IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Novem

ber

2009 IG

CSE M

ath

em

ati

cs

(4400)

Mark

Schem

e –

Paper

4H

Exc

ept

for

quest

ions

* w

here

the m

ark

sch

em

e s

tate

s oth

erw

ise t

he c

orr

ect

answ

er,

unle

ss c

learl

y obta

ined b

y an i

nco

rrect

meth

od,

should

be

take

n t

o im

ply

a c

orr

ect

meth

od.

[* Q

uest

ions

2(b

), 2

1 a

nd 2

2]

Tri

al and im

pro

vem

ent

meth

ods

for

solv

ing

equati

ons

score

no m

ark

s, e

ven if

they

lead t

o a

corr

ect

solu

tion.

Q

Work

ing

Answ

er

Mark

N

ote

s

1.

30

26

350 ..

2

M1

for

350.2

6

1167.5

333

A1

Acc

ept

1dp o

r bett

er

Als

o a

ccept

35

1167

. o

r 15

17513

T

ota

l 2 m

ark

s

2.

(a)

n(n

4

) 2

B2

B1 f

or

fact

ors

whic

h,

when e

xpanded

and s

implifi

ed,

give

tw

o t

erm

s, o

ne o

f w

hic

h is

corr

ect

exc

ept

(n +

2)(

n

2)

and s

imilar

SC B

1 f

or

n(n

4

n)

(b)

M1 f

or

5x +

2 =

8

5x =

8

2 o

r 5x =

2

8

or

5x =

6 o

r 5x =

6

3

M2

511

oe

A1

dep o

n M

2

Do n

ot

acc

ept

56

Tota

l 5 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

3.

(a)(

i)

62

2

B1

cao

(

ii)

alt

ern

ate

angl

es

B1

Acc

ept

‘alt

ern

ate

’ but

not

‘Z a

ngl

es’

(b)

2

62

180

""

or

2

62

180

or

59

2

M1

121

A1

cao

Tota

l 4 m

ark

s

4.

(a)

1

(0.4

+ 0

.5)

2

M1

0.1

A1

Als

o a

ccept

11.0

(b)

0.4

× 8

0 o

r 4

080

.n

2

M1

32

A1

cao

Tota

l 4 m

ark

s

5.

(a)

100

3500

161

2

M1

for

3500

161

oe inc

0.0

46

4.6

A1

cao

(b)

1% =

$2

5338 .

or

65 s

een

or

5.2

% (

of

am

ount)

= 3

38

3

M1

“65” ×

100

M1

M2 f

or

100

25338 .

or

052

0338

.

6500

A1

Tota

l 5 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

6.

(a)

Refl

ect

ion in t

he lin

e y

= 4

2

B2

B1 f

or

refl

ect

ion,

refl

ect

s etc

B1 f

or

y =

4 o

r eg

‘dott

ed lin

e’

but,

if

giv

en,

equati

on

must

be c

orr

ect

(b

) Enla

rgem

ent

wit

h s

cale

fact

or

1½

,

centr

e (

1,6

)

3

B3

B1 f

or

enla

rgem

ent,

enla

rge e

tc

B1 f

or

1½

oe

B1 f

or

(1,6

)

These

mark

s

are

in

dependent

but

aw

ard

no

mark

s if

answ

er

is

not

a s

ingl

e

transf

orm

ati

on.

(S

eco

nd

transf

orm

ati

on

may

be im

plied)

Tota

l 5 m

ark

s

7.

1 +

9 +

2 o

r 12 o

r 5 s

een

3

M1

May

be im

plied b

y 1 c

orr

ect

answ

er

5

10

45

A2

A1 f

or

one c

orr

ect

Tota

l 3 m

ark

s

8.

Arc

s of

equal ra

dii >

½A

B,

centr

es

A,

B,

whic

h inte

rsect

tw

ice

2

M1

Perp

endic

ula

r bis

ect

or

wit

hin

guid

elines

A1

Tota

l 2 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

9.

