)luvw fhuwl¿fdwlrq iurp mathematical formulae and … · 2020-07-13 · 7 as mathematics 7 pure...

132
Pearson Edexcel Level 3 Advanced Subsidiary and Advanced GCE Mathematics and Further Mathematics Mathematical formulae and statistical tables )LUVW FHUWL¿FDWLRQ IURP AdvanceG 6XEVLGLDU\ *&( LQ 0DWKHPDWLFV 0$ $GYDQFHG *&( LQ 0DWKHPDWLFV 0$ Advanced Subsidiary G&( LQ )XUWKHU 0DWKHPDWLFV )0 )LUVW FHUWL¿FDWLRQ IURP $GYDQFHG *&( LQ )XUWKHU 0DWKHPDWLFV )0 This copy is the property of Pearson. It is not to be removed from the examination room or marked in any way. Q$

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Page 1: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Pear

son

Edex

cel L

evel

3

Adv

ance

d Su

bsid

iary

and

A

dvan

ced

GC

E M

athe

mat

ics

and

Furt

her M

athe

mat

ics

Mat

hem

atic

al fo

rmul

ae a

nd s

tatis

tical

tabl

es

A

dvan

ce

Adv

ance

d Su

bsid

iary

G

This

cop

y is

the

prop

erty

of P

ears

on.

It is

not

to b

e re

mov

ed fr

om th

e ex

amin

atio

n ro

om o

r mar

ked

in a

ny w

ay.

Q

Page 2: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Abo

ut P

ears

on e

mpl

oyee

s in

mor

e th

an

Page 3: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Ref

eren

ces

to th

ird p

arty

mat

eria

l mad

e in

this

sam

ple

asse

ssm

ent

acce

pt re

spon

sibi

lity

for t

he c

onte

nt o

f mat

eria

ls, w

hich

may

be

subj

ect

All

info

rmat

ion

in th

is d

ocum

ent i

s co

rrec

t at t

ime

of p

ublic

atio

n.

Page 4: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Con

tent

s

7 A

S M

ATH

EMAT

ICS

7 Pu

re M

athe

mat

ics

9 St

atis

tics

A LE

VEL

MAT

HEM

ATIC

S

33

Stat

istic

s 39

M

echa

nics

Page 5: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Con

tent

s co

ntin

ued

53

Stat

istic

s

STAT

ISTI

CA

L TA

BLE

S

t F

Page 6: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Intr

oduc

tion

or A

Lev

el F

urth

er M

athe

mat

ics

pape

rs m

ay b

e re

quire

d to

use

the

form

ulae

that

wer

e in

trod

uced

in A

S or

A L

evel

Mat

hem

atic

s pa

pers

.

need

to u

se fo

rmul

ae in

trod

uced

in th

e ap

prop

riate

Pur

e M

athe

mat

ics

pape

rs fo

r the

Page 7: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

A

S M

athe

mat

ics

Men

sura

tion

Surf

ace

area

of s

pher

e =

4r

Are

a of

cur

ved

surf

ace

of c

one

= r ×

Bin

omia

l ser

ies

(a +

n = a

n + n 1

an _

b + n 2

an _

b +

. . . +

n r

an _ r br +

. . . +

bn

(n

whe

re

n r =

n Cr =

n

rn

r!!(

)!_

Page 8: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

a x =

log

logb bx a

ex ln

a = a

x

Firs

t Prin

cipl

es

f =

lim

fx

hfx

h(

)()

_+

Page 9: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

STAT

ISTI

CS

Prob

abili

tyP(

A_

Stan

dard

dev

iatio

nSt

anda

rd d

evia

tion

= = IQ

R =

Q3

_ QFo

r a s

et o

f n va

lues

x, x

, . . .

xi, .

. . x

n

S xx =

(x

i _ =

x i _

Stan

dard

dev

iatio

n =

S nxx o

r (x i

x n _ x

n

Page 10: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Stat

istic

al ta

bles

Th

to

Ran

dom

Num

bers

(to

MEC

HA

NIC

S

v =

u +

at

s =

ut +

1 2at

s =

vt _ 1 2

at

v =

u +

s = 1 2

(u +

Page 11: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Men

sura

tion

Surf

ace

area

of s

pher

e =

4r

Are

a of

cur

ved

surf

ace

of c

one

= r ×

Arit

hmet

ic s

erie

s

S n = 1 2

n(a

+ =

1 2 +

(n _

Page 12: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Bin

omia

l ser

ies

(a +

n = a

n + n 1

an _

b + n 2

an _

b +

. . . +

n r

an _ r br +

. . . +

bn

(n

whe

re

n r =

n Cr =

n

rn

r!!(

)!

+

n =

+ n

x + nn(

)112

x +

. . . +

nn

nr r

()...(

)...

11

12

xr +

. . .

( x

<

, n

a x =

log

logb bx a

ex ln

a = a

x

_

×__

+. .

. . . .

_ ××

×

Page 13: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Geo

met

ric s

erie

s

S n = a

r rn(

)1 1

S =

ar

1 fo

r r

<

sin(

A ±

= s

in A

cos

B ±

cos

A s

in B

cos(

A ±

= c

os A

cos

B +

sin

A s

in B

tan(

A ±

= tan

tan

tan

tan

AB

AB

1 (

A ±

B

(k +

1 2

sin

A +

sin

B =

si

n A

B2

cos

AB

2

sin

A _ s

in B

=

cos A

B2

sin

AB

2

__ _

+ +__

Page 14: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

cos A

+ c

os B

=

cosA

B2

cos

AB

2

cos A

_ cos

B =

_ si

nA

B2

sin

AB

2

sin

co

s

_

tan

whe

re

is m

easu

red

in ra

dian

s

+_

+_

Page 15: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Firs

t Prin

cipl

es

f =

lim

fx

hfx

h(

)()

tan

kx

k s

ec k

x

sec k

x

k sec

kx ta

n kx

cot k

x

_ k cos

ec k

x

cose

c kx

_ k c

osec

kx c

ot kx

fx

gx()

()

fxgx

fxgx

gx

()()

()()

(())2

_

_

+

Page 16: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

(+

sec

kx

1 k

tan

kx

tan

kx

1 k

ln s

ec kx

cot k

x

1 k

ln s

in kx

cose

c kx

_ 1 k

ln c

osec

kx +

cot

kx ,

1 k

ln t

an(1 2

sec k

x

1 k

ln s

ec kx

+ ta

n kx

, 1 k

ln t

an(1 2

kx +

1 4

udv dx

dx =

uv

_ vdu dx

dx

Page 17: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Num

eric

al M

etho

ds

y d

x 1 2

h{(y

+ y

n+

+ y

+ . .

. + y

n_, w

here

h =

ba

n

The

New

ton

=

x

= x

n _ fx

fxn n

()

()

b a

_

Page 18: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

STAT

ISTI

CS

Prob

abili

tyP(

A_

P(A

= +

_ P

(A

P(A

=

=

PBAPA

PBAPA

PBAPA

()()

()()

()()

For i

ndep

ende

nt e

vent

s A

and

B,

=

=

P(A

=

+

Page 19: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Stan

dard

dev

iatio

n

Stan

dard

dev

iatio

n = =

IQR

= Q

3 _ Q

For a

set

of n

valu

es x

, x, .

. . x

i, . . .

xn

S xx =

(x

i _ =

x i _

Stan

dard

dev

iatio

n =

S nxx o

r

(x i

n

_ xx n

Page 20: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

utio

ns

For a

rand

om s

ampl

e of

n ob

serv

atio

ns fr

om

Xn

~,

XB

inom

ial B

px

_n _ x

Mea

nnp

Varia

nce

_

_

n x

Page 21: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Stat

istic

al ta

bles

to

to

Ran

dom

Num

bers

(to

Page 22: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

MEC

HA

NIC

S

v =

u +

at

s =

ut +

1 2at

s =

vt _ 1 2

at

v =

u +

s = 1 2

(u +

Page 23: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

3 A

S Fu

rthe

r Mat

hem

atic

s

form

ulae

list

ed fo

r

Sum

mat

ions

n

r =r

= 1 6

n(n

+ +

n

r =r3 =

1 4n

(n +

Page 24: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Mat

rix tr

ansf

orm

atio

ns a

bout

cos

sin

sin

cos

y =

(tan

Are

a of

a s

ecto

r

A =

1 2r

d

Com

plex

num

bers

+ i

n = rn (c

os

+ i s

in

The

root

s of

n =

=

e2ki n

, for

k =

,

, , .

