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List MF19

List of formulae and statistical tables

Cambridge International AS & A Level

Mathematics (9709) and Further Mathematics (9231)

For use from 2020 in all papers for the above syllabuses.

CST319

*2508709701* 2

PURE MATHEMATICS

Mensuration

Volume of sphere =

34
3 r

Surface area of sphere =

2 4r

Volume of cone or pyramid =

1 3 base area height××

Area of curved surface of cone =

slant heightr×

Arc length of circle

r= (in radians)

Area of sector of circle

21
2 r= (in radians)

Algebra

For the quadratic equation

2

0ax bx c++=:

2 4

2bb acxa±=

For an arithmetic series:

(1) n uand=+, 11 22
() {2(1)} n

Snalnand=+= +

For a geometric series:

1n n uar (1 ) (1)1 n n arSrr =z,

11aSrr

Binomial series:

12233
() 12 3 nn n n n n nn nab a ab a b ab b +=+ + + ++ K , where n is a positive integer and !( )!n n r rn r= 23
(1) (1)(2)(1 ) 12! 3! n nn nn nxnx x x+=++ + +K, where n is rational and 1x< 3

Trigonometry

sintancos T 22
cos sin 1+, 22

1tan sec+,

22
cot 1 cosec+ sin( ) sin cos cos sinAB A B A B±± cos( ) cos cos sin sinAB A B A B±m tan tantan( )1tantanABABAB±±m sin2 2sin cosAAA

22 2 2

cos2 cos sin 2cos 1 1 2sinAAA A A 2

2tantan21tanAAA

Principal values:

1 2 1 sinx 1 2 , 0 င 1 cosx 111
22
tanx S<<

Differentiation

f( )x f( )x n x 1n nx lnx 1 x e x e x sinx cosx cos x sinx tanx 2 secx secx sec tanxx cosec x cosec cotxx cotx 2 cosecx 1 tanx 2 1 1 x+ uv dd dduvvu xx u v 2 dd dduvvu xx v If f( )xt= and g( )yt= then ddd dddyyx xtt=÷ 4

Integration

(Arbitrary constants are omitted; a denotes a positive constant.) f( )x f( ) dxx n x 1 1 n x n (1)n 1 x lnx e x e x sinx cosx cos x sinx 2 secx tanx 22
1 xa+ 1 1tanx aa 22
1 xa

1ln2xa

axa ()xa> 22
1 ax

1ln2ax

aax+ xa< ddddddvuuxuvvx xx= f()dlnf()f( )x xxx=

Vectors

If 123
aaa=++aijk and 123
bbb=++bijk then

11 22 33

.cosab ab ab=+ +=ab a b 5

FURTHER PURE MATHEMATICS

Algebra

Summations:

1 2 1 (1) n r rnn 21
6 1 (1)(21) n r rnn n 32 21
4 1 (1) n r rnn

Maclaurin's series:

2 f( ) f(0) f (0) f (0) f (0)2! ! r r xxxxr=+ + ++ +KK 2 eexp()12! ! r x xxxxr==+++++KK (all x) 231
ln(1 ) ( 1)23 rr xx xxxr 35 21
sin ( 1)3! 5! (2 1)! r r xx xxxr =++++KK (all x) 24 2
cos 1 ( 1)2! 4! (2 )! r r xx xxr=+++

KK (all x)

35 211

tan ( 1)35 21 rr xx xxxr =++++KK (-1 င x င 1) 35 21
sinh3! 5! (2 1)! r xx xxxr =+ + + + ++KK (all x) 24 2
cosh 12! 4! (2 )! r xx xxr=+ + + + +

KK (all x)

35 211

tanh35 21 r xx xxxr =+ + + + ++KK (-1 < x < 1)

Trigonometry

If 1 2 tantx= then: 2

2sin1txt=+

and 2 2

1cos1txt

Hyperbolic functions

22
cosh sinh 1xx, sinh2 2sinh coshxxx, 22
cosh2 cosh sinhxxx+ 12 sinh ln 1()xxx 12 cosh ln 1()xxx =+ (x စ 1) 11 2

1tanh ln (| | 1)1xxxx

6

Differentiation

f( )x f( )x 1 sinx 2 1 1 x 1 cosx 2 1 1 x sinhx coshx coshx sinhx tanhx 2 sechx 1 sinhx 2 1 1 x+ 1 coshx 2 1 1x 1 tanhx 2 1 1 x

Integration

(Arbitrary constants are omitted; a denotes a positive constant.) f( )x f( ) dxx sec x 11 24
ln|sec tan | ln| tan()|xx x+=+ 1 2 x< cosecx 1 2 ln|cosec cot | ln| tan |()xxx+= (0 )x<< sinhx coshx coshx sinhx 2 sechx tanhx 22
1 ax 1 sin x a xa< 22
1 xa 1 cosh x a ()xa> 22
1 ax+ 1 sinh x a 7

MECHANICS

Uniformly accelerated motion

vuat=+, 1 2 ()suvt=+, 21
2 sut at=+, 22

2vu as=+

FURTHER MECHANICS

Motion of a projectile

Equation of trajectory is:

2 22
tan2cosgxyxV T

Elastic strings and springs

x Tl 2 2 x El

Motion in a circle

For uniform circular motion, the acceleration is directed towards the centre and has magnitude 2 r or 2 v r

Centres of mass of uniform bodies

Triangular lamina:

2 3 along median from vertex

Solid hemisphere of radius r:

3 8 r from centrequotesdbs_dbs14.pdfusesText_20