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

r 3

Surface area of sphere = 4r2

Volume

of

cone

or

pyramid

=

1 3

?

base

area

?

height

Area of curved surface of cone = r ? slant height

Arc length of circle = r ( in radians)

Area

of

sector

of

circle

=

1 2

r

2

( in radians)

Algebra For the quadratic equation ax2 + bx + c = 0 :

For an arithmetic series:

x = -b ? b2 - 4ac 2a

un = a + (n -1)d ,

Sn

=

1 2

n(a

+

l)

=

1 2

n{2a

+

(n

- 1)d }

For a geometric series:

un = ar n-1 ,

Sn

=

a(1 - 1-

rn) r

(r 1) ,

S

=

a 1- r

( r a)

( x < a)

4

FURTHER PURE MATHEMATICS

Algebra Summations:

n

r

=

1 2

n(n

+ 1)

,

r =1

n

r2

=

1 6

n(n

+ 1)(2n

+ 1)

,

r =1

n

r3

=

1 4

n2 (n

+ 1)2

r =1

Maclaurin's series:

f(x) = f(0) + x f (0) + x2 f (0) + K + xr f (r) (0) +K

2!

r!

ex = exp(x) = 1 + x + x2 +K + xr +K

2!

r!

ln(1 + x) = x - x2 + x3 -K + (-1)r+1 xr + K

23

r

sin

x

=

x

-

x3 3!

+

x5 5!

-K+

(-1)r

x 2 r +1 (2r +1)!

+

K

cos x = 1 - x2 + x4 -K + (-1)r x2r + K

2! 4!

(2r)!

tan -1

x

=

x-

x3 3

+

x5 5

-K + (-1)r

x 2 r +1 2r +1

+

K

sinh

x

=

x

+

x3 3!

+

x5 5!

+K+

x 2 r +1 (2r +1)!

+

K

cosh x = 1 + x2 + x4 + K + x2r + K

2! 4!

(2r)!

tanh -1

x

=

x+

x3 3

+

x5 5

+K+

x2 2r

r +1

+1

+

K

(all x) (?1 < x 1)

(all x) (all x) (?1 x 1) (all x) (all x) (?1 < x < 1)

Trigonometry

If

t

=

tan

1 2

x

then:

sin x = 2t 1+ t2

and

cos x = 1 - t2

1+ t2

Hyperbolic functions cosh2 x - sinh2 x 1 ,

sinh 2x 2sinh x cosh x ,

sinh-1 x = ln(x + x2 +1)

cosh 2x cosh2 x + sinh2 x

cosh-1 x = ln(x + x2 -1) (x 1)

tanh -1

x

=

1 2

ln

1 1

+ -

x x

(| x | < 1)

5

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