. ., n

_

_cos

sin

sin

cos

22

22_

Page 25: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

=

+ x

f +

x2 2!

f +

. . . +

x rr !f

+ . .

.

ex =

=

+ x

+ x2 2!

+ . .

. + x rr ! +

. . .

fo

r all x

ln(

+

= x

_ x2 2

+ x3 3

_ . . . +

(_r

x rr

+ . .

.

(_ < x

sin

x =

x _ x

3 3! +

x5 5!

_ . . . +

(_r x rr21

21

()!

+ . .

.

for a

ll x

cos x

=

_ x2 2!

+ x4 4!

_ . . . +

(_r x rr2 2()!

+ . .

.

for a

ll x

arct

an x

= x

_ x3 3

+ x5 5

_ . . . +

(_r xrr2

1

21 +

. . .

(_

x

+

++

+

Page 26: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Vect

ors

a ×

b =

a

bsi

n n

=

=

a.( b

× c

=

a b c

a b c

a 3 b 3 c 3

= b

.( c ×

a =

c.( a

× b

If A

is th

e po

int w

ith p

ositi

on v

ecto

r a =

ai +

aj +

a3k

and

the

dire

ctio

n ve

ctor

b

b =

bi +

bj +

b3k

A w

ith d

irect

ion

vect

or b

has

car

tesi

an e

quat

ion

xa

b1

1 =

ya

b2

2 =

za

b3

3(=

i a b

j a b

k a 3 b 3___

__

_

ab 3

a 3b ab

a 3b ab 3

ab

Page 27: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

A w

ith n

orm

al v

ecto

r n =

ni +

nj +

n3k

has

car

tesi

an e

quat

ion

nx

+ n

y +

n 3 +

d =

w

here

d =

_ a.n

colli

near

poi

nts

A, B

and

C h

as v

ecto

r equ

atio

n

r = a

+

( b _ a

+

c _ a

=

_ _

a +

b +

c

a an

d pa

ralle

l to

b an

d c

has

equa

tion

r = a

+ s

b +

t c

The

perp

endi

cula

r dis

tanc

e of

(,

, fr

om n

x +

ny

+ n 3

+ d

=

is

nn

nd

nn

n1

23

22

2+

++

++

3

Page 28: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Hyp

erbo

lic fu

nctio

ns

cosh

x _ s

inh

x =

sinh

x

= si

nh x

cos

h x

cosh

=

cos

h x

+ s

inh

x

arco

sh x

= ln

{x +

x2

1

(x

arsi

nh x

= ln

{x +

x2

1

arta

nh x

= 1 2

ln 1 1

x x

(x

<

_ +

_+

Page 29: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

arcs

in x

1

12 x

arcc

os x

_ 1

12 x

arct

an x

1

12 x

sinh

x

co

sh x

cosh

x

si

nh x

tanh

x

se

ch x

arsi

nh x

1

12 x_

+ +_

Page 30: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

arco

sh x

11

2 x

arta

nh x

1

12 x_

_

Page 31: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

(+

cons

tant

; a >

d

xsi

nh x

cosh

x

cosh

x

si

nh x

tanh

x

ln

cosh

x

1 22

ax

ar

csin

x a

( x

12

2a

x

1 a

arct

an x a

1 22

xa

ar

cosh

x a

, ln

{x +

x

_ a

( x >

1 22

ax

ar

sinh

x a

, ln

{x +

x

+ a

_ + _ +

Page 32: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

+ co

nsta

nt; a

>

dx

12

2a

x

1 2aln

ax

ax

= 1 a

arta

nh x a

(

x <

12

2x

a

1 2aln

xa

xa

__+

_+_

Page 33: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

STAT

ISTI

CS

For a

dis

cret

e ra

ndom

var

iabl

e X

x i with

pro

babi

litie

s P(

X =

x i =

=

x i P

(X =

x i =

=

(x

i _ P

(X =

xi =

x

P(X

= x

i _

XM

ean

Varia

nce

Bin

omia

l B(n

,

px

_ n _ x

np _

Pois

son

Po(

e_

i

x!x

n x

Page 34: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Con

tinuo

us d

istr

ibut

ions

For a

con

tinuo

us ra

ndom

var

iabl

e X

f =

=

x

Var

=

=

(x _

=

x

d

x _

For a

func

tion

=

Cum

ulat

ive

dist

ribut

iF(

x =

P(X

x

=

d

t

XM

ean

Varia

nce

Nor

mal

(a

+

(b _

1 21 12

x

1 2

e_ 1 2 1b

a_

_

_

x

Page 35: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

For a

set

of n

pairs

of v

alue

s (x

i, y i

S xx =

(x

i _ x =

x i

_ ()

x i2

S yy =

(y

i _ y

=

y i _ (

)y ni2

S xy =

(x

i _ x

i _ y

x i yi _ (

)()

xy

nii

n

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Cor

rela

tion

and

rco

nti

r =S SSxy xxyy

=(

)()

()

()

xy

xy

ii

ii

22

=

xy

xy

n

xx n

yy n

ii

ii

ii

()(

)

()

()

22

22

{

{ {

{

_y_

ii

__ x

x

y

__

_

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y on

x is b

=S Sxy xx

=(

)()

()

xy

xi

i

i2

y on

x is

y =

a +

bx

whe

re a

= y

_ bx

= S yy

_ SSxy xx

= S yy

(_ r

r s =

_ 6

12

2d

nn(

)_

x__ y

_ x

Page 38: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Non

para

met

ric te

sts

Goo

dnes

sof

()

OE

Eii

i

2 ~

Stat

istic

al ta

bles

to to

to to

Ran

dom

Num

bers

(to

_v

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MEC

HA

NIC

S

Cen

tres

of m

ass 2 3

Circ

ular

arc

, rad

ius r

rsin

from

cen

tre

Sect

or o

f circ

le, r

adiu

s r 23rsin

from

cen

tre

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4 A

Leve

l Fur

ther

Mat

hem

atic

s

Sum

mat

ions

n

r =r

= 1 6

n(n

+ +

n

r =r3 =

1 4n

(n +

Page 41: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Mat

rix tr

ansf

orm

atio

ns a

bout

y =

(tan

cos

sin

sin

cos

22

22

Are

a of

a s

ecto

r

A =

1 2r

d

Com

plex

num

bers

+ i

n = rn (c

os

+ i s

in

The

root

s of

n =

=

e2ki n

, for

k =

,

, , .

. . ,

n_

_

_

cos

sin

sin

cos

Page 42: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

=

+ x

f +

x2 2!

f +

. . . +

x rr !f +

. . .

ex =

=

+ x

+ x2 2!

+ . .

. +x rr !

+ . .

.

for a

ll x

ln(

+

= x

_ x2 2

+ x3 3

_ . . . +

(_r

x rr

+. .

.(_ <

x

sin

x =

x_ x

3 3! +

x5 5!

_ . . . +

(_r x rr21

21

()!

+ . .

.

for a

ll x

cos x

=

_ x2 2!

+ x4 4!

_ . . . +

(_r x rr2 2()!

+ . .

.

for a

ll x

arct

an x

= x

_ x3 3

+ x5 5

_ . . . +

(_r xrr2

1

21 +

. . .

