LIST OF FORMULAE STATISTICAL TABLES for Mathematics …

1 Cambridge Pre-U Revised Syllabus MINISTRY OF EDUCATION, SINGAPORE. in collaboration with UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE. General certificate of Education Advanced Level List MF26. LIST OF FORMULAE . AND. STATISTICAL TABLES . for Mathematics and further Mathematics For use from 2017 in all papers for the H1, H2 and H3 Mathematics and H2 further Mathematics syllabuses. CST310.. This document consists of 11 printed pages and 1 blank page. UCLES & MOE 2015. PURE Mathematics . Algebraic series Binomial expansion: n n n . (a + b) n = a n + a n 1b + a n 2b 2 + a n 3b3 + + b n , where n is a positive integer and 1 2 3 . n n! =. r r!(n r )! Maclaurin expansion: x2 x n (n). f( x) = f(0) + x f (0) + f (0) + + f (0) + . 2! n! n(n 1) 2 n(n 1) (n r + 1) r (1 + x) n = 1 + nx + x + + x + ( x < 1). 2! r! x2 x3 xr ex =1+ x + + + + + (all x). 2! 3! r! x3 x5 ( 1) r x 2 r +1. sin x = x + + + (all x). 3! 5! (2r + 1)! x2 x4 ( 1) r x 2 r cos x = 1 + + + (all x).

2 2! 4! (2r )! x2 x3 ( 1) r +1 x r ln(1 + x) = x + + + ( 1< x 1). 2 3 r Partial fractions decomposition Non-repeated linear factors: px + q A B. = +. (ax + b)(cx + d ) (ax + b) (cx + d ). Repeated linear factors: px 2 + qx + r A B C. = + +. (ax + b)(cx + d ) 2. (ax + b) (cx + d ) (cx + d ) 2. Non-repeated quadratic factor: px 2 + qx + r A Bx + C. = + 2. (ax + b)( x + c ). 2 2. (ax + b) ( x + c 2 ). 2. Trigonometry 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) . 1 tan A tan B. sin 2 A 2 sin A cos A. cos 2 A cos 2 A sin 2 A 2 cos 2 A 1 1 2 sin 2 A. 2 tan A. tan 2 A . 1 tan 2 A. sin P + sin Q 2 sin 12 ( P + Q) cos 12 ( P Q). sin P sin Q 2 cos 12 ( P + Q) sin 12 ( P Q). cos P + cos Q 2 cos 12 ( P + Q) cos 12 ( P Q). cos P cos Q 2 sin 12 ( P + Q) sin 12 ( P Q). Principal values: 12 sin 1x 1. 2. ( x 1). 0 cos 1x ( x 1). 12 < tan 1 x < 12 . Derivatives f(x) f ( x). 1. sin 1 x 1 x 2.

3 1. cos 1 x . 1 x 2. 1. tan 1 x 1 + x2. cosec x cosec x cot x sec x sec x tan x 3. Integrals (Arbitrary constants are omitted; a denotes a positive constant.). f(x) f( x) dx 1 1 x . tan 1 . x + a2. 2. a a . x . (x < a). 1. sin 1 . a x 2 2. a . 1 1 x a . ln ( x > a). x a 2 2. 2a x + a . 1 1 a+x . ln ( x <a). a x 2 2. 2a a x . tan x ln(sec x) ( x < 12 ). cot x ln(sin x) (0< x < ). cosec x ln(cosec x + cot x) (0< x < ). sec x ln(sec x + tan x) ( x < 12 ). Vectors a + b The point dividing AB in the ratio : has position vector + . Vector product: a1 b1 a 2 b3 a 3 b2 .. a b = a 2 b2 = a 3 b1 a1b3 . a b a b a b . 3 3 1 2 2 1 . 4. Numerical methods f ( x)dx 2 (b a)[f (a) + f (b)]. b . 1. Trapezium rule (for single strip): a b a +b . f ( x)dx 6 (b a ) f (a ) + 4f + f (b) . 1. Simpson's rule (for two strips): a 2 . The Newton-Raphson iteration for approximating a root of f(x) = 0: f ( x1 ). x2 = x1 , f ( x1 ). where x1 is a first approximation.

