4. Differentiation f(x) f (x). tan kx k sec2 kx 1. sin 1 x 1 x2. 1. cos 1 x . 1 x2. 1. tan 1 x 1 + x2. sec x sec x tan x cot x cosec2 x cosec x cosec x cot x sinh x cosh x cosh x sinh x tanh x sech2 x 1. sinh 1 x 1 + x2. 1. cosh 1 x x2 1. 1. tanh 1 x 1 x2. Integration (+ constant; a > 0 where relevant). f(x) . f( x ) dx 1. sec2 kx tan kx k tan x ln|sec x|. cot x ln|sin x|. cosec x ln|cosec x + cot x| = ln|tan( 12 x)|. sec x ln|sec x + tan x| = ln|tan( 12 x + 1.)

8 4. )|. sinh x cosh x cosh x sinh x tanh x ln cosh x 1 x . sin 1 (|x| < a). a2 x2 a . 1 1 x . tan 1 . a + x2. 2. a a . 1 x . cosh 1 or ln{x + x2 a2 } (x > a). x a 2 2 a . 1 x . sinh 1 or ln{x + x2 + a2 }. a +x 2 2 a . 1 1 a+x 1 x . ln = tanh 1 (|x| < a). a x 2 2. 2a a x a a . 1 1 x a ln x a 2 2. 2a x + a dv du u dx d=. x uv v . dx dx 5. Area of a sector . A = 12 r 2 d (polar coordinates). dy dx . x dt y dt dt 1. A= 2. (parametric form). Arc length 2. dy . s= 1 + dx (cartesian coordinates). dx . 2 2. dx dy . s= + dt dt dt.

9 (parametric form). 2. dr . s= r2 + d . d . (polar form). Surface area of revolution S x = 2 y ds .. Sy =2 x ds Numerical solution of equations f ( xn ). The Newton-Raphson iteration for solving f(x) = 0: xn + 1 = xn . f ( xn ). 6. MECHANICS. Motion in a circle Transverse velocity: v = r . Transverse acceleration: v = r . v2. Radial acceleration: r 2 =.. r PROBABILITY. Probability P(A B) = P(A) + P(B) P(A B). P(A B) = P(A) P(B | A). P( B A) P( A). P(A | B) =. P( B A) P( A) + P( B A ) P( A ). P( A j ) P( B A j ). Bayes' Theorem: P(Aj | B) =.

10 P( Ai ) P( B Ai ). Discrete distributions For a discrete random variable X taking values xi with probabilities pi Expectation (mean): E(X) = = xi pi Variance: Var(X) = 2 = (xi )2 pi = xi2 pi 2. For a function g(X) : E(g(X)) = g(xi) pi The probability generating function ( ) of X is GX (t) = E(tX), and E(X) = G X (1), Var(X) = G X (1) + G X (1) {G X (1)}2. For Z = X + Y, where X and Y are independent: GZ (t) = GX (t) GY (t). The moment generating function ( ) of X is MX (t) = E(etX), and E(X) = M X (0), E(X n) = M (Xn ) (0), Var(X) = M X (0) { M X (0)}2.