(_

x

++

++

Page 43: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Vect

ors

a ×

b =

a

bsi

n n

=

=

a.( b

× c

=

a b c

a b c

a 3 b 3 c 3

= b

.( c ×

a =

c.( a

× b

If A

is th

e po

int w

ith p

ositi

on v

ecto

r a =

ai +

aj +

a3k

and

the

dire

ctio

n ve

ctor

b

b =

bi +

bj +

b3k

A w

ith d

irect

ion

vect

or b

has

car

tesi

an e

quat

ion

xa

b1

1 =

ya

b2

2 =

za

b3

3(=

__

_

i a b

j a b

k a 3 b 3

ab 3

a 3b ab

a 3b ab 3

ab

___

Page 44: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

A w

ith n

orm

al v

ecto

r n =

ni +

nj +

n3k

has

car

tesi

an e

quat

ion

nx

+ n

y +

n 3 +

d =

w

here

d =

_ a.n

colli

near

poi

nts

A, B

and

C h

as v

ecto

r equ

atio

n

r = a

+

( b_ a

+

c _ a

=

_ _

a +

b +

c

a an

d pa

ralle

l to

b an

d c

has

equa

tion

r = a

+ s

b +

t c

The

perp

endi

cula

r dis

tanc

e of

(,

, fr

om n

x +

ny

+ n 3

+ d

=

is

nn

nd

nn

n1

23

22

2+

++

++

3

Page 45: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Hyp

erbo

lic fu

nctio

ns

cosh

x_ s

inh

x =

sinh

x

= si

nh x

cos

h x

cosh

=

cos

h x

+ s

inh

x

arco

sh x

= ln

{x +

x2

1

(x

arsi

nh x

= ln

{x +

x2

1

arta

nh x

= 1 2

ln1 1

x x

(x

<

_ +

+ _

Page 46: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Con

ics

Ellip

sePa

rabo

laH

yper

bola

Hyp

erbo

la

Stan

dard

Fo

rmy

= 4

axxy

= c

Para

met

ric

Form

(a co

s si

n (a

t

(a se

c ta

n

(± a c

osh

sinh

Ecce

ntric

itye

<

b =

a _ e

b

= a

(e _

e =

Foci

(±c,

±

x =

±x

= _ a

x =

±x

+ y

= ±

c

Asy

mpt

otes

none

none

= ±

a ea e

x a2 2y b2 2

+=

= x a2 2

y b2 2 y bx a

_

ctc t,

Page 47: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

arcs

in x

1

12 x

arcc

os x

_ 1

12 x

arct

an x

11

2 x

sinh

x

cosh

xco

sh x

si

nh x

tanh

x

sech

x

arsi

nh x

11

2 x_ _

++

Page 48: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

arco

sh x

11

2 x

arta

nh x

1

12 x_

_

Page 49: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

(+

cons

tant

; a >

dx

sinh

x

cosh

x

cosh

x

sinh

x

tanh

x

ln co

sh x

1 22

ax

arcs

in x a

(

x <

12

2a

x1 a

arct

anx a

1 22

xa

arco

sh x a

, ln

{x +

x

_ a

( x >

1 22

ax

arsi

nh x a

, ln

{x +

x

+ a

_ + _ +

Page 50: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

+ co

nsta

nt; a

>

d

x

12

2a

x1 2a

lna

xa

x =

1 aar

tanh

x a

( x

<

12

2x

a1 2a

lnx

ax

a

__+

_+_

Page 51: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

s =

dy dx d

x

s =

dx dt

dy dt2

2dt

s =

rdr d

22

d

+

+

+

Page 52: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Surf

ace

area

of r

evol

utio

n

s x =

ydy dx

2dx

s x =

y dx dt

dy dt2

2

s x =

r sin

r

dr d2

2d

+

dt+ +

Page 53: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

STAT

ISTI

CS

For a

dis

cret

e ra

ndom

var

iabl

e X

x i with

pro

babi

litie

s P(

X =

x i =

=

x i P

(X =

x i

=

= (x

i_ P

(X =

x i =

x P

(X =

x i _

For a

func

tion

=

i P(X

= x i

X is

GX

= E

(tX a

nd

= G

a

nd

= G

+ G

_ [G

For Z

= X

+ Y,

whe

re X

and

YG

Z=

GX

× G

Y

i

XX

XX

Page 54: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

XM

ean

Varia

nce

P.G

.F.

Bin

omia

l B(n

, px

_ n _ x

np _

_ p +

n

Pois

son

Po(

e_

e(t

_

Geo

met

ricon

. .

._

x _

r, r +

. . .

pr

_x _ r

x!x

1 p12pp_

__ _

r

ptpt

11(

)_

__

_r p

rp

p()

12

x r1 1

ptpt

11(

)

n x

Page 55: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Con

tinuo

us d

istr

ibut

ions

For a

con

tinuo

us ra

ndom

var

iabl

e X

f =

=

x

Var

=

=

(x _

=

x

d

x _

For a

func

tion

=

Cum

ulat

ive

dist

ribut

iF(

x =

P(X

x

=

dt

XM

ean

Varia

nce

Nor

mal

e

[a

(a +

(b

_

_x

1b

a_1 2

1 12

1 2

__ 1 2

x

Page 56: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

For a

set

of n

pairs

of v

alue

s (x

i, y i

S xx =

(x

i _ x

=

x i _ (

)x i2

S yy =

(y

i _ y

=

y i _ (

)y ni2

S xy =

(x

i _ x

i _ y

x i yi _ (

)()

xy

nii

n

Page 57: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

The

prod

uct m

omen

t cor

rela

tioci

ent i

s

r =S SSxy xxyy

=(

)()

()

()

xy

xy

ii

ii

22

=

xy

xy

n

xx n

yy n

ii

ii

ii

()(

)

()

()

22

22

{

{ {

{

_y_

_i

i

xy

x _

_

__

Page 58: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

y on

x is b

= S Sxy xx

= (

)()

()

xy

xi

i

i2

y on

x is

y =

a +

bx

whe

re a

= y

_ bx

= S yy

_ SSxy xx

= S

yy (

_ r

r s =

_ 6

12

2d

nn(

)__

__

xx

y

Page 59: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

For i

ndep

ende

nt ra

ndom

var

iabl

es X

and

Y

, V

ar(a

X ±

= a

+

b

Page 60: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

is

kno

wn

For a

rand

om s

ampl

e X

, X, .

. ., X

n of n

inde

pend

ent o

bser

vatio

ns fr

om a

mea

n an

d va

rianc

e

X

is a

n un

bias

ed e

stim

ator

of

, with

Var

(X =

2 n

S

is a

n un

bias

ed e

stim

ator

of

, whe

re S

= (

)X

Xni

2

1

For a

rand

om s

ampl

e of

n ob

serv

atio

ns fr

om

, ,

n ~

__ X _

Page 61: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

For a

rand

om s

ampl

e of

nx o

bser

vatio

ns fr

om N

x, a

nd, i

ndep

ende

ntly

, a

rand

om s

ampl

e of

ny o

bser

vatio

ns fr

om

y,

(

)(

)X

~

is

not

kno

wn

For a

rand

om s

ampl

e of

n ob

serv

atio

ns fr

om

()

nS

12

2 ~

X S

n ~

t n _

(a

lso

valid

in m

atch

ed

x_

y

n__ _

Yx

_ y

+x n x

n y

y

_

Page 62: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

For a

rand

om s

ampl

e of

nx o

bser

vatio

ns fr

om N

x

and,

inde

pend

ently

, a

rand

om s

ampl

e of

ny o

bser

vatio

ns fr

om N

y,

If

=

=

(un

(

)(

)

Sn

nx

y2

11

~

whe

re

S = (

)(

)n

Sn

Sn

nx

xy

y

xy

11

2

x

p

y

y~

F n x_

y_

xy

+p

__

+

S x S y

x

X _ Y

x _

y+

t n x+n

y__

_

Page 63: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Non

para

met

ric te

sts

Goo

dnes

sof

()

