GCE AS and A Level MATHEMATICS FORMULA BOOKLET

GCE AS and A Level MATHEMATICS

FORMULA BOOKLET

From September 2017

? WJEC CBAC Ltd.

Issued 2017

Pure Mathematics

Mensuration Surface area of sphere = 4r 2 Area of curved surface of cone = r slant height

Arithmetic Series

Sn =

1 2

n(a + l) =

1 2

n

[2a

+

(n

1)d]

Geometric Series a(1 r n )

Sn = 1 r a

S = 1 r for | r | < 1

Summations

n

r2

1 6

n(n

1)(2n

1)

r 1

n

r3

1 4

n2 (n 1)2

r 1

Binomial Series

n r

r

n

1

n r

11

(a

b)n

an

n 1

a

n1b

n 2

a

n2b

2

n r

a

nr

b

r

bn

(n NN)

where

n r

nCr

n! r!(n r)!

(1 x)n 1 nx n(n 1) x2 n(n 1)(n r 1) xr ( x 1, n RR)

1.2

1.2r

Logarithms and exponentials e xln a a x

Complex Numbers

{r(cos i sin )}n r n (cosn i sin n )

The roots of

z n

1

are given by

2k i

z e n , for

k

0,1, 2,, n 1

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Maclaurin's and Taylor's Series

f (x) f (0) xf (0) x2 f (0) xr f (r) (0)

2!

r!

f (x) f (a) (x a) f (a) (x a)2 f (a) (x a)r f (r) (a)

2!

r!

f (a x) f (a) xf (a) x2 f (a) xr f (r) (a)

2!

r!

ex exp(x) 1 x x2 xr for all x

2!

r!

ln(1 x) x x2 x3 (1)r1 xr (1 x 1)

23

r

sin x x x3 x5 (1)r x2r1 for all x

3! 5!

(2r 1)!

cos x 1 x2 x4 (1)r x2r for all x

2! 4!

(2r)!

tan1 x x x3 x5 (1)r x2r1 (1 x 1)

35

2r 1

sinh x x x3 x5 x2r1 for all x

3! 5!

(2r 1)!

cosh x 1 x2 x4 x2r for all x

2! 4!

(2r)!

tanh1 x x x3 x5 x2r1 (1 x 1)

35

2r 1

Hyperbolic Functions cosh2 x sinh2 x 1 sinh 2x 2sinh xcosh x cosh 2x cosh2 x sinh2 x

cosh1 x ln{x x2 1}

(x 1)

sinh1 x ln{x x2 1}

t

anh1

x

1 2

ln 1 1

x x

( x 1)

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Trigonometric Identities sin(A B) sin AcosB cos Asin B

cos(A B) cos AcosB sin Asin B

t

an(A

B)

t an 1 t

A tanB an AtanB

(

A

B

(k

1 2

)

)

For

t

t

an

1 2

A:

sin A 2t , 1 t2

cos

A

1 1

t t

2 2

sin A sin B 2sin A B cos A B

2

2

sin A sin B 2cos A B sin A B

2

2

cos A cosB 2cos A B cos A B

2

2

cos A cosB 2sin A B sin A B

2

2

Vectors The resolute of a in the direction of b is a.b b

The point dividing AB in the ratio : is a b

The equation of a plane in Cartesian form is n1x n2 y n3z k

(b a).n

The perpendicular distance between two skew lines is D

, where a and b are

n

position vectors of points on each line and n is a mutual perpendicular to both lines.

The perpendicular distance between a point and a line is D ax1 by1 c , where the a2 b2

coordinates of the point are (x1, y1) and the equation of the line is given by ax by c .

The perpendicular distance between a point and a plane is D n1 n2 n3 k , where

n2 n2 n2

1

2

3

, , are the coordinates of the point and n1x n2 y n3z k is the equation of the plane.

Matrix transformations

Anticlockwise

rotation

through

about

O:

cos sin

sin cos

Reflection in the line

y (tan )x :

cos 2 sin 2

sin 2 cos2

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Differentiation

Function f (x) g(x) tan x sec x cot x cosec x sin 1 x

cos1 x

tan1 x sinh x cosh x t anh x sinh 1 x

cosh1 x

tanh1 x

Derivative f (x)g(x) f (x)g(x) ( g ( x)) 2 sec2x

sec x tanx cosec2 x cosecx cot x

1 1 x2 1 1 x2

1 1 x2 cosh x sinh x sech2 x

1 1 x2

1 x2 1

1 1 x2

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