LIST OF FORMULAE STATISTICAL TABLES for …

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:

(a + b)n

=

an

+

n 1

a

n

-1b

+

n 2

a

n

-2b

2

+

n 3

a

n

-3b3

+ + bn , where

n

is

a

positive

integer

and

n r

=

n! r!(n -

r)!

Maclaurin expansion:

f(x) = f(0) + x f (0) + x 2 f (0) + + x n f (n) (0) +

2!

n!

(1 + x)n = 1 + nx + n(n - 1) x 2 + + n(n - 1)(n - r + 1) x r +

2!

r!

ex =1+ x + x2 + x3 ++ xr +

2! 3!

r!

sin x = x - x3 + x5 - + (-1) r x 2r+1 +

3! 5!

(2r + 1)!

cos x = 1 - x 2 + x 4 - + (-1) r x 2r +

2! 4!

(2r)!

ln(1 + x) = x - x 2 + x3 - + (-1) r+1 x r +

23

r

( x < 1)

(all x) (all x) (all x) (-1< x 1)

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 (ax + b)(x 2 + c 2 ) (ax + b) (x 2 + c 2 )

2

Trigonometry

sin( A ? B) sin A cos B ? cos Asin B

cos(A ? B) cos A cos B sin Asin B

tan( A ? B) tan A ? tan B 1 tan A tan B

sin 2A 2 sin A cos A

cos 2A cos 2 A - sin 2 A 2 cos 2 A -1 1- 2 sin 2 A

tan 2A 2 tan A 1- tan 2 A

sin

P

+

sin

Q

2 sin

1 2

(P

+

Q)

cos

1 2

(P

-

Q)

sin

P

-

sin

Q

2

cos

1 2

(P

+

Q)

sin

1 2

(P

-

Q)

cos

P

+

cos

Q

2

cos

1 2

(P

+

Q)

cos

1 2

(P

-

Q)

cos

P

-

cos

Q

-2

sin

1 2

(P

+

Q)

sin

1 2

(P

-

Q)

Principal values:

-

1 2

sin-1x

1 2

0 cos-1x

-

1 2

<

tan-1 x

<

1 2

( x 1) ( x 1)

Derivatives

f(x) sin -1 x

cos -1 x

tan -1 x cosec x sec x

f (x)

1 1- x2

- 1 1- x2

1 1+ x2 ? cosec x cot x sec x tan x

3

Integrals (Arbitrary constants are omitted; a denotes a positive constant.)

f(x)

f(x) dx

1 x2 + a2

1 a2 - x2

1 tan -1 x a a sin -1 x

a

1 x2 -a2

1 a2 - x2

tan x

1 ln x - a 2a x + a 1 ln a + x 2a a - x

ln(sec x)

cot x cosec x

ln(sin x) - ln(cosec x + cot x)

sec x

ln(sec x + tan x)

( x < a)

( x > a)

( x ................
................

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