OE

Eii

i

2 ~

Stat

istic

al ta

bles

to to

to to

Ran

dom

Num

bers

(to t

to

Fto

_v

Page 64: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

MEC

HA

NIC

S

Cen

tres

of m

ass 2 3

Circ

ular

arc

, rad

ius r

rsin

from

cen

tre

Sect

or o

f circ

le, r

adiu

s r 23rsin

from

cen

tre

Solid

hem

isph

ere,

radi

us

3 8r f

rom

cen

tre

Hem

isph

eric

al s

hell,

radi

us

1 2r f

rom

cen

tre

1 4h

abov

e th

e ba

se o

n th

e lin

e fr

om

cent

re o

f bas

e to

ver

tex

C1 3

h ab

ove

the

base

on

the

line

from

cen

tre

of b

ase

to

vert

ex

Page 65: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

Mot

ion

in a

circ

lev =

r v =

r _ r

= _ v

r2

• •

Page 66: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

5 St

atis

tical

Tab

les

The

tabu

late

d va

lue

is

whe

re X

has

a b

inom

ial d

istr

ibut

ion

with

inde

x n an

d

para

met

er p

n =

5

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

0

1 2 3 4

•0•

1681

11

60

n

= 5, x

= 0

0•05

03

0•

0778

0•

0312

0 • 42

84

0 • 76

48

0 • 94

60

0 • 99

47

0 • 5

282

0 • 8

369

0 • 9

692

0 • 9

976

0 • 33

70

0 • 68

26

0 • 91

30

0 • 98

98

0 •

2562

0 •

5931

0 •

8688

0 •

9815

0 • 18

75

0 • 50

00

0 • 81

25

0 • 96

88

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

1 2 3

n

= 5, x

= 0 4

0 • 7

738

0 • 59

05

0

• 443

7

0 •

3277

0 • 2

373

0 • 9

774

0 • 9

988

1 • 0

000

1 • 0

000

0 • 91

85

0 • 99

14

0 • 99

95

1 • 00

00

0 •

8352

0 •

9734

0 •

9978

0 •

9999

0 • 73

73

0 • 94

21

0 • 99

33

0 • 99

97

0 •

6328

0 •

8965

0 •

9844

0 •

9990

Page 67: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n

= 6

p =

0

• 05

0

• 10

0 •

15

0 •

20

0 • 2

5

n

= 6, x

= 0 1 2 3 4 5

0 • 7

351

0 • 53

14

0

• 377

1

0 •

2621

0 • 1

780

0 • 9

672

0 • 9

978

0 • 9

999

1 • 0

000

1 • 0

000

0 • 88

57

0 • 98

42

0 • 99

87

0 • 99

99

1 • 00

00

0 •

7765

0 •

9527

0 •

9941

0 •

9996

1 •

0000

0 • 65

54

0 • 90

11

0 • 98

30

0 • 99

84

0 • 99

99

p =

0 • 53

39

0 • 8

306

0 • 96

24

0 • 99

54

0 • 99

98

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

n

= 6, x

= 0 1 2 3 4 5

0 • 1

176

0 • 07

54

0

• 046

7

0 •

0277

0 • 0

156

0 • 4

202

0 • 7

443

0 • 9

295

0 • 9

891

0 • 9

993

0 • 31

91

0 • 64

71

0 • 88

26

0 • 97

77

0 • 99

82

0 •

2333

0 •

5443

0 •

8208

0 •

9590

0 •

9959

0 • 16

36

0 • 44

15

0 • 74

47

0 • 93

08

0 • 99

17

0 •

1094

0 •

3438

0 •

6563

0 •

8906

0 •

9844

Page 68: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n =

7p

=

0 • 0

5

0 • 1

0

0 • 15

0

• 20

0

• 25

n

= 7, x

= 0 1 2 3 4 5 6

0 • 6

983

0 • 47

83

0

• 320

6

0 •

2097

0 • 1

335

0 • 9

556

0 • 9

962

0 • 9

998

1 • 0

000

1 • 0

000

0 • 85

03

0 • 97

43

0 • 99

73

0 • 99

98

1 • 00

00

0 •

7166

0 •

9262

0 •

9879

0 •

9988

0 •

9999

0 • 57

67

0 • 85

20

0 • 96

67

0 • 99

53

0 • 99

96

1 • 0

000

0 • 44

49

0 • 75

64

0 • 92

94

0 • 98

71

0 • 99

87

1 • 00

00

1

• 000

0

1 •

0000

0 • 9

999

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 50

n = 7

, x =

0 1 2 3 4 5 6

0 • 0

824

0 • 04

90

0

• 028

0

0 •

0152

0 • 0

078

0 • 3

294

0 • 6

471

0 • 8

740

0 • 9

712

0 • 9

962

0 • 23

38

0 • 53

23

0 • 80

02

0 • 94

44

0 • 99

10

0 •

1586

0 •

4199

0 •

7102

0 •

9037

0 •

9812

0 • 10

24

0 • 31

64

0 • 60

83

0 • 84

71

0 • 96

43

0 • 9

998

0 • 06

25

0 • 22

66

0 • 50

00

0 • 77

34

0 • 93

75

0 • 99

94

0

• 998

4

0 •

9963

0 • 9

922

Page 69: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n

= 8

p =

0

• 05

0 •

10

0 •

15

0 • 2

0

0 • 2

5

n = 8

, x =

0 1 2 3 4 5 6 7

0 • 6

634

0 • 43

05

0

• 272

5

0 •

1678

0 • 1

001

0 • 9

428

0 • 9

942

0 • 9

996

1 • 0

000

1 • 0

000

0 • 81

31

0 • 96

19

0 • 99

50

0 • 99

96

1 • 00

00

0 •

6572

0 •

8948

0 •

9786

0 •

9971

0 •

9998

0 • 50

33

0 • 79

69

0 • 94

37

0 • 98

96

0 • 99

88

0 •

3671

0 •

6785

0 •

8862

0 •

9727

0 •

9958

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

0 • 99

99

1 • 00

00

0 •

9996

1 •

0000

Page 70: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0 •

35

0 •

40

0 • 4

5

0 • 5

0

1 2 3

n

= 8, x

= 0 4 5 6 7

0 • 0

576

0 • 03

19

0

• 016

8

0 •

0084

0 • 0

039

0 • 2

553

0 • 5

518

0 • 8

059

0 • 9

420

0 • 9

887

0 • 16

91

0 • 42

78

0 • 70

64

0 • 89

39

0 • 97

47

0 •

1064

0 •

3154

0 •

5941

0 •

8263

0 •

9502

0 • 06

32

0 • 22

01

0 • 47

70

0 • 73

96

0 • 91

15

0 •

0352

0 • 14

450 •

3633

0 • 63

670 •

8555

0 • 9

987

0 • 9

999

0 • 99

64

0 • 99

98

0 •

9915

0 •

9993

0 • 98

19

0 • 99

83

0 •

9648

0 • 99

61

n =

8 (c

ontin

ued)

Page 71: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

n = 9

, x =

0 1 2 3 4 5

6 7 8

0 • 6

302

0 • 38

74

0

• 231

6

0 •

1342

0 • 0

751

0 • 9

288

0 • 9

916

0 • 9

994

1 • 0

000

1 • 0

000

0 • 77

48

0 • 94

70

0 • 99

17

0 • 99

91

0 • 99

99

0 •

5995

0 •

8591

0 •

9661

0 •

9944

0 •

9994

0 • 43

62

0 • 73

82

0 • 91

44

0 • 98

04

0 • 99

69

0 •

3003

0 •

6007

0 •

8343

0 •

9511

0 •

9900

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

0 • 99

97

1 • 00

00

1 • 00

00

n

= 9

0 • 99

87

0 • 99

99

1 • 00

00

Page 72: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

n = 9

, x =

0 1 2 3 4 5 6 7 8

0 • 0

404

0 • 02

07

0

• 010

1

0 •

0046

0 • 0

020

0 • 1

960

0 • 4

628

0 • 7

297

0 • 9

012

0 • 9

747

0 • 12

11

0 • 33

73

0 • 60

89

0 • 82

83

0 • 94

64

0 •

0705

0 •

2318

0 •

4826

0 •

7334

0 •

9006

0 • 03

85

0 • 14

95

0 • 36

14

0 • 62

14

0 • 83

42

0 •

0195

0 •

0898

0 •

2539

0 •

5000

0 •

7461

0 • 9

957

0 • 9

996

1 • 0

000

0 • 98

88

0 • 99

86

0 • 99

99

0 •

9750

0 •

9962

0 •

9997

0 • 95

02

0 • 99

09

0 • 99

92

0 •

9102

0 • 98

050 •

9980

n =

9 (c

ontin

ued)

Page 73: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n =

10

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5 n

= 10,

x =

0 1 2 3 4 5 6 7 8 9

0 • 5

987

0 • 34

87

0

• 196

9

0 •

1074

0 • 0

563

0 • 9

139

0 • 9

885

0 • 9

990

0 • 9

999

1 • 0

000

0 • 73

61

0 • 92

98

0 • 98

72

0 • 99

84

0 • 99

99

0 •

5443

0 •

8202

0 •

9500

0 •

9901

0 •

9986

0 • 37

58

0 • 67

78

0 • 87

91

0 • 96

72

0 • 99

36

0 •

2440

0 •

5256

0 •

7759

0 •

9219

0 •

9803

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9999

1 •

0000

1 •

0000

1 •

0000

0 • 99

91

0 • 99

99

1 • 00

00

1 • 00

00

0 •

9965

0 •

9996

1 •

0000

1 •

0000

Page 74: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0 n

= 10,

x =

0 1 2 3 4 5 6 7 8 9

0 • 0

282

0 • 01

35

0

• 006

0

0 •

0025

0 • 0

010

0 • 1

493

0 • 3

828

0 • 6

496

0 • 8

497

0 • 9

527

0 • 08

60

0 • 26

16

0 • 51

38

0 • 75

15

0 • 90

51

0 •

0464

0 •

1673

0 •

3823

0 •

6331

0 •

8338

0 • 02

33

0 • 09

96

0 • 26

60

0 • 50

44

0 • 73

84

0 •

0107

0 •

0547

0 •

1719

0 •

3770

0 •

6230

0 • 9

894

0 • 9

984

0 • 9

999

1 • 0

000

0 • 97

40

0 • 99

52

0 • 99

95

1 • 00

00

0 •

9452

0 •

9877

0 •

9983

0 •

9999

0 • 89

80

0 • 97

26

0 • 99

55

0 • 99

97

n

= 10

(con

tinue

d)

