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MATHEMATICS 0580/04Paper 4 (Extended) For examination from 2020MARK SCHEME

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Page 2 of 12© UCLES 2017

Gen

eric

Mar

king

Pri

ncip

les

Thes

e ge

nera

l mar

king

prin

cipl

es m

ust b

e ap

plie

d by

all

exam

iner

s whe

n m

arki

ng c

andi

date

ans

wer

s. Th

ey sh

ould

be

appl

ied

alon

gsid

e th

e sp

ecifi

c co

nten

t of t

he

mar

k sc

hem

e or

gen

eric

leve

l des

crip

tors

for a

que

stio

n. E

ach

ques

tion

pape

r and

mar

k sc

hem

e w

ill a

lso

com

ply

with

thes

e m

arki

ng p

rinci

ples

.

GEN

ERIC

MA

RK

ING

PR

INC

IPLE

1:

Mar

ks m

ust b

e aw

arde

d in

line

with

:

•th

e sp

ecifi

c co

nten

t of t

he m

ark

sche

me

or th

e ge

neric

leve

l des

crip

tors

for t

he q

uest

ion

•th

e sp

ecifi

c sk

ills d

efin

ed in

the

mar

k sc

hem

e or

in th

e ge

neric

leve

l des

crip

tors

for t

he q

uest

ion

•th

e st

anda

rd o

f res

pons

e re

quire

d by

a c

andi

date

as e

xem

plifi

ed b

y th

e st

anda

rdis

atio

n sc

ripts

.

GEN

ERIC

MA

RK

ING

PR

INC

IPLE

2:

Mar

ks a

war

ded

are

alw

ays w

hole

mar

ks (n

ot h

alf m

arks

, or o

ther

frac

tions

).

GEN

ERIC

MA

RK

ING

PR

INC

IPLE

3:

Mar

ks m

ust b

e aw

arde

d po

sitiv

ely:

•m

arks

are

aw

arde

d fo

r cor

rect

/val

id a

nsw

ers,

as d

efin

ed in

the

mar

k sc

hem

e. H

owev

er, c

redi

t is g

iven

for v

alid

ans

wer

s whi

ch g

o be

yond

the

scop

e of

the

sylla

bus a

nd m

ark

sche

me,

refe

rrin

g to

you

r Tea

m L

eade

r as a

ppro

pria

te •

mar

ks a

re a

war

ded

whe

n ca

ndid

ates

cle

arly

dem

onst

rate

wha

t the

y kn

ow a

nd c

an d

o •

mar

ks a

re n

ot d

educ

ted

for e

rror

s •

mar

ks a

re n

ot d

educ

ted

for o

mis

sion

s •

answ

ers s

houl

d on

ly b

e ju

dged

on

the

qual

ity o

f spe

lling

, pun

ctua

tion

and

gram

mar

whe

n th

ese

feat

ures

are

spec

ifica

lly a

sses

sed

by th

e qu

estio

n as

indi

cate

d by

the

mar

k sc

hem

e. T

he m

eani

ng, h

owev

er, s

houl

d be

una

mbi

guou

s.

GEN

ERIC

MA

RK

ING

PR

INC

IPLE

4:

Rul

es m

ust b

e ap

plie

d co

nsis

tent

ly e

.g. i

n si

tuat

ions

whe

re c

andi

date

s hav

e no

t fol

low

ed in

stru

ctio

ns o

r in

the

appl

icat

ion

of g

ener

ic le

vel d

escr

ipto

rs.

GEN

ERIC

MA

RK

ING

PR

INC

IPLE

5:

Mar

ks sh

ould

be

awar

ded

usin

g th

e fu

ll ra

nge

of m

arks

def

ined

in th

e m

ark

sche

me

for t

he q

uest

ion

(how

ever

; the

use

of t

he fu

ll m

ark

rang

e m

ay b

e lim

ited

acco

rdin

g to

the

qual

ity o

f the

can

dida

te re

spon

ses s

een)

.



GEN

ERIC

MA

RK

ING

PR

INC

IPLE

6:

Mar

ks a

war

ded

are

base

d so

lely

on

the

requ

irem

ents

as d

efin

ed in

the

mar

k sc

hem

e. M

arks

shou

ld n

ot b

e aw

arde

d w

ith g

rade

thre

shol

ds o

r gra

de d

escr

ipto

rs in

m

ind.

