A Level Further Mathematics A (H245) Formulae …

Oxford Cambridge and RSA

A Level Further Mathematics A (H245)

Formulae Booklet

A Level Further Mathematics A

*9990860740*

INSTRUCTIONS ? Do not send this Booklet for marking. Keep it in the centre or recycle it.

INFORMATION ? This document has 16 pages.

Copyright Information

OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (.uk) after the live examination series.

If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity.

For queries or further information please contact The OCR Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.

OCR is part of Cambridge University Press & Assessment, which is itself a department of the University of Cambridge.

? OCR 2022 [603/1325/0] DC (SLM) 320372/1

A Level Further Mathematics A

CST322

2

Pure Mathematics

Arithmetic series

Sn

=

1 2

n (a + l)

=

1 2

n

"2a

+

(n - 1) d,

Geometric series

a (1 - rn) Sn = 1 - r

S3

=

a 1-r

for

r

11

Binomial series

(a + b) where

n n

= Cr

an =

+n CnrC=1 aJLKKnrnNPOO-

1b =

+ n C2an-2b2

n! r! (n - r) !

+

f

+

n

Cr

an-rbr

+

f

+

bn

(n ! N),

(1

+

x) n

=

1

+

nx

+

n

(n 2!

1)

x2

+

f

+

n

(n

-

1)

f (n r!

-

r

+

1)

xr

+

f

^ x 1 1, n ! Rh

Series

/ / n r2

r=1

=

1 6

n (n + 1) (2n + 1),

n

r3

r=1

=

1 4

n2

(n

+

1)

2

Maclaurin series

f

(x)

=

f

(0)

+

f

l(0)

x

+

f

m (0) 2!

x2

+

f

+

f

(r) (0) r!

xr

+

f

ex

=

exp (x)

=

1

+

x

+

x2 2!

+

f

+

xr r!

+

f

for

all

x

ln (l

+

x)

=

x

-

x2 2

+

x3 3

-

f

+

(-1) r+1

xr r

+

f (-1

1

x

#

1)

sin

x

=

x

-

x3 3!

+

x5 5!

-

f

+

(-1)

r

x 2r + 1 (2r + 1)

!

+

f

for

all

x

cos

x

=

1

-

x2 2!

+

x4 4!

-

f

+

(-1)

r

x2r (2r)

!

+

f

for

all

x

(1

+

x) n

=

1

+

nx

+

n

(n 2!

1)

x2

+

f

+

n

(n

-

1)

f (n r!

-

r

+

1)

xr

+

f

^ x 1 1, n ! Rh

Matrix transformations

Reflection

in

the

line

y

=

!x

:

JLKK!

0 1

! 10NPOO

Anticlockwise rotation through i about O : JLKKcsoinsii -cosisniiNPOO

? OCR 2022

A Level Further Mathematics A

3

Rotations through i about the coordinate axes. The direction of positive rotation is taken to be anticlockwise when looking towards the origin from the positive side of the axis of rotation.

10 0

Rx = > 0 cos i -sin i H

0 sin i cos i

cos i 0 sin i

Ry = > 0 1 0 H

-sin i 0 cos i

cos i -sin i 0

Rz = > sin i cos i 0 H

0 01

Differentiation

f (x) tan kx sec x cot x cosec x

arcsin x or sin-1x

arccos x or cos-1x arctan x or tan-1x

Quotient rule

y = uv,

dy dx

=

v

du dx

-u v2

dv dx

Differentiation from first principles

f

l(x)

=

f (x lim h"0

+

h) h

- f (x)

Integration

y

f l(x) f (x)

dx

=

ln

f (x)

+

c

yf

l(x)^f (x)hn

dx

=

n

1 +

1^f (x)hn+1

+

c

Integration

by

parts

y

u

dv dx

dx

=

uv -

y

v

du dx

d

x

f l(x) k sec2kx sec x tan x - cosec 2 x -cosec x cot x

1 1 - x2 -1 1 - x2

1 1 + x2

? OCR 2022

A Level Further Mathematics A

Turn over

4

The mean value of f (x) on the interval [a, b] is 1 y bf (x) dx b-a a

Area

of

sector

enclosed

by

polar

curve

is

1 2

y

r

2di

f (x)

1 a2 -x2

1 a2 +x2

1 a2 +x2

1 x2 -a2

y f(x)dx

sin-1

JKK L

x a

NOO P

^ x 1 ah

1 a

tan

-1

JKK L

x a

NOO P

sinh

-1

JKK L

x a

NOO P

or

ln (x +

cosh

-1

JKK L

x a

NOO P

or

ln (x +

x2 + a2) x2 - a2)

(x 2 a)

Numerical methods

y Trapezium rule:

b a

ydx

.

1 2

h {(y0

+ yn)

+ 2 (y1

+ y2

+ f + yn-1)

},

where

h

=

b-a n

The Newton-Raphson iteration for solving

f

(x)

=

0

:

xn+1

=

xn

-

f (xn) f l(xn)

Complex numbers

Circles: z - a = k

Half lines: arg (z - a) = a

Lines: z - a = z - b

De Moivre's theorem: {r (cos i + i sin i)} n = rn (cos ni + i sin ni)

Roots

of

unity:

The

roots

of

zn

=

1

are

given

by

z

=

exp JLKK2rn k

iNOO P

for

k

=

0, 1,

2, f,

n-1

Vectors and 3-D coordinate geometry

Cartesian equation of the line through the point A with position vector a = a1i + a2 j + a3k in direction

u

=

u1i + u2 j + u3k

is

x - a1 u1

=

y - a2 u2

=

z

- a3 u3

^=

mh

Cartesian equation of a plane n1 x + n2 y + n3 z + d = 0

JKa1 NO JKb1 NO i a1 b1 JKa2 b3 - a3 b2 NO

Vector

product:

a

#

b

=

K K L

a2 a3

O O P

#

K K L

b2 b3

O O P

=

j k

a2 a3

b2 b3

=

K K L

a3 a1

b1 b2

-

a1 a2

b3 b1

O O P

? OCR 2022

A Level Further Mathematics A

5

The distance between skew lines is D = ^b - ah. n , where a and b are position vectors of points on each line and n

n is a mutual perpendicular to both lines

The

distance between a

point

and

a line

is

D

=

ax1 + by1 - c a2 +b2

,

where

the coordinates

of

the point

are (x1,

y1) and

the equation of the line is given by ax + by = c

The distance between a point and a plane is D = b . n - p , where b is the position vector of the point and the n

equation of the plane is given by r . n = p

Small angle approximations

sin i

.

i,

cos i

.

1-

1 2

i

2,

tan i

.

i

where

i

is

small

and

measured

in

radians

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 1 " tan A tan B

(A ! B ! (k + 21) r)

Hyperbolic functions cosh2x - sinh2x = 1 sinh-1x = ln [x + (x2 + 1)] cosh-1x = ln [x + (x2 - 1)], x $ 1

tanh-1x

=

1 2

ln JLKK11 -+ xxNPOO,

-1

1

x

1

1

Simple harmonic motion

x = A cos (~t) + B sin (~t)

x = R sin (~t + {)

Statistics

Probability

P (A , B) = P (A) + P (B) - P (A + B)

P (A + B) = P (A) P (B A) = P (B) P (A B)

or

P(A

B)

=

P (A + B) P (B)

? OCR 2022

A Level Further Mathematics A

Turn over

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download