0 • 82

810 •

9453

0 • 98

930 •

9990

Page 75: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n

= 1

2p

=

0 • 0

5

0 • 1

0

0 • 15

0

• 20

0

• 25

n = 1

2, x

= 0 1 2 3 4 5 6 7 8 9 10 11

0 • 5

404

0 • 28

24

0

• 142

2

0 •

0687

0 • 0

317

0 • 8

816

0 • 9

804

0 • 9

978

0 • 9

998

1 • 0

000

0 • 65

90

0 • 88

91

0 • 97

44

0 • 99

57

0 • 99

95

0 •

4435

0 •

7358

0 •

9078

0 •

9761

0 •

9954

0 • 27

49

0 • 55

83

0 • 79

46

0 • 92

74

0 • 98

06

0 •

1584

0 • 39

070 •

6488

0 • 84

240 •

9456

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

99

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9993

0 •

9999

1 •

0000

1 •

0000

1 •

0000

0 • 99

61

0 • 99

94

0 • 99

99

1 • 00

00

1 • 00

00

0 •

9857

0 • 99

720 •

9996

1 • 00

001 •

0000

1 • 0

000

1 • 00

00

1

• 000

0

1 •

0000

1 • 0

000

Page 76: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n

= 12

(con

tinue

d)

p =

0

• 30

0

• 35

0

• 40

0

• 45

0

• 50

n = 1

2, x

= 0 1 2 3 4 5 6 7 8 9 10 11

0 • 0

138

0 • 00

57

0

• 002

2

0 •

0008

0 • 0

002

0 • 0

850

0 • 2

528

0 • 4

925

0 • 7

237

0 • 8

822

0 • 04

24

0 • 15

13

0 • 34

67

0 • 58

33

0 • 78

73

0 •

0196

0 •

0834

0 •

2253

0 •

4382

0 •

6652

0 • 00

83

0 • 04

21

0 • 13

45

0 • 30

44

0 • 52

69

0 •

0032

0 •

0193

0 •

0730

0 •

1938

0 •

3872

0 • 9

614

0 • 9

905

0 • 9

983

0 • 9

998

1 • 0

000

0 • 91

54

0 • 97

45

0 • 99

44

0 • 99

92

0 • 99

99

0 •

8418

0 •

9427

0 •

9847

0 •

9972

0 •

9997

0 • 73

93

0 • 88

83

0 • 96

44

0 • 99

21

0 • 99

89

1 • 0

000

0 • 61

280 •

8062

0 • 92

700 •

9807

0 • 99

68

1 • 00

00

1

• 000

0

0 •

9999

0 • 9

998

Page 77: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n

= 1

5p

=

0 • 0

5

0 • 1

0

0 • 15

0

• 20

0

• 25

n = 1

5, x

= 0 1 2 3 4 5 6 7 8 9 10 11 12

13 14

0 • 4

633

0 • 20

59

0

• 087

4

0 •

0352

0 • 0

134

0 • 8

290

0 • 9

638

0 • 9

945

0 • 9

994

0 • 9

999

0 • 54

90

0 • 81

59

0 • 94

44

0 • 98

73

0 • 99

78

0 •

3186

0 •

6042

0 •

8227

0 •

9383

0 •

9832

0 • 16

71

0 • 39

80

0 • 64

82

0 • 83

58

0 • 93

89

0 •

0802

0 •

2361

0 •

4613

0 •

6865

0 •

8516

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

97

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9964

0 •

9994

0 •

9999

1 •

0000

1 •

0000

0 • 98

19

0 • 99

58

0 • 99

92

0 • 99

99

1 • 00

00

0 •

9434

0 •

9827

0 •

9958

0 •

9992

0 •

9999

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 • 00

001 •

0000

Page 78: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0

• 40

0

• 45

0 •

50

n = 1

5, x

= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0 • 0

047

0 • 00

16

00 •

0005

•0

0001

•0

000

0•0

017

0•00

05

0•

001

07

•003

7 0•

004

24

•017

6

0

•0

1204

•0

592

0 • 0

353

0 • 1

268

0 • 2

969

0 • 5

155

0 • 7

216

0 • 01

42

0 • 06

17

0 • 17

27

0 • 35

19

0 • 56

43

0 • 26

08

0 • 00

520 •

0271

0 • 09

05

0 • 21

730 •

4032

0 •

1509

0 • 8

689

0 • 9

500

0 • 9

848

0 • 9

963

0 • 9

993

0 • 75

48

0 • 88

68

0 • 95

78

0 • 98

76

0 • 99

72

0 •

6098

0 •

7869

0 •

9050

0 •

9662

0 •

9907

0 • 45

22

0 • 65

35

0 • 81

82

0 • 92

31

0 • 97

45

0 •

3036

0 •

5000

0 •

6964

0 •

8491

0 •

9408

0 • 9

999

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

95

0 • 99

99

1 • 00

00

1 • 00

00

0 •

9981

0 •

9997

1 •

0000

1 •

0000

0 • 99

37

0 • 99

89

0 • 99

99

1 • 00

00

n

= 15

(con

tinue

d)

0 • 98

24

0 • 99

63

0 • 99

95

1 • 00

00

Page 79: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 •

20

0 • 2

5

n = 2

0, x

= 0 1 2 3 4 5 106 7 8 9

0 • 3

585

0 • 12

16

0

• 038

8

0 •

0115

0 • 0

032

0 • 7

358

0 • 9

245

0 • 9

841

0 • 9

974

0 • 9

997

0 • 39

17

0 • 67

69

0 • 86

70

0 • 95

68

0 • 98

87

0 •

1756

0 •

4049

0 •

6477

0 •

8298

0 •

9327

0 • 06

92

0 • 20

61

0 • 41

14

0 • 62

96

0 • 80

42

0 •

0243

0 •

0913

0 •

2252

0 •

4148

0 •

6172

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

76

0 • 99

96

0 • 99

99

1 • 00

00

1 • 00

00

0 •

9781

0 •

9941

0 •

9987

0 •

9998

1 •

0000

0 • 91

33

0 • 96

79

0 • 99

00

0 • 99

74

0 • 99

94

n

= 2

0

0 • 78

58

0 • 89

82

0 • 95

91

0 • 98

61

0 • 99

61

Page 80: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

11 12 13

n = 2

0, x

= 10 14 15 16 17 18

1 • 0

000

1 • 00

00

1

• 000

0

0 •

9994

0 • 9

961

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

0 • 99

99

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9991

0 •

9998

1 •

0000

1 •

0000

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

n

= 20

(con

tinue

d)

Page 81: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

n = 2

0, x

= 0 51 2 3 4 106 7 8 9

0 • 0

008

0 • 00

02

0•

0 • 00

000

0000

•000

0

0•0

001

0

•000

0

0

•0

0009

•000

2

0

•0

0049

•001

3

0

•0

0189

•005

9

0

•055

3

0 • 0

076

0 • 0

355

0 • 1

071

0 • 2

375

0 • 4

164

0•

0 • 00

21

0 • 01

21

0 • 04

44

0 • 11

82

0 •

2454

02

07

0 • 00

050 •

0036

0 • 01

60 0

• 051

00 •

1256

0•

012

99

•0

577

0•

025

20

•1

316

0•

041

43

•2

517

0

•0

5914

•411

9

0•

075

07

0 • 6

080

0 • 7

723

0 • 8

867

0 • 9

520

0 • 9

829

0 • 41

66

0 • 60

10

0 • 76

24

0 • 87

82

0 • 94

68

5881

0 • 25

000 •

4159

0 • 59

560 •

7553

0 • 87

25

n

= 20

(con

tinue

d)