MA

RK

SC

HE

ME

NO

TE

S

The

follo

win

g no

tes a

re in

tend

ed to

aid

inte

rpre

tatio

n of

mar

k sc

hem

es in

gen

eral

, but

indi

vidu

al m

ark

sche

mes

may

incl

ude

mar

ks a

war

ded

for s

peci

fic re

ason

s ou

tsid

e th

e sc

ope

of th

ese

note

s.

Type

s of m

ark

M

Met

hod

mar

k, a

war

ded

for a

val

id m

etho

d ap

plie

d to

the

prob

lem

.A

A

ccur

acy

mar

k, a

war

ded

for a

cor

rect

ans

wer

or i

nter

med

iate

step

cor

rect

ly o

btai

ned.

For

acc

urac

y m

arks

to b

e gi

ven,

the

asso

ciat

ed M

etho

d m

ark

mus

t be

earn

ed o

r im

plie

d.B

M

ark

for a

cor

rect

resu

lt or

stat

emen

t ind

epen

dent

of M

etho

d m

arks

.

Whe

n a

part

of a

que

stio

n ha

s tw

o or

mor

e ‘m

etho

d’ st

eps,

the

M m

arks

are

in p

rinci

ple

inde

pend

ent u

nles

s the

sche

me

spec

ifica

lly sa

ys o

ther

wis

e; a

nd si

mila

rly

whe

re th

ere

are

seve

ral B

mar

ks a

lloca

ted.

The

not

atio

n ‘d

ep is

use

d to

indi

cate

that

a p

artic

ular

M o

r B m

ark

is d

epen

dent

on

an e

arlie

r mar

k in

the

sche

me.

Abb

revi

atio

ns

cao

co

rrec

t ans

wer

onl

yde

p

depe

nden

tFT

follo

w th

roug

h af

ter e

rror

isw

igno

re su

bseq

uent

wor

king

nfw

w

not f

rom

wro

ng w

orki

ngoe

or e

quiv

alen

tSC

Spec

ial C

ase

soi

se

en o

r im

plie

d



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

1(a)

(i)48

2M

1 fo

r 372

1(a)

(ii)

32.4

[0]

1

1(a)

(iii)

30132

M1

for

.()ii

their 72

7284

−−

oe

1(a)

(iv)

243

M2

for .. 08

192

oeor

M1

for

reco

gnis

ing

19.2

is 8

0%

1(b)

660

3M

2 fo

r 100

550210

##

+ 5

50

oe

or M

1 fo

r 100

550210

##

oe

1(c)

663.

9[0]

2M

1 fo

r 550

× 1

.019

10

oe

1(d)

1.5[

0]3

M2

for

.[]

550

63830

10

oe

or M

1 fo

r 55

0 ×

m10

= 6

38.3

[0]



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

2(a)

(i)40

01

2(a)

(ii)

702

M1

for u

pper

qua

rtile

= 4

20 o

r low

er q

uarti

le =

35

0

2(a)

(iii)

405

to 4

101

2(a)

(iv)

170

2B

1 fo

r 30

seen

2(b)

(i)M

id-v

alue

s 40,

80,

125

, 200

so

iM

1

Σfx

with

cor

rect

freq

uenc

ies a

nd x

’s in

cor

rect

inte

rval

s or o

n bo

unda

ries o

f cor

rect

in

terv

als

M1

÷ 20

0M

1D

ep o

n se

cond

M1

106

nfw

wA

1SC

2 fo

r cor

rect

ans

wer

with

out w

orki

ng

2(b)

(ii)

Cor

rect

his

togr

am4

B1

for c

orre

ct w

idth

san

d B

1 fo

r eac

h re

ctan

gle

of c

orre

ct h

eigh

t at 0

.8, 1

.6,

1.6

(up

to B

3)A

fter 0

scor

ed, S

C1

for 3

cor

rect

freq

uenc

y de

nsiti

es se

en

2(b)

(iii)

10712

3840

oe

is

w 1339

480

R TS SV XW W

3M

2 fo

r [2×

] 10424

10380

#c

m o

e

or M

1 fo

r ,

10424

10380

seen



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

3(a)

9

10.

52

B1

for e

ach

3(b)

Fully

cor

rect

cur

ve5

SC4

for c

orre

ct c

urve

, but

bra

nche

s joi

ned

B3

FT fo

r 9 o

r 10

poin

ts p

lotte

dor

B

2 FT

for 7

or 8

poi

nts p

lotte

dor

B

1 FT

for 5

or 6

poi

nts p

lotte

dan

d B

1 fo

r tw

o se

para

te b

ranc

hes n

ot to

uchi

ng o

r cu

tting

y-a

xis

3(c)

2.1

to 2

.68.

