OCR 2023 GCE MATHEMATICS B MEI H640/03: PURE MATHEMATICS AND COMPREHENSION A LEVEL QUESTION PAPER & MARK SCHEME (MERGED)

2 © OCR 2023 H640/03 Jun23 = e n+1 2 Formulae A Level Mathematics B (MEI) (H640) Arithmetic series S = 1 n^a+lh = 1 n"2a+^n-1hd, n 2 2 Geometric series Sn = a^1-r nh 1 -r a 3 1 -r for r 1 1 Binomial series ^a+bh n = a n + nC1 a n-1 b+ nC2 a n-2 b 2 +f+ nCr a n-r b r +f+ b n ^n e Nh, n Jn N n! where Cr = n Cr = K r O = r!^n -rh! L P ^1+xh n = 1+nx+ n^n -1h x 2 +f+ n^n - 1hf^n -r+1h x r +f ^x 1 1, n e Rh 2! Differentiation f^xh r! fl^xh tan kx k sec2 kx sec x sec x tan x cot x -cosec2 x cosec x -cosec x cot x u dy v du -u dv Quotient Rule y = v , dx = dx dx v 2 Differentiation from first principles fl^xh = lim f^x+hh-f^xh h"0 h Integration c fl^xh d f^xh dx = ln f^xh +c ;fl^xhaf^xhk n dx = 1 af^xhk n+1 +c Integration by parts ; u dv dx = uv -; v du dx dx dx Small angle approximations sin i ≈ i, cosi ≈ 1 - 1 i 2 , tan i ≈ i where i is measured in radians S 3 © OCR 2023 H640/03 Jun23 Turn over 2 i i P 2 2 2 2 2 2 Trigonometric identities sin^A!Bh = sinAcosB! cosAsinB cos^A!Bh = cosA cos B " sin Asin B tanA!tanB 1 " tan A tan B Numerical methods aA!B ! ^k+ 1hrk Trapezium rule: ; b ydx ≈ 1 h"^y +y h+2^y +y +f+ y h,, where h = b-a a 2 0 n 1 2 n-1 n The Newton‑Raphson iteration for solving f^xh = 0: x Probability n+1 = xn - f^xnh fl^xnh P^A j Bh = P^Ah +P^Bh - P^A k Bh P^AkBh = P^AhP^B Ah = P^BhP^A Bh or Sample variance P^A Bh = P^AkBh P^Bh 2 1 2 ^/xih 2 2 s = n 1 Sxx where Sxx = /^xi - - xh = /x 2 - n = /x 2 -n - x Standard deviation, s = variance The binomial distribution If X + B^n, ph then P^X = rh = nCr p r q n-r where q = 1 -