Page 82: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

11 12

n = 2

0, x

= 10 13 14 15 16 17 18

0 • 9

829

0 • 94

68

0

• 872

5

0 •

7507

0 • 5

881

0 • 9

949

0 • 9

987

0 • 9

997

1 • 0

000

1 • 0

000

0 • 98

04

0 • 99

40

0 • 99

85

0 • 99

97

1 • 00

00

0 •

9435

0 •

9790

0 •

9935

0 •

9984

0 •

9997

0 • 86

92

0 • 94

20

0 • 97

86

0 • 99

36

0 • 99

85

0 •

7483

0 •

8684

0 •

9423

0 •

9793

0 •

9941

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

0 • 99

97

1 • 00

00

1 • 00

00

0 •

9987

0 •

9998

1 •

0000

n

= 20

(con

tinue

d)

Page 83: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

n = 2

5, x

= 0 1 2 3 4 5 106 7 8 9

0 • 2

774

0 • 07

18

0

• 017

2

0 •

0038

0 • 0

008

0 • 6

424

0 • 8

729

0 • 9

659

0 • 9

928

0 • 9

988

0 • 27

12

0 • 53

71

0 • 76

36

0 • 90

20

0 • 96

66

0 •

0931

0 •

2537

0 •

4711

0 •

6821

0 •

8385

0 • 02

74

0 • 09

82

0

• 234

0

0 •

4207

0 •

6167

0 • 00

70

0 • 03

21

0 • 09

62

0 • 21

37

0 • 37

83

0 • 9

998

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

05

0 • 99

77

0 • 99

95

0 • 99

99

1 • 00

00

0 •

9305

0 •

9745

0 •

9920

0 •

9979

0 •

9995

0 • 78

00

0 • 89

09

0 • 95

32

0 • 98

27

0 • 99

44

n

= 2

5

0 • 56

11

0 • 72

65

0 • 85

06

0 • 92

87

0 • 97

03

Page 84: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

11 12

n = 2

5, x

= 10 13 14 15 16 17 18 19 20 21 22

1 • 0

000

1 • 00

00

0

• 999

5

0 •

9944

0 • 9

703

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9999

1 •

0000

1 •

0000

1 •

0000

1 •

0000

0 • 99

85

0 • 99

96

0 • 99

99

1 • 00

00

1 • 00

00

0 •

9893

0 •

9966

0 •

9991

0 •

9998

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

n

= 25

(con

tinue

d)

Page 85: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0 n

= 25,

x =

0 1 2 3 4 5 106 7 8 9

0 • 0

001

0 • 00

00

0•

0 • 00

000

0000

•000

0

0•0

000

0

•000

0

0

•0

0001

•000

0

0

•0

0005

•000

1

0•

000

23

•0

005

0

•0

0086

0 • 0

016

0 • 0

090

0 • 0

332

0 • 0

905

0 • 1

935

0 • 00

03

0 • 00

21

0 • 00

97

0 • 03

20

0 • 08

26

0020

0 • 00

010 •

0004

0 • 00

240 •

0095

0 • 02

94

0•

002

58

•0

073

0•

006

39

•0

216

0•

013

40

•0

539

0

•0

2424

•114

8

0•

038

43

0 • 3

407

0 • 5

118

0 • 6

769

0 • 8

106

0 • 9

022

0 • 17

34

0 • 3

061

0 • 46

68

0 • 63

03

0 • 77

12

2122

0 • 07

360 •

1536

0 • 27

350 •

4246

0 • 58

58

n

= 25

(con

tinue

d)

Page 86: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0

• 40

0

• 45

0

• 50

11 12 13

n = 2

5, x

= 10 14 15 16 17 18 19 20 21 22

0 • 9

022

0 • 77

12

0

• 585

8

0 •

3843

0 • 2

122

0 • 9

558

0 • 9

825

0 • 9

940

0 • 9

982

0 • 9

995

0 • 87

46

0 • 93

96

0 • 97

45

0 • 99

07

0 • 99

71

0 •

7323

0 •

8462

0 •

9222

0 •

9656

0 •

9868

0 • 54

26

0 • 69

37

0 • 81

73

0 • 90

40

0 • 95

60

0 •

3450

0 •

5000

0 •

6550

0 •

7878

0 •

8852

0 • 9

999

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

92

0 • 99

98

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9957

0 •

9988

0 •

9997

0 •

9999

1 •

0000

0 • 98

26

0 • 99

42

0 • 99

84

0 • 99

96

0 • 99

99

0 •

9461

0 •

9784

0 •

9927

0 •

9980

0 •

9995

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

0 •

9999

1 •

0000

n

= 2

5 (c

ontin

ued)

Page 87: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

n = 3

0, x

= 0 1 2 3 4 5 106 7 8 9

0 • 2

146

0 • 04

24

0

• 007

6

0 •

0012

0 • 0

002

0 • 5

535

0 • 8

122

0 • 9

392

0 • 9

844

0 • 9

967

0 • 18

37

0 • 41

14

0 • 64

74

0 • 82

45

0 • 92

68

0 •

0480

0 •

1514

0 •

3217

0 •

5245

0 •

7106

0 • 01

05

0 • 04

42

0 • 12

27

0 • 25

52

0 • 42

75

0 •

0020

0 •

0106

0 •

0374

0 •

0979

0 •

2026

0 • 9

994

0 • 9

999

1 • 0

000

1 • 0

000

1 • 0

000

0 • 97

42

0 • 99

22

0 • 99

80

0 • 99

95

0 • 99

99

0 •

8474

0 •

9302

0 •

9722

0 •

9903

0 •

9971

0 • 60

70

0 • 76

08

0 • 87

13

0 • 93

89

0 • 97

44

n

= 3

0

0 • 34

81

0 • 51

43

0 • 67

36

0 • 80

34

0 • 89

43

Page 88: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

11 12 13

n = 3

0, x

= 10 14 15 16 17 18 19 20 21 22

23 24 25

1 • 0

000

0 • 99

99

0

• 997

1

0 •

9744

0 • 8

943

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9992

0 •

9998

1 •

0000

1 •

0000

1 •

0000

0 • 99

05

0 • 99

69

0 • 99

91

0 • 99

98

0 • 99

99

0 •

9493

0 •

9784

0 •

9918

0 •

9973

0 •

9992

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9998

0 •

9999

1 •

0000

1 •

0000

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

n

= 30

(con

tinue

d)

1 • 00

00

1 • 00

001 •

0000

1 •

0000

1 •

0000

Page 89: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

n = 3

0, x

= 0 1 2 3 4 5 106 7 8 9

0 • 0

000

0 • 00

00

00 •

0000

•0

0000

•0

000

0•0

000

0•00

00

0•

000

00

•000

0

0•0

0000

•0

000

0

•0

0002

•0

000

0 • 0

003

0 • 0

021

0 • 0

093

0 • 0

302

0 • 0

766

0 • 00

00

0 • 00

03

0 • 00

19

0 • 00

75

0 • 02

33

0 • 00

11

0 • 00

000 •

0000

0 • 00

03

0 • 00

150 •

0057

0 •

0002

0 • 1

595

0 • 2

814

0 • 4

315

0 • 5

888

0 • 7

304

0 • 05

86

0 • 12

38

0 • 22

47

0 • 35

75

0 • 50

78

0 •

0172

0 •

0435

0 •

0940

0 •

1763

0 •

2915

0 • 00

40

0 • 01

21

0 • 03

12

0 • 06

94

0 • 13

50

n

= 30

(con

tinue

d)

0 • 00

07

0 • 00

26

0 • 00

81

0 • 02

14

0 • 04

94

Page 90: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0

• 40

0

• 45

0

• 50

11 12 13

n = 3

0, x

= 10 14 15 16 17 18 19 20 21 22 23 24 25

0 • 7

304

0 • 50

78

0

• 291

5

0 •

1350

0 • 0

494

0 • 8

407

0 • 9

155

0 • 9

599

0 • 9

831

0 • 9

936

0 • 65

48

0 • 78

02

0 • 87

37

0 • 93

48

0 • 96

99

0 •

4311

0 •

5785

0 •

7145

0 •

8246

0 •

9029

0 • 23

27

0 • 35

92

0 • 50

25

0 • 64

48

0 • 76

91

0 •

1002

0 •

1808

0 •

2923

0 •

4278

0 •

5722

0 • 9

979

0 • 9

994

0 • 9

998

1 • 0

000

1 • 0

000

0 • 98

76

0 • 99

55

0 • 99

86

0 • 99

96

0 • 99

99

0 •

9519

0 •

9788

0 •

9917

0 •

9971

0 •

9991

0 • 86

44

0 • 92

86

0 • 96

66

0 • 98

62

0 • 99

50

0 •

7077

0 •

8192

0 •

8998

0 •

9506

0 •

9786

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9998

1 •

0000

1 •

0000

1 •

0000

1 •

0000

0 • 99

84

0 • 99

96

0 • 99

99

1 • 00

00

1 • 00

00

n

= 30

(con

tinue

d)