5 to

92

B1

for e

ach

3(d)

2, 3

, 5, 7

2SC

1 fo

r cor

rect

4 v

alue

s and

no

mor

e th

an o

ne

extra

pos

itive

inte

ger o

r ± 2

, ± 3

, ± 5

, ± 7

or 3

cor

rect

val

ues a

nd n

o ex

tras

3(e)

(– 2

, – 1

2)

1

3(f)

(i)20

+ x

2 = x

3M

1fo

r mul

tiplic

atio

n by

x

x3 – x

2 – 2

0 =

0A

1fo

r no

erro

rs o

r om

issi

ons

3(f)

(ii)

Fully

cor

rect

cur

ve y

= x

22

SC1

for U

-sha

ped

para

bola

, ver

tex

at o

rigin

3(f)

(iii)

3.1

to 3

.61

3(f)

(iv)

3.[0

] to

3.1

or F

T th

eir a

nsw

er to

(f)(

iii)

1FT

dep

on

(f)(

iii) >

0

Que

stio

nA

nsw

erM

arks

Part

ial M

arks

4(a)

(i)C

orre

ct im

age

(2, –

5) (4

, –5)

(4

, –2)

2SC

1 fo

r ref

lect

ion

in y

= 0

or 3

cor

rect

poi

nts n

ot jo

ined

4(a)

(ii)

Cor

rect

imag

e (–

3, 1

) (–6

, 1)

(–6,

–1)

2SC

1 fo

r rot

atio

n 90° c

lock

wis

e an

y ce

ntre

or 3

cor

rect

poi

nts n

ot jo

ined

4(b)

Tran

slat

ion

by 1 9J LK KN PO O

2B

1 fo

r eac

h



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

5(a)

(i)[y

=] 21

(80

– 2x

) o

eM

1fo

r 40

– x

is e

noug

h

A =

thei

r 21

(80

– 2x

) × x

oe

M1

for 21

(80

– x)

or 4

0 –

2x fo

r the

ir 21

(80

– 2x

)

A =

40x

– x2 a

nd x

2 – 4

0x +

A =

0A

1fo

r no

erro

rs o

r om

issi

ons

5(a)

(ii)

(x –

30)

(x –

10)

B2

B1

for x

(x –

30)

–10

(x –

30)

[= 0

]or

x(x

– 1

0) –

30(

x –

10) [

= 0]

or S

C1

for (

x +

a)(x

+ b

) whe

re a

b =

300

or

a +

b =

– 40

30, 1

0B

1

5(a)

(iii)

()

()(

)40

41200

2−

− o

r bet

ter

B1

Or f

or (x

– 2

0)2

p =

– – 4

0 an

d r =

2(1

) B

1M

ust s

ee

rp

q+

or

rp

q-

or b

oth

or fo

r 20 ±

200

5.86

34.1

4B

2If

B0,

SC

1 fo

r 5.9

or 5

.857

to 5

.858

and

34.1

or 3

4.14

…or

5.8

6 an

d 34

.14

seen

in w

orki

ngor

–5.

86 a

nd –

34.1

4 as

fina

l ans

wer

s

5(b)

(i)x

x200

10200

-+

M2

Or M

1 fo

r x200 o

r x10

200+

so

i

()

()

()

xx

xx

xx

10200

10200

102000

=+

+-

+A

1N

o er

rors

or o

mis

sion

s

5(b)

(ii)

16 (m

in) 4

0 (s

)3

B2

for .027o o

r 0.2

78 o

r 0.2

777

to 0

.277

8 or

185 [h

] o

e

or .