(a)

Corr

ect

lin

e

2

B2

Must

be a

sin

gle s

traig

ht

line p

ass

ing

thro

ugh

at

least

3 o

f (0

, 2),

(3,

0),

(6

, 2),

(9,

4)

B1 f

or

a s

ingl

e s

traig

ht

line w

ith a

posi

tive

gra

die

nt

pass

ing

thro

ugh e

ither

(0,

2)

or

(3,

0)

or

for

3 o

f 4 p

oin

ts (

0,

2),

(3,

0),

(6

, 2),

(9,

4)

corr

ect

wit

h a

t m

ost

1

poin

t in

corr

ect

Allow

± 2

mm

(b

) Lin

es

x =

3 a

nd x

= 6

dra

wn

3

B1

Lin

es

y =

2 a

nd y

= 4

dra

wn

B1

R s

how

n

B1

Condone o

mis

sion o

f la

bel

Acc

ept

shadin

g in

or

shadin

g out,

if

consi

stent

Aw

ard

3 m

ark

s fo

r co

rrect

labelled

rect

angl

e,

eve

n if

not

shaded

Aw

ard

2 m

ark

s fo

r a c

orr

ect

unsh

aded

rect

angl

e w

ithout

a c

orr

ect

label

SC

B1 f

or

regi

on b

ounded b

y 2 <

x <

4

and 3

< y

< 6

T

ota

l 5 m

ark

s

M1

for

findin

g at

least

3 p

roduct

s x ×

f

consi

stentl

y w

ithin

inte

rvals

(in

c end

poin

ts)

10.

(a)

6.2

C

5 ×

23 +

15 ×

3 +

25 ×

2 +

35 ×

3

= 1

15 +

45 +

50 +

105

3

M1

(dep)

for

use

of

at

least

3 c

orr

ect

half

way

valu

es

315

A1

cao isw

aft

er

315

(b)

6.1

A

19

4

7

3

B3

B1 f

or

each

valu

e c

ao

Tota

l 6 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

11.

(a)

64 =

26 a

nd 8

0 =

24 ×

5

or

1,2

,4,8

,16,3

2,6

4

and 1

,2,4

,5,8

,10,1

6,2

0,4

0,8

0

or

24

2

M1

Need n

ot

be p

roduct

of

pow

ers

;

acc

ept

pro

duct

s or

list

s ie

2,2

,2,2

,2,2

and 2

,2,2

,2,5

Pri

me f

act

ors

may

be s

how

n a

s fa

ctor

trees

or

repeate

d d

ivis

ion

16

A1

cao

(b)

26 ×

5 o

e e

g 2

4 ×

4 ×

5,1

6 ×

4 ×

5

or

64,1

28,1

92,2

56,3

20

and 8

0,1

60,2

40,3

20

2

M1

320

A1

cao

Tota

l 4 m

ark

s

12.

(a)

p2

4p +

7p

28

2

M1

for

4 c

orr

ect

term

s ig

nori

ng

sign

s or

for

3 t

erm

s w

ith c

orr

ect

sig

ns

p

2 +

3p

28

A1

cao

(b)

12x

5y

6

2

B2

B1 f

or

any

two p

art

s co

rrect

(c)

9q

4

2

B2

B1 f

or

eit

her

9 o

r q

4

Tota

l 6 m

ark

s

13.

(a)

12

15

18

2

M1

for

12

15

(1.2

5)

oe o

r 12

18

(1.5

) oe s

een

22.5

A1

cao

(b)

eg

12

15

20

, 15

12

20

,15

20

12

2

M1

for

eg

20 ÷

1.2

5,

20 ×

0.8

, 3.

112

16

A1

cao

Tota

l 4 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

14.

(a)

8

(8)

12

10

8

12

2

B2

for

all c

orr

ect

(B1 f

or

3 c

orr

ect

)

(b)

Poin

ts

2

B1

Allow

+ ½

sq f

t fr

om

table

if

at

least

B1

score

d in (

a)

Curv

e

B1

ft if

B1 f

or

poin

ts

Aw

ard

for

singl

e c

urv

e (

not

line

segm

ents

) w

hic

h d

oes

not

mis

s m

ore

th

an o

ne p

lott

ed p

oin

t by

more

than ½

sq

uare

T

ota

l 4 m

ark

s

15.

