OCR AS Level Further Mathematics A (H235) Formulae Booklet.

Matrix transformations Reflection in the line y x: 0 1 10 ! ! ! = JKKL NOOP Anticlockwise rotation through i about O: cos sin sin cos i i i i - JKKL NOOP Rotations through i about the coordinate axes. ...

Matrix transformations Reflection in the line y x: 0 1 10 ! ! ! = JKKL NOOP Anticlockwise rotation through i about O: cos sin sin cos i i i i - JKKL NOOP Rotations through i about the coordinate axes. The direction of positive rotation is taken to be anticlockwise when looking towards the origin from the positive side of the axis of rotation. cos sin sin cos 100 0 0 R x i i i i = - > H cos sin sin cos 0 010 R 0 y i i i i = > - H cos sin sin cos 0 0 001 R z i i i = i > - H Differentiation from first principles f ( ) x lim f f ( ) ( ) h x h x h 0 = + - " l Complex numbers Circles: z a - = k Half lines: arg( ) z a - = a Lines: z a - = z b -3 © OCR 2020 Further Mathematics A Turn over Vectors and 3-D coordinate geometry Cartesian equation of the line through the point A with position vector a i = + a a 1 2 j k +a3 in direction u i = + u u 1 2j k +u3 is u x a u y a u z a 1 1 2 2 3 - = - = - 3^= mh Vector product: a b aaa bbb aaa bbb a b a b a b a b a b a b ijk 1 2 3 1 2 3 1 2 3 1 2 3 2 3 3 2 3 1 1 3 1 2 2 1 # # = = = - - - JKKKKL JKKKKL JKKKKL NOOOOP NOOOOP NOOOOP Statistics Standard deviation n x x xn x 2 2 - 2 = - /^ h / or f f x x f fx x 2 2 - 2 = - ^ h / / // Discrete distributions X is a random variable taking values xi in a discrete distribution with P^X x = = i i h p Expectation: n = = E( ) X /x p i i Variance: v2 2 = = Var( ) X x /( ) i i - = n n p x / i i 2p - 2 P X ( ) = x E( ) X Var( ) X Binomial B(n, p) n ( ) x p p x n 1- -x JKKL NOOP np np( ) 1-p Uniform distribution over 1, 2, …, n U(n) 1n n 2 +1 n 12 1 ^ 2 -1h Geometric distribution Geo(p) ( ) 1-p p x-1 1p p 1 p 2 - Poisson Po(m) e x! -m mx m m Non-parametric tests Goodness-of-fit test and contingency tables: ( ) E O E i i i v 2 2 + | - /4 © OCR 2020 Further Mathematics A Correlation and regression For a sample of n pairs of observations (xi, yi) S x ( ) x n x x xx i i i 2 2 2 = = - - a k / / / , S y ( ) y y n y yy i i i 2 2 2 = - = - a k / / / , S x ( ) x y ( ) y x y n x y xy i i i i i i = - / - = / - / / Product moment correlation coefficient: r S S S x x n y y n x y n x y xx yy xy i i i i i i i i 2 2 2 2 = = - - - JKKKL JKKKL a a N OOOP NOOOP k k RSSSST VWWWWX / / / / / / / The regression coefficient of y on x is ( ( ) )( ) b S S x x x x y y xx xy i i i = = 2 - - - / / Least squares regression line of y on x is y a = +bx where a y = -bx Spearman’s rank correlation coefficient: ( ) r n n d 1 1