AS A Level 9231 9709 Mathematics MF19 2020

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*

PURE MATHEMATICS

Mensuration

Volume of sphere =

4

3

¦Ðr 3

Surface area of sphere = 4¦Ðr 2

Volume of cone or pyramid = 13 ¡Á base area ¡Á height

Area of curved surface of cone = ¦Ðr ¡Á slant height

Arc length of circle = r¦È

( ¦È in radians)

Area of sector of circle = 12 r 2¦È

( ¦È in radians)

Algebra

For the quadratic equation ax 2 + bx + c = 0 :

x=

?b ¡À b 2 ? 4ac

2a

For an arithmetic series:

un = a + (n ? 1)d ,

S n = 12 n( a + l ) = 12 n{2a + (n ? 1) d }

For a geometric series:

un = ar n ?1 ,

Sn =

a(1 ? r n )

1? r

(r ¡Ù 1) ,

S¡Þ =

a

1? r

(

r a)

(

x < a)

FURTHER PURE MATHEMATICS

Algebra

Summations:

n

¡Æ

r =1

n

¡Æ

r = 12 n(n + 1) ,

r =1

n

¡Ær

r 2 = 16 n(n + 1)(2n + 1) ,

3

r =1

= 14 n 2 (n + 1) 2

Maclaurin¡¯s series:

f( x) = f(0) + x f ¡ä(0) +

x2

xr

f ¡ä¡ä(0) + K + f ( r ) (0) + K

r!

2!

e x = exp( x) = 1 + x +

(all x)

x 2 x3

xr

+ ? K + (?1) r +1 + K

2

3

r

(¨C1 < x ? 1)

x3 x5

x 2 r +1

+ ? K + (?1) r

+K

3! 5!

(2r + 1)!

(all x)

x2 x4

x2r

+

? K + (?1) r

+K

2! 4!

(2r )!

(all x)

ln(1 + x) = x ?

sin x = x ?

x2

xr

+K + +K

r!

2!

cos x = 1 ?

tan ?1 x = x ?

x3 x5

x 2 r +1

+ ? K + (?1) r

+K

3

5

2r + 1

sinh x = x +

(¨C1 ? x ? 1)

x3 x5

x 2 r +1

+ +K +

+K

3! 5!

(2r + 1)!

(all x)

x2 x4

x2r

+

+K+

+K

2! 4!

(2r )!

(all x)

cosh x = 1 +

tanh ?1 x = x +

x3 x5

x 2 r +1

+ +K +

+K

3

5

2r + 1

(¨C1 < x < 1)

Trigonometry

If t = tan 12 x then:

sin x =

Hyperbolic functions

cosh 2 x ? sinh 2 x ¡Ô 1 ,

2t

1+ t2

cos x =

and

1? t2

1+ t2

cosh 2 x ¡Ô cosh 2 x + sinh 2 x

sinh 2 x ¡Ô 2sinh x cosh x ,

sinh ?1 x = ln( x + x 2 + 1)

cosh ?1 x = ln( x + x 2 ? 1)

?1+ x ?

tanh ?1 x = 12 ln ?

?

?1? x ?

5

(x ? 1)

(| x | < 1)

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