(a)(

i)

2 ×

58

116

2

B1

cao

(

ii)

eg

angl

e a

t th

e c

entr

e

= 2

× a

ngl

e a

t th

e c

ircu

mfe

rence

B1

Thre

e k

ey

poin

ts m

ust

be m

enti

oned

1.

angl

e a

t ce

ntr

e/m

iddle

/O

/ori

gin

2.

twic

e/double

/ 2

× o

r h

alf/

1/ 2

as

ap

pro

pri

ate

3

. a

ng

le a

t cir

cu

mfe

ren

ce

/ e

dg

e/

perim

ete

r/arc

(NOT

e.g

. an

gle

B, a

ngle

ABC

, a

ng

le a

t to

p,

an

gle

at o

uts

ide

) (b

)(i)

180

58

122

2

B1

cao

(

ii)

eg

sum

of

opposi

te a

ngl

es

of

a c

yclic

quadri

late

ral = 1

80°

B1

Acc

ept

reaso

n w

hic

h incl

udes

‘opposi

te’

and ‘

cycl

ic’

and n

oth

ing

inco

rrect

Als

o a

ward

if

(b)(

i) is

corr

ect

and

reaso

n is

giv

en a

s ‘a

ngl

e a

t th

e c

entr

e

= 2

× a

ngl

e a

t th

e c

ircu

mfe

rence

’ oe

Ignore

addit

ional re

aso

n(s

)

T

ota

l 4 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

16.

(a)

2

B2

for

com

ple

tely

corr

ect

dia

gra

m,

inc

labels

(acc

ept

clear

abbre

viati

ons

eg

S and H

)

(B1 f

or

bra

nch

es

wit

h a

t le

ast

3 c

orr

ect

pro

babilit

ies

in

the c

orr

ect

pla

ce)

M1

for

one

corr

ect

pro

duct

SC M

1 f

or

103

107

or

107

103

or

103

103

(b)

107

×

93 +

103

×

97 +

103

×

92

(=

9021

+

9021

+

906

)

or

107

×

93 +

103

(=

9021

+ 1

03)

3

M1

for

com

ple

tely

co

rrect

exp

ress

ion

M2 f

or

1107

×96

SC

M2 f

or

1107

×107

SC M

1 (

dep)

for

sum

of

above

pro

duct

s or

for

103

103

107

for

meth

od

mark

s ft

fr

om

their

tr

ee

dia

gram

, pro

vided

pro

babilit

ies

< 1

9048

A1

for

9048

oe inc

158

or

for

0.5

3

or

for

answ

er

roundin

g to

0.5

3

Tota

l 5 m

ark

s

103

96 93 97 92

Fir

st c

hoco

late

Sec

ond

cho

cola

te

so

ftc

entr

e

har

d

cen

tre

107

soft

cent

re

soft

cent

re

hard

cent

re

hard

cent

re

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

17.

(a)

)2.0

1(

)2.0

1(6.

8 o

r 8.

0

32

.10

2

M1

for

corr

ect

subst

ituti

on

12.9

oe

A1

(b)

)(

)(

en

eT

11

5

M1

rem

ove

s fr

act

ions

en

neT

T

M1

expands

bra

ckets

nT

eT

en

M

1

collect

s te

rms

nT

Tn

e)

(

M1

fact

ori

ses

nT

nT

A1

for

nT

nT

oe

Tota

l 7 m

ark

s

18.

8.3

2

7.2

2

= 6

8.8

9

51.8

4 =

17.0

5

5

M1

for

8.3

2

7.2

2

22

27

38

..

= 4

.129…

M

1

for

22

27

38

..

ta

n a

nd

93129

4

.

..."

."

M

2

M1 f

or

tan a

nd

..."

."

.

129

4

93

Acc

ept

CD

rounded o

r tr

unca

ted t

o a

t le

ast

1 d

p (

4.1

2916…

)

46.6

A1

Acc

ept

answ

er

roundin

g to

46.6

(4

.1

46.4

3…

4.1

2

46.5

7…

4.1

3

46.6

4…

)

T

ota

l 5 m

ark

s

Alt

ern

ati

ve m

eth

ods

for

Q18 a

ppear

on t

he n

ext

tw

o p

age

s.

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Quest

ion 1

8 A

ltern

ati

ve m

eth

ods

Meth

od 1

W

ork

ing

Answ

er

Mark

N

ote

s

8.3

2

7.2

2

= 6

8.8

9

51.8

4 =

17.0

5

5

M1

for

8.3

2

7.2

2

22

27

38

..

= 4

.129…

22

93

129

4.

.= 5

.679…

M1

for

22

27

38

..

cos

and

".

"

. 679

5

93

M

2

M1 f

or

cos

and

936

79

5

.

".

"

Acc

ept

BC r

ounded o

r tr

unca

ted t

o a

t le

ast

1 d

p (

5.6

7978…

)

46.6

A1

Acc

ept

answ

er

roundin

g to

46.6

Tota

l 5 m

ark

s

Meth

od 2

W

ork

ing

Answ

er

Mark

N

ote

s

8.3

2

7.2

2

= 6

8.8

9

51.8

4 =

17.0

5

5

M1

for

8.3

2

7.2

2

22

27

38

..