166o

or 1

6.7

or 1

6.66

to 1

6.67

or 350

[min

]

or M

1 fo

r 200

0 ÷

80(8

0 +

10) o

r 80200 –

90200



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

6(a)

(i)21p

1

6(a)

(ii)

21p

– 31r

1

6(a)

(iii)

p + 32r

1

6(b)

r + 23p

2M

1 fo

r cor

rect

uns

impl

ified

ans

wer

or

for c

orre

ct ro

ute

or fo

r rec

ogni

sing

OU

as p

ositi

on v

ecto

r

6(c)

6 n

fww

3B

2 fo

r (2k

)2 + ([

–]k)

2 = 1

80

oeor

M1

for (

2k)2 +

([–]

k)2

oe

Que

stio

nA

nsw

erM

arks

Part

ial M

arks

7(a)

22

M1

for 2

x +

1 =

1 +

4

7(b)

x21-

oe

fin

al a

nsw

er2

M1

for y

– 1

= 2

x or

y 2 =

x +

21

or x

= 2

y +

1

7(c)

4x2 +

4x

+ 5

fin

al a

nsw

er3

M1

for (

2x +

1)2 +

4an

d B

1 fo

r [(2

x +

1)2 =

] 4x2 +

2x

+ 2x

+ 1

or

bette

r

7(d)

2 o

r 1.4

1 or

1.4

14...

.1

7(e)

–11



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

8(a)

70.5

and

289.

54

B3

for o

ne c

orre

ct v

alue

or 2

cor

rect

val

ues n

ot

roun

ded

to 1

dec

imal

pla

ce

or M

2 fo

r cos

–131 cm

or M

1 fo

r cos

x = 31

If 0

scor

ed S

C1

for t

wo

solu

tions

whi

ch su

m to

36

0°

8(b)

90°

180°

270°

–1

–0.50

0.51

360°

x

y2

B1

for c

orre

ct sh

ape

but i

nacc

urat

e am

plitu

de o

r pe

riod



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

9(a)

45.[0

] or 4

5.01

to 4

5.02

nf

ww

4M

2 fo

r 552 +

702 –

2 ×

55 ×

70 co

s 40

or M

1 fo

r cor

rect

impl

icit

equa

tion

A1

for 2

026.

[...]

9(b)

84.9

or 8

4.90

to 8

4.91

4B

1 fo

r ang

le B

DC

= 4

0 s

oi

M2

for

()

sin

sintheir

3270

40

or M

1 fo

r cor

rect

impl

icit

equa

tion

9(c)

4060

or 4

063

to 4

064

nfw

w3

M2

for 21

(55 ×

70 si

n 40)

+ 21

(70 ×

thei

r (b)

sin (

180

– th

eir 4

0 –

32))

oe

or M

1 fo

r cor

rect

met

hod

for o

ne o

f the

tria

ngle

ar

eas

9(d)

35.4

or 3

5.35

...

nfw

w2

M1

for s

in 40

= 55

distance

or b

ette

r

or fo

r = 21

(55 ×

70 si

n 40)

= (7

0 ×

dist

ance

) ÷ 2

or b

ette

r



Que

stio

nA

nsw

erM

arks

Part

ial M

arks

10(a

)14

137

to 1

4 137

.2 o

r 14 1

392

M1

for 34

× π

× 1

53

10(b

)(i)

104 0

00 o

r 103

600

to 1

03 70

03

M2

for π

× 2

52 × 6

0 –

14 14

0or

M1

for π

× 2

52 × 6

0FT

π ×

37 5

00 =

117

809.

…al

low

thei

r ans

wer

as l

ong

as it

roun

ds to

14 1

40

10(b

)(ii)

52.8

or 5

2.75

to 5

2.81

…2

M1

for t

heir

(b)(

i) ÷

(π ×

252 )

or 1

4 140

÷ (π

× 2

52 )FT

π ×

252 =

196

3…(a

llow

use

of t

heir

ans

wer

as l

ong

as it

roun

ds to

14

140)

or 7

.198

to 7

.201

…

10(c

)()

()

xx

512

22

+M

1

[sla

nt h

eigh

t =] 1

3xA

1

π(5x

)2 or π

(5x)

(13x

)M

1A

ccep

t 25π

x2

π(5x

)2 + π

(5x)

(13x

) = 4πr

2M

1

r2 = 490 rr

x2 = 245 x2

A1

With

all

step

s sho

wn

and

no e

rror

s see

n

Que

stio

nA

nsw

erM

arks

Part

ial M

arks

11(a

)(0

, 16)

(4, –

16)

6M

1 fo

r 3x2 o

r 12x

A1

corr

ect 3

x2 – 1

2xB

1 se

tting

thei

r dy/

dx =

0M

1 fo

r fac

toris

ing

thei

r dy/

dxA

1 x

= 0

and

x =

4A

1 (0

, 16)

and

(4, –

16)

11(b

)(0

, 16)

max

imum

with

cor

rect

reas

on(4

, –16

) min

imum

with

cor

rect

reas

on3

B2

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