Mathematical Formulae and Statistical Tables

List MF20

List of Formulae and Statistical Tables

Cambridge Pre-U Mathematics (9794) and Further Mathematics (9795)

For use from 2017 in all papers for the above syllabuses.

CST317

PURE MATHEMATICS

Mensuration Surface area of sphere = 4r2

Area of curved surface of cone = r ? slant height

Trigonometry a2 = b2 + c2 - 2bc cos A

Arithmetic series

un = a + (n - 1)d

Sn =

1 n(a + l) =

2

1 2

n{2a

+

(n

-

1)d}

Geometric series un = arn - 1

a(1 - rn ) Sn = 1 - r

a S = 1 - r for r < 1

Summations

n

r2 = 1 n(n +1)(2n +1) 6 r =1

n

= r3 1 n2 (n + 1)2 4 r =1

Binomial series

n r

+

r

n +

1

= nr ++11

(a

+

b)n=

a

n+

n

1

a

n-1b

+

n

2

a

n-2b 2+

...

+

n

r

a

n-rb

r+

...

+

b

n

,

(n

),

where

n = r

(1+ x) n=1+ nx + n (n -1) x2 + ... + n (n -1)...(n - r +1) xr + ... ( x < 1, n )

1.2

1.2...r

= n Cr

n! r!(n - r)!

Logarithms and exponentials ex ln a = ax

Complex numbers {r(cos + i sin )}n = rn(cos n + i sin n) ei = cos + i sin

2 ki

The roots of zn = 1 are given by z = e n , for k = 0, 1, 2, ..., n ? 1

2

Maclaurin's series

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

2!

r !

ex = exp(x) = 1 + x + x2 + ... + xr + ... for all x

2!

r !

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

23

r

sin x = x - x3 + x5 - ... + (-1)r x2r+1 + ... for all x

3! 5!

(2r +1)!

cos x = 1 - x2 + x4 - ... + (-1)r x2r + ... for all x

2! 4!

(2r )!

tan-1 x = x - x3 + x5 - ... + (-1)r x2r+1 + ... (-1 x 1)

35

2r +1

x3 x5

x 2 r +1

sinh x = x + + + ... +

+ ... for all x

3! 5!

(2r +1)!

x2 x4

x2r

cosh x = 1 + + + ... +

+ ... for all x

2! 4!

(2r )!

tanh-1 x = x + x3 + x5 + ... + x2r+1 + ... (-1 < x < 1)

35

2r +1

Hyperbolic functions cosh2 x - sinh2 x = 1 sinh 2x = 2 sinh x cosh x cosh 2x = cosh2x + sinh2x

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

sinh-1 x = ln {x + x2 + 1 }

tanh-1 x =

1 2

ln

1 1

+ -

x x

(|x| < 1)

Coordinate geometry ah + bk + c

The perpendicular distance from (h, k) to ax + by + c = 0 is a2 + b2

The

acute

angle

between

lines

with

gradients

m1

and

m2

is

tan?1

m1 1+

- m2 m1m2

3

Trigonometric identities

sin(A ? B) = sin A cos B ? cos A sin B

cos(A ? B) = cos A cos B sin A sin B

tan(A ? B) = tan A ? tan B (A ? B (k + 1 ))

1 tan A tan B

2

For t = tan

1 A : sin A =

2

2t 1+ t2

, cos A =

1- t2 1+ t2

sin A + sin B = 2 sin A + B cos A - B

2

2

sin A - sin B = 2 cos A + B sin A - B

2

2

cos A + cos B = 2 cos A + B cos A - B

2

2

cos A - cos B = ?2 sin A + B sin A - B

2

2

Vectors The resolved part of a in the direction of b is a.b

b

The point dividing AB in the ratio : is ?a + b +?

i a1 b1 a2b3 - a3b2

Vector product: a ? b = |a||b| sin= n^

j

a2 = b2

a3b1

-

a1b3

k a3 b3 a1b2 - a2b1

If A is the point with position vector a = a1i + a2j + a3k and the direction vector b is given by

b = b1i + b2j + b3k, then the straight line through A with direction vector b has cartesian equation

x= - a1 y= - a2 z - a3 (= )

b1

b2

b3

The plane through A with normal vector n = n1i + n2j + n3k has cartesian equation n1x + n2y + n3z + d = 0 where d = -a.n

The plane through non-collinear points A, B and C has vector equation r = a + (b - a) + (c - a) = (1 - - )a + b + c

The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc

The perpendicular distance of (, , ) from n1x + n2y + n3z + d = 0 is n1 + n2 + n3 + d n12 + n22 + n32

Matrix transformations

Anticlockwise

rotation

through

about

O:

cos sin

-sin

cos

Reflection

in

the

line

y

=

(tan

)x:

cos 2

sin

2

sin 2

-

cos

2

4

Differentiation f(x) tan kx sin-1 x

cos-1 x

tan-1 x

f(x) k sec2 kx

1

1- x2 -1

1- x2 1 1+ x2

sec x cot x cosec x sinh x cosh x tanh x sinh-1 x

cosh?1 x

tanh-1 x

sec x tan x - cosec2 x - cosec x cot x cosh x sinh x sech2 x

1

1+ x2

1

x2 -1

1 1- x2

Integration (+ constant; a > 0 where relevant)

f(x)

f( x) dx

sec2 kx

tan x cot x cosec x

sec x

sinh x cosh x tanh x

1 tan kx k

ln|sec x|

ln|sin x|

-ln|cosec x + cot x| = ln|tan( 1 x)| 2

ln|sec x + tan x| = ln|tan( 1 x + 2

1 4

)|

cosh x

sinh x

ln cosh x

1 a2 - x2

sin?1

x a

(|x| < a)

1 a2 + x2

1 a

tan?1

x a

1 x2 - a2

cosh?1

x a

or

ln{x

+

x2 - a2 }

(x > a)

1 a2 + x2

sinh?1

x a

or

ln{x

+

x2 + a2 }

1 a2 - x2

1 ln 2a

a+x a- x

=1 a

tanh?1

x a

(|x| < a)

1 x2 - a2

1 ln x - a 2a x + a

u dv d=x uv - v du dx

dx

dx

5

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