= 4

.129…

22

93

129

4.

.= 5

.679…

M1

for

22

27

38

..

sin a

nd

".

"

".

"

679

5

129

4

M2

M1 f

or

sin a

nd

".

"

".

"

129

4

679

5

Acc

ept

CD

rounded o

r tr

unca

ted t

o a

t le

ast

1 d

p (

4.1

2916…

) and B

C r

ounded o

r tr

unca

ted t

o a

t le

ast

1 d

p (

5.6

7978…

)

46.6

A1

Acc

ept

answ

er

roundin

g to

46.6

Tota

l 5 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Meth

od 3

W

ork

ing

Answ

er

Mark

N

ote

s

Corr

ect

meth

od f

or

findin

g A

5

M1

eg f

or

3.8

2.7

cos

A (

A =

29.8

3…

°)

".

cos"

..

..

829

38

111

23

81

11

22

M

1

for

corr

ect

Cosi

ne R

ule

expre

ssio

n f

or

calc

ula

ting

BC

cos

and

".

"

. 679

5

93

M

2

M1 f

or

cos

and

936

79

5

.

".

"

Acc

ept

BC r

ounded o

r tr

unca

ted t

o a

t le

ast

1 d

p (

5.6

7978…

)

46.6

A1

Acc

ept

answ

er

roundin

g to

46.6

Tota

l 5 m

ark

s

Meth

od 4

W

ork

ing

Answ

er

Mark

N

ote

s

Corr

ect

meth

od f

or

findin

g A

5

M1

eg f

or

3.8

2.7

cos

A (

A =

29.8

3…

°)

".

cos"

..

..

829

38

111

23

81

11

22

M

1

for

corr

ect

Cosi

ne R

ule

expre

ssio

n f

or

calc

ula

ting

BC

"68

.5"

"8.

29

sin"

3.8

sin

B

M2

for

corr

ect

expre

ssio

n f

or

sinB

M1

for

corr

ect

sta

tem

ent

of

Sine R

ule

eg

"68

.5"

"8.

29

sin"

3.8sin

B

46.6

A1

Acc

ept

answ

er

roundin

g to

46.6

Tota

l 5 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

19.

(a)

3t2

1

0t

2

B2

B1 f

or

3t2

or

10t

Ignore

furt

her

dif

fere

nti

ati

on s

een in b

ody

or

on

answ

er

line

(b)

6t

10 =

20

2

M1

for

linear

exp

ress

ion incl

udin

g eit

her

6t

or

10

5

A1

ft f

rom

“6t

10” =

20 if

M1 s

core

d

Tota

l 4 m

ark

s

20.

(a)

14

1

B1

cao

(b)

9

1

B1

cao

(c)(

i)

6

3

2

3

B2

B1 f

or

2 c

orr

ect

(

ii)

11

B1

cao

Tota

l 5 m

ark

s

21.

12 ×

12

= 1

8(d

1

8)

12 ×

12

= 1

8x

4

M1

or

for

r2 =

12

2 +

(18

r)2

1

44 =

18d

324

x =

8

M1

or

for

r2 =

144 +

324

18r

18r

+ r

2

1

8d =

468

(d=)8

+18

M1

or

for

36r

= 4

68

26

A1

dep o

n a

ll m

eth

od m

ark

s

T

ota

l 4 m

ark

s

Alt

ern

ati

ve m

eth

ods

for

Q21 a

ppear

on t

he n

ext

page

.

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Quest

ion 2

1 A

ltern

ati

ve m

eth

ods

Meth

od 1

W

ork

ing

Answ

er

Mark

N

ote

s

Com

ple

te,

corr

ect

meth

od

for

findin

g A

OM

or

BO

Mor

OA

B o

r O

BA

4

M1

eg

18

12

tan

ALM

69

.33

ALM

69

.33

2A

OM 3

8.