0 • 99

19

0 • 99

740 •

9993

0 •

9998

1 •

0000

Page 91: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

n

= 4

0p

=

0 • 0

5

0 • 1

0

0 • 15

0

• 20

0

• 25

n = 4

0, x

= 0 1 2 3 4 5 106 7 8 9 11 12 13 14 15

0 • 1

285

0 • 01

48

0

• 001

5

0 •

0001

0 • 0

000

0 • 3

991

0 • 6

767

0 • 8

619

0 • 9

520

0 • 9

861

0 • 08

05

0 • 22

28

0 • 42

31

0 • 62

90

0 • 79

37

0 •

0121

0 •

0486

0 •

1302

0 •

2633

0 •

4325

0 • 00

15

0 • 00

79

0 • 02

85

0 • 07

59

0 • 16

13

0 •

0001

0 •

0010

0 •

0047

0 •

0160

0 •

0433

0 • 9

966

0 • 9

993

0 • 9

999

1 • 0

000

1 • 0

000

0 • 90

05

0 • 95

81

0 • 98

45

0 • 99

49

0 • 99

85

0 •

6067

0 •

7559

0 •

8646

0 •

9328

0 •

9701

0 • 28

59

0 • 43

71

0 • 59

31

0 • 73

18

0 • 83

92

0 •

0962

0 •

1820

0 •

2998

0 •

4395

0 •

5839

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

96

0 • 99

99

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9880

0 •

9957

0 •

9986

0 •

9996

0 •

9999

0 • 91

25

0 • 95

68

0 • 98

06

0 • 99

21

0 • 99

71

0 •

7151

0 •

8209

0 •

8968

0 •

9456

0 •

9738

Page 92: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0

• 15

0

• 20

0

• 25

16 17

n = 4

0, x =

15 18 19 20 21 22 23 24 25 26

2

7 2

8

29

3

0

31

32

1 • 0

000

1 • 00

00

0

• 999

9

0 •

9971

0 • 9

738

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

0 • 99

90

0 • 99

97

0 • 99

99

1 • 00

00

1 • 00

00

0 •

9884

0 •

9953

0 •

9983

0 •

9994

0 •

9998

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

n

= 40

(con

tinue

d)

Page 93: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

n =

40, x

= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 • 0

000

0 • 00

00

00 •

0000

•0

0000

•0

000

0•0

000

0•00

00

0•

000

00

•000

0

0 •0

0000

•0

000

0

•0

0000

•0

000

0 • 0

000

0 • 0

001

0 • 0

006

0 • 0

026

0 • 0

086

0 • 00

00

0 • 00

00

0 • 00

00

0 • 00

01

0 • 00

03

0 • 00

13

0 •

0000

0 • 00

000 •

0000

0 •

0000

0 • 00

01

0 • 00

00

0 • 0

238

0 • 0

553

0 • 1

110

0 • 1

959

0 • 3

087

0 • 00

44

0 • 01

24

0 • 0

303

0 • 06

44

0 • 12

15

0 •

0006

0 •

0021

0 •

0061

0 •

0156

0 •

0352

0 • 00

01

0 • 00

02

0 • 00

09

0 • 00

27

0 • 00

74

0 •

0000

0 •

0000

0 •

0001

0 •

0003

0 •

0011

0 • 4

406

0 • 5

772

0 • 7

032

0 • 8

074

0 • 8

849

0 • 20

53

0 • 31

43

0 • 44

08

0 • 57

21

0 • 69

46

0 •

0709

0 •

1285

0 •

2112

0 •

3174

0 •

4402

n =

40

(con

tinue

d)

0 • 01

79

0 • 03

86

0 • 0

751

0 • 13

26

0 • 21

42

0 •

0032

0 •

0083

0 •

0192

0 •

0403

0 •

0769

Page 94: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

16 17 18

n = 4

0, x

= 15 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0 • 8

849

0 • 69

46

0

• 440

2

0 •

2142

0 • 0

769

0

• 936

7

0

• 968

0

0

• 985

2

0

• 993

7

0

• 997

6

0 • 79

78

0 • 87

61

0 • 93

01

0 • 96

37

0 • 98

27

0 •

5681

0 •

6885

0 •

7911

0 •

8702

0 •

9256

0 • 31

85

0 • 43

91

0 • 56

51

0 • 68

44

0 • 78

70

0 •

1341

0 •

2148

0 •

3179

0 •

4373

0 •

5627

0 • 9

991

0 • 9

997

0 • 9

999

1 • 0

000

1 • 0

000

0 • 99

25

0 • 99

70

0 • 99

89

0 • 99

96

0 • 99

99

0 •

9608

0 •

9811

0 •

9917

0 •

9966

0 •

9988

0 • 86

69

0 • 92

33

0 • 95

95

0 • 98

04

0 • 99

14

0 •

6821

0 •

7852

0 •

8659

0 •

9231

0 •

9597

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9996

0 •

9999

1 •

0000

1 •

0000

1 •

0000

0 • 99

66

0 • 99

88

0 • 99

96

0 • 99

99

1 • 00

00

0 •

9808

0 •

9917

0 • 99

68

0 • 99

89

0 • 99

97

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

0 •

9999

1 •

0000

n =

40 (c

ontin

ued)

Page 95: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0

• 15

0

• 20

0

• 25

n = 5

0, x

= 0 61 2 3 4 5 7 8 9 10 11 12 13 14 15

0 • 0

769

0 • 00

52

0

• 000

3

0 •

0000

0 • 0

000

0 • 2

794

0 • 5

405

0 • 7

604

0 • 8

964

0 • 9

622

0 • 03

38

0 • 11

17

0 • 25

03

0 • 43

12

0 • 61

61

0 •

0029

0 •

0142

0 •

0460

0 •

1121

0 •

2194

0 • 00

02

0 • 00

13

0 • 00

57

0 • 01

85

0 • 04

80

0 •

0000

0 •

0001

0 •

0005

0 •

0021

0 •

0070

0 • 9

882

0 • 9

968

0 • 9

992

0 • 9

998

1 • 0

000

0 • 77

02

0 • 87

79

0 • 94

21

0 • 97

55

0 • 99

06

0 •

3613

0 •

5188

0 •

6681

0 •

7911

0 •

8801

0 • 10

34

0 • 19

04

0 • 30

73

0 • 44

37

0 • 58

36

0 •

0194

0 •

0453

0 •

0916

0 •

1637

0 •

2622

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

68

0 • 99

90

0 • 99

97

0 • 99

99

1 • 00

00

0 •

9372

0 •

9699

0 •

9868

0 •

9947

0 •

9981

0 • 71

07

0 • 81

39

0 • 88

94

0 • 93

93

0 • 96

92

n

= 5

0

0 • 38

16

0 • 51

10

0 • 63

70

0 • 74

81

0 • 83

69

Page 96: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

16 17 18

n = 5

0, x

= 15 19 20 21 22 23 24 25 26 27

28 29 30

1 • 0

000

1 • 00

00

0

• 998

1

0 •

9692

0 • 8

369

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9993

0 •

9998

0 •

9999

1 •

0000

1 •

0000

0 • 98

56

0 • 99

37

0 • 99

75

0 • 99

91

0 • 99

97

0 •

9017

0 •

9449

0 •

9713

0 •

9861

0 •

9937

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

0 • 99

99

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9974

0 •

9990

0 •

9996

0 •

9999

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

n

= 50

(con

tinue

d)

1 • 00

00

1 • 00

001 •

0000

1 •

0000

1 •

0000

Page 97: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 05

0

• 10

0 •

15

0 • 2

0

0 • 2

5

31 32

n = 5

0, x

= 30 33 34 35 36 37 38

1 • 0

000

1 • 00

00

1

• 000

0

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

n

= 50

(con

tinue

d)