4 Euler Method with step size h: y 2 = y1 + hf ( x1 , y1 ). Improved Euler Method with step size h: u 2 = y1 + hf ( x1 , y1 ). y 2 = y1 +. h [f (x1 , y1 ) + f (x2 , u 2 )]. 2. 5. PROBABILITY AND STATISTICS. Standard discrete distributions Distribution of X P( X = x) Mean Variance n x Binomial B(n,p ) p (1 p ) n x np np (1 p ). x . Poisson Po( ) x e . x! 1 1 p Geometric Geo(p) (1 p)x 1p p p2. Standard continuous distribution Distribution of X Mean Variance 1 1. Exponential e x 2. Sampling and testing Unbiased estimate of population variance: n ( x x ) 2 1 2 ( x ) 2 . s2 = = . n 1 n 1 x n . n . Unbiased estimate of common population variance from two samples: ( x1 x1 ) 2 + ( x 2 x 2 ) 2. s2 =. n1 + n 2 2. Regression and correlation Estimated product moment correlation coefficient: x y xy . ( x x )( y y ) n r= =. { ( x x ) }{ ( y y ) }. 2 2. 2 ( x ) 2. x . 2 ( y ) 2. y .. n n . Estimated regression line of y on x : ( x x )( y y ).

5 Y y = b( x x ), where b =. ( x x ) 2. 6. THE NORMAL DISTRIBUTION FUNCTION. If Z has a normal distribution with mean 0 and variance 1 then, for each value of z, the table gives the value of (z) , where (z ) = P(Z z). For negative values of z use ( z) = 1 ( z) . 1 2 3 4 5 6 7 8 9. z 0 1 2 3 4 5 6 7 8 9. ADD. 4 8 12 16 20 24 28 32 36. 4 8 12 16 20 24 28 32 36. 4 8 12 15 19 23 27 31 35. 4 7 11 15 19 22 26 30 34. 4 7 11 14 18 22 25 29 32. 3 7 10 14 17 20 24 27 31. 3 7 10 13 16 19 23 26 29. 3 6 9 12 15 18 21 24 27. 3 5 8 11 14 16 19 22 25. 3 5 8 10 13 15 18 20 23. 2 5 7 9 12 14 16 19 21. 2 4 6 8 10 12 14 16 18. 2 4 6 7 9 11 13 15 17. 2 3 5 6 8 10 11 13 14. 1 3 4 6 7 8 10 11 13. 1 2 4 5 6 7 8 10 11. 1 2 3 4 5 6 7 8 9. 1 2 3 4 4 5 6 7 8. 1 1 2 3 4 4 5 6 6. 1 1 2 2 3 4 4 5 5. 0 1 1 2 2 3 3 4 4. 0 1 1 2 2 2 3 3 4. 0 1 1 1 2 2 2 3 3. 0 1 1 1 1 2 2 2 2. 0 0 1 1 1 1 1 2 2. 0 0 0 1 1 1 1 1 1. 0 0 0 0 1 1 1 1 1. 0 0 0 0 0 1 1 1 1. 0 0 0 0 0 0 0 1 1.

6 0 0 0 0 0 0 0 0 0. Critical values for the normal distribution If Z has a normal distribution with mean 0 and variance 1 then, for each value of p, the table gives the value of z such that P(Z z) = p. p z 7. CRITICAL VALUES FOR THE t-DISTRIBUTION. If T has a t-distribution with degrees of freedom then, for each pair of values of p and , the table gives the value of t such that P(T t) = p. p =1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 8. CRITICAL VALUES FOR THE 2 -DISTRIBUTION. If X has a 2 -distribution with degrees of freedom then, for each pair of values of p and , the table gives the value of x such that P(X x) = p. p =1 3. 1571 3. 9821 2. 3932 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 70 80 90 100 9. WILCOXON SIGNED RANK TEST. P is the sum of the ranks corresponding to the positive differences, Q is the sum of the ranks corresponding to the negative differences, T is the smaller of P and Q.

7 For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the level of significance indicated. Critical values of T. Level of significance One Tail Two Tail n=6 2 0. 7 3 2 0. 8 5 3 1 0. 9 8 5 3 1. 10 10 8 5 3. 11 13 10 7 5. 12 17 13 9 7. 13 21 17 12 9. 14 25 21 15 12. 15 30 25 19 15. 16 35 29 23 19. 17 41 34 27 23. 18 47 40 32 27. 19 53 46 37 32. 20 60 52 43 37. For larger values of n , each of P and Q can be approximated by the normal distribution with mean 1. 4. n(n + 1) and variance 1. 24. n(n + 1)(2n + 1) . 10. BLANK PAGE. 11. This booklet is the property of SINGAPORE EXAMINATIONS AND ASSESSMENT BOARD. 12.