67

22

18

12

AL

468

= 2

1.6

3 c

m

3846

.0

468

2

576

468

468

cos

ALB

ALB

38

.67

69

.33

ALM

69

.33

2A

OM

38

.67

18

12

tan

ALM

69

.33

ALM

OA

M

69

.33

290

= 2

2.6

2°

Corr

ect

num

eri

cal

exp

ress

ion f

or

lengt

h o

f O

A

or

OM

M1

eg

38

.67

sin

12

or

76

.134

sin

62

.22

sin

24

(= 1

3)

or

38

.67

tan

12

or

62

.22

tan

12

(= 5

)

Length

of

OA

or

OM

use

d t

o

find d

iam

ete

r

M

1

eg

2 ×

“13” o

r 2 ×

(18

“5”)

dep o

n b

oth

pre

vio

us M

1s

26

A1

dep o

n a

ll m

eth

od m

ark

s Acc

ept

answ

er

roundin

g to

26.0

Tota

l 4 m

ark

s

Meth

od 2

Work

ing

Answ

er

Mark

N

ote

s

AM

= 1

2,

OM

= 5

, O

A =

13

and

13 +

5 =

18 o

r 18

5 =

13

4

M3

for

use

of

Pyt

hago

rean t

riple

5-1

2-1

3

or

use

of

OM

= 5

Pyt

hagora

s to

obta

in

13

12

52

2 f

or

OA

and

just

ific

ati

on

26

A1

dep o

n M

3

Tota

l 4 m

ark

s

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Q

Work

ing

Answ

er

Mark

N

ote

s

22.

y =

3x +

4

7

B1

for

corr

ect

rearr

ange

ment

34

43

22

)(

xx

M

1

for

corr

ect

subst

ituti

on

34

16

12

12

92

2x

xx

x

or

34

16

24

92

2x

xx

B1

(indep)

for

corr

ect

expansi

on o

f

(3x +

4)2

eve

n if

unsi

mplifi

ed

)0

(18

24

10

2x

x

B1

for

corr

ect

sim

plifi

cati

on

Condone o

mis

sion o

f ‘=

0’

)0

)(62

)(35(

xx

or

)0)(3

)(35(

xx

or

)0)(3

)(610

(x

x

or

20

1296

24

or

10

324

12

or

10324

1012

or

581

56

B1

for

corr

ect

fact

ori

sati

on

Condone o

mis

sion o

f ‘=

0’

or

for

corr

ect

subst

ituti

on into

the

quadra

tic

form

ula

and c

orr

ect

eva

luati

on o

f ‘b

2

4ac’

or

for

usi

ng

square

com

ple

tion

corr

ect

ly a

s fa

r as

indic

ate

d

53x

or

3x

A1

for

both

valu

es

of

x

53x

, 54

5y

3x

, 5

y

A1

for

com

ple

te,

corr

ect

solu

tions

Need n

ot

be e

xplici

tly

pair

ed

T

ota

l 7 m

ark

s

TO

TA

L F

OR

PA

PER

: 1

00 M

AR

KS

Note

The m

ark

sch

em

e f

or

an a

ltern

ati

ve m

eth

od f

or

Q22 i

s on t

he n

ext

page

.

IGCSE

Math

em

ati

cs (

4400)

Paper

4H

Nove

mber

2009

Quest

ion 2

2 A

ltern

ati

ve m

eth

od

W

ork

ing

Answ

er

Mark

N

ote

s

3

4y

x

7

B1

for

corr

ect

rearr

ange

ment

34

3

42

2

yy

M

1

for

corr

ect

subst

ituti

on

34

9

16

44

22

yy

yy

or

306

916

44

22

yy

yy

o

r 34

9

16

82

2

yy

y

or

306

916

82

2y

yy

B1

(indep)

for

corr

ect

exp

ansi

on o

f

(y

4)2

eve

n if

unsi

mplifi

ed

)0(

290

810

2y

y

B1

for

corr

ect

sim

plifi

cati

on

Condone o

mis

sion o

f ‘=

0’

)0)(5

)(29

5(y

y

)0)(

10

2)(

29

5(y

y

)0)(5

)(58

10

(y

y

or

2011664

8 o

r 102

916

4

or

10

2916

104

or

5729

52

B1

for

corr

ect

fact

ori

sati

on

Condone o

mis

sion o

f ‘=

0’

or

for

corr

ect

subst

ituti

on into

the

quadra

tic

form

ula

and c

orr

ect

eva

luati

on o

f ‘b

2

4ac’

or

for

usi

ng s

quare

com

ple

tion

corr

ect

ly a

s fa

r as

indic

ate

d

545

y o

r 5

y

A1

for

both

valu

es

of

y

53x

, 54

5y

3x

, 5

y

A1

for

com

ple

te,

corr

ect

solu

tions

T

ota

l 7 m

ark

s