Page 98: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

n = 5

0, x

= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0 • 0

000

0 • 00

00

00 •

0000

•0

0000

•000

0

0•0

000

0

•000

0

0

•0

0000

•000

0

0

•0

0000

•000

0

0•

000

00

•0

000

0

•0

0000

0 • 0

000

0 • 0

000

0 • 0

000

0 • 0

002

0 • 0

007

0 • 00

00

0 • 00

00

0 • 00

00

0 • 00

00

0 • 00

01

0000

0 • 00

000 •

0000

0 • 00

000 •

0000

0 • 00

00

0•

000

00

•0

000

0•

000

00

•0

000

0•

000

00

•0

000

0

•0

0001

•000

0

0•

000

02

0 • 0

025

0 • 0

073

0 • 0

183

0 • 0

402

0 • 0

789

0 • 00

02

0 • 00

08

0 • 00

25

0 • 00

67

0 • 01

60

0000

0 • 00

000 •

0001

0 • 00

020 •

0008

0 • 00

22

0•

000

06

•0

000

0•

000

18

•0

002

0•

000

45

•0

005

0

•0

0104

•001

3

0•

002

20

0 • 1

390

0 • 2

229

0 • 3

279

0 • 4

468

0 • 5

692

•003

3

0 • 03

42

0 • 06

61

0 • 11

63

0 • 18

78

0 • 28

01

0 •

0057

0 • 01

330 •

0280

0 • 05

400 •

0955

n

= 50

(con

tinue

d)

Page 99: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

16 17 18

n = 5

0, x

= 15 19 20 21 22 23 24 25 26 27

28 29 30

0 • 5

692

0 • 28

01

0

• 095

5

0 •

0220

0 • 0

033

0 • 6

839

0 • 7

822

0 • 8

594

0 • 9

152

0 • 9

522

0 • 38

89

0 • 50

60

0 • 62

16

0 • 72

64

0 • 81

39

0 •

1561

0 •

2369

0 •

3356

0 •

4465

0 •

5610

0 • 04

27

0 • 07

65

0 • 12

73

0 • 19

74

0 • 28

62

0 •

0077

0 •

0164

0 •

0325

0 •

0595

0 •

1013

0 • 9

749

0 • 9

877

0 • 9

944

0 • 9

976

0 • 9

991

0 • 88

13

0 • 92

90

0 • 96

04

0 • 97

93

0 • 99

00

0 •

6701

0 •

7660

0 •

8438

0 •

9022

0 •

9427

0 • 39

00

0 • 50

19

0 • 61

34

0 • 71

60

0 • 80

34

0 •

1611

0 •

2399

0 •

3359

0 •

4439

0 •

5561

0 • 9

997

0 • 9

999

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

55

0 • 99

81

0 • 99

93

0 • 99

97

0 • 99

99

0 •

9686

0 •

9840

0 •

9924

0 •

9966

0 •

9986

0 • 87

21

0 • 92

20

0 • 95

56

0 • 97

65

0 • 98

84

n

= 50

(con

tinue

d)

0 • 66

41

0 • 76

010 •

8389

0 •

8987

0 •

9405

Page 100: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

p =

0

• 30

0

• 35

0 •

40

0 • 4

5

0 • 5

0

31 32 33

n = 5

0, x

= 30 34 35 36 37 38

1 • 0

000

0 • 99

99

0

• 998

6

0 •

9884

0 • 9

405

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9995

0 •

9998

0 •

9999

1 •

0000

1 •

0000

0 • 99

47

0 • 99

78

0 • 99

91

0 • 99

97

0 • 99

99

0 •

9675

0 •

9836

0 •

9923

0 •

9967

0 •

9987

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9995

0 •

9998

1 •

0000

n

= 50

(con

tinue

d)

Page 101: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

The

valu

es

in th

e ta

ble

are

thos

e w

hich

a ra

ndom

var

iabl

e e

xcee

ds w

ith

prob

abili

ty p

; tha

t is,

pp

• •

• •

• •

• •

• •

• •

• •

• •

• •

• 3 •

• •

• 3 •

Page 102: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

The

tabu

late

d va

lue

is P

(X

w

here

X h

as a

Poi

sson

dis

trib

utio

n w

ith p

aram

eter

=

0 • 5

1 • 0

1 • 5

2 • 0

2 • 5

x =

0 1 2 3 4 5 106 7 8 9

0 • 6

065

0 • 36

79

0

• 223

1

0 •

1353

0 • 0

821

0 • 9

098

0 • 9

856

0 • 9

982

0 • 9

998

1 • 0

000

0 • 73

58

0 • 91

97

0 • 98

10

0 • 99

63

0 • 99

94

0 •

5578

0 •

8088

0 •

9344

0 •

9814

0 •

9955

0 • 40

60

0 • 67

67

0 • 85

71

0 • 94

73

0 • 98

34

0 • 28

730 •

5438

0 • 75

760 •

8912

0 • 95

80

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

99

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

0 •

9991

0 •

9998

1 •

0000

1 •

0000

1 •

0000

0 • 99

55

0 • 99

89

0 • 99

98

1 • 00

00

1 • 00

00

0 •

9858

0 • 99

580 •

9989

0 • 99

970 •

9999

Page 103: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

=

0 • 5

1 • 0

1 • 5

2 • 0

2 • 5

11

x =

10 12 13 14 15

16 17 18 19

1 • 0

000

1 • 00

00

1

• 000

0

1 •

0000

0 • 9

999

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 • 00

001 •

0000

1 • 00

001 •

0000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 • 00

001 •

0000

1 • 00

00

Page 104: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

=

3 • 0

3 • 5

4 • 0

4 • 5

5 • 0

x =

0 1 2 3 4 5 6 7 8 9 10

0 • 0

498

0 • 03

02

0

• 018

3

0 •

0111

0 • 0

067

0 • 1

991

0 • 4

232

0 • 6

472

0 • 8

153

0 • 9

161

0 • 13

59

0 • 32

08

0 • 53

66

0 • 72

54

0 • 85

76

0 •

0916

0 •

2381

0 •

4335

0 •

6288

0 •

7851

0 • 06

11

0 • 17

36

0 • 34

23

0 • 53

21

0 • 70

29

0 •

0404

0 • 12

470 •

2650

0 • 44

050 •

6160

0 • 9

665

0 • 9

881

0 • 9

962

0 • 9

989

0 • 9

997

0 • 93

47

0 • 97

33

0 • 99

01

0 • 99

67

0 • 99

90

0 •

8893

0 •

9489

0 •

9786

0 •

9919

0 •

9972

0 • 83

11

0 • 91

34

0 • 95

97

0 • 98

29

0 • 99

33

0 •

7622

0 • 86

660 •

9319

0 • 96

820 •

9863

Page 105: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

=

3 • 0

3 • 5

4 • 0

4 • 5

5 • 0

x =

10 11 12 13 14 15 16 17 18 19

0 • 9

997

0 • 99

90

0

• 997

2

0 •

9933

0 • 9

863

0 • 99

76

0 • 99

92

0 •

9945

0 • 99

80

0

•999

7

0•9

993

0 • 9

999

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

0 • 99

97

0 • 99

99

1 • 00

00

1 • 00

00

1 • 00

00

0 • 99

99

1 • 00

00

0 • 99

91

0 • 99

97

0 • 99

991 •

0000

1 •

0000

0 •

9998

0 • 99

99

1 • 0

000

1 • 0

000

1 • 0

000

1 • 0

000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 •

0000

1 •

0000

1 •

0000

1 • 00

00

1 • 00

00

1 • 00

00

1 • 00

00

1 •

0000

1 • 00

001 •

0000

1 • 00

00

Page 106: )LUVW FHUWL¿FDWLRQ IURP Mathematical formulae and … · 2020-07-13 · 7 AS MATHEMATICS 7 Pure Mathematics 9 Statistics 0HFKDQLFV A LEVEL MATHEMATICS 3XUH 0DWKHPDWLFV 6WDWLVWLFV

=

5 • 5

6 • 0

6 • 5

7 • 0

7 • 5

x =

0 1 2 3 4 5 6 7 8 9 10

0 • 0

041

0 • 00

25

0

• 001

5

0 •

0009

0 • 0

006

0 • 0

266

0 • 0

884

0 • 2

017

0 • 3

575

0 • 5

289

0 • 01

74

0 • 06

20

0 • 15

12

0 • 28

51

0 • 44

57

0 •

0113

0 •

0430

0 •

1118

0 •

2237

0 •

3690

0 • 00

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7

2

• 34

1 • 9

5

1

• 38

2

• 51

2 • 3

2

2

• 18

1 • 7

9

1

• 00

21

vv