OCR A Level Further Mathematics A (H245) Formulae Booklet. RATED A+

Pure Mathematics Arithmetic series S n n = + 2 1 ( ) a l = + 2 1 n a "2 1 ( ) n d - , Geometric series ( ) S r a r 11 n n = - - S r a 3 = 1 - for r 1 1 Binomial series ( ) a b + = n n a a + + n n C C 1 ...

Pure Mathematics Arithmetic series S n n = + 2 1 ( ) a l = + 2 1 n a "2 1 ( ) n d - , Geometric series ( ) S r a r 11 n n = - - S r a 3 = 1 - for r 1 1 Binomial series ( ) a b + = n n a a + + n n C C 1 - - 1b a n n 2 2 2 b a + + f f nCr n r - b b r n + + ( ) n ! N , where !( )! ! C C nr r n r n n r r n = = = - JKKL NOOP ( ) ! ( ) ! ( ) ( ) x nx , n n x r n n n r 1 1 x x n 2 1 1 1 + = n r + + - 2 + + f - - f + +f ^ 1 1 ! Rh Series r n( ) n n ( ) 16 1 2 1 r n 2 1 = + + /= , r n 4 1 ( ) n 1 r n 3 1 2 2 = + /= Maclaurin series ( ) ( ) ( ) ! ( ) ! ( ) f f x x f f f x r 0 0 x 0 0 2 ( ) r 2 r = + l + + m f f + + ( ) e exp x x ! ! x xr 1 2 x 2 r = = + + + + f f + for all x ln( ) l x x x x ( ) ( ) xr x 2 3 1 1 r 1 2 3 r 1 + = - + -f f + - + + - 1 # sin x x 3 5 x x ! ! ( ) 1 r ( ) 2 1 xr ! 3 5 2 1 r = - + -f f + - + + + for all x cos x 1 2 4 x x ! ! ( ) 1 r ( ) 2xr ! 2 4 2r = - + -f f + - + for all x ( ) ! ( ) ! ( ) ( ) x nx , n n x r n n n r 1 1 x x n 2 1 1 1 + = n r + + - 2 + + f - - f + +f ^ 1 1 ! Rh 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 NOOP3 © OCR 2020 Further Mathematics A Turn over 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 f( ) x f l( ) x tankx k k sec2 x sec x sec t x x an cot x -cosec2x cosec x -cosec c x x ot arcsin x or sin-1x 1 x 1 2 - arccos x or cos-1x 1 x 1 2 - - arctan x or tan-1x 1 x 1 2 + Quotient rule , dd dd dd y uv yx v v ux u vx = = 2 - Differentiation from first principles f ( ) x lim f f ( ) ( ) h x h x h 0 = + - " l Integration ( ) ( ) f ( ) y fl xx d l x x = + n f c f f ( ) x x ( ) d f x ( ) n x c 1 n 1 n 1 = y l ^ ^ h h + + + Integration by parts dd d dd u d vx x uv v ux y y = - x4 © OCR 2020 Further Mathematics A The mean value of f( ) x on the interval [a, b] is f( ) x x b a 1 d b a - y Area of sector enclosed by polar curve is 2 1 y r2di f( ) x y f( ) x x d a x 1 2 2 - sin xa x a -1 1 JKKL ^ NOOP h a x 1 2 2 + tan a a 1 -1J x KKL NOOP a x 1 2 2 + sinh xa -1 JKKL NOOP or ln( ) x x + + 2 2 a x a 1 2 2 - cosh xa -1 JKKL NOOP or ln( ) x x + - 2 2 a x ( ) 2 a Numerical methods Trapezium rule: y x d h y {( y y ) ( 2 y y ) } b a 2 n n 1 y . 0 1 + + + + 2 1 f+ - , where h = b a -n The Newton-Raphson iteration for solving ( ) : ( ) ( ) f ff x x x x x 0 n n n n = = +1 - l Complex numbers Circles: z a - = k Half lines: arg( ) z a - = a