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Cambridge International AS & A Level

MATHEMATICS 9709/04Paper 4 Mechanics For examination from 2020MARK SCHEME

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of 10© 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)

.


of 10© UCLES 2017

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.

Mar

k Sc

hem

e N

otes

Mar

ks a

re o

f the

follo

win

g th

ree

type

s.

M

Met

hod

mar

k, g

iven

for a

val

id m

etho

d ap

plie

d to

the

prob

lem

. Met

hod

mar

ks c

an st

ill b

e gi

ven

even

if th

ere

are

num

eric

al e

rror

s, al

gebr

aic

slip

s or

erro

rs in

uni

ts. H

owev

er th

e m

etho

d m

ust b

e ap

plie

d to

the

spec

ific

prob

lem

, e.g

. by

subs

titut

ing

the

rele

vant

qua

ntiti

es in

to a

form

ula.

Cor

rect

use

of

a fo

rmul

a w

ithou

t the

form

ula

bein

g qu

oted

ear

ns th

e M

mar

k an

d in

som

e ca

ses a

n M

mar

k ca

n be

impl

ied

from

a c

orre

ct a

nsw

er.

A

Acc

urac

y m

ark,

giv

en fo

r an

accu

rate

ans

wer

or a

ccur

ate

inte

rmed

iate

step

follo

win

g a

corr

ect m

etho

d. A

ccur

acy

mar

ks c

anno

t be

give

n un

less

the

rele

vant

met

hod

mar

k ha

s als

o be

en g

iven

.B

M

ark

for a

cor

rect

stat

emen

t or s

tep.

DM

or D

B

M m

arks

and

B m

arks

are

gen

eral

ly in

depe

nden

t of e

ach

othe

r. Th

e no

tatio

n D

M o

r DB

mea

ns a

par

ticul

ar M

or B

mar

k is

dep

ende

nt o

n an

ear

lier M

or

B m

ark

(indi

cate

d by

*).

Whe

n tw

o or

mor

e st

eps a

re ru

n to

geth

er b

y th

e ca

ndid

ate,

the

earli

er m

arks

are

impl

ied

and

full

cred

it is

giv

en.

•A

or B

mar

ks a

re g

iven

for c

orre

ct w

ork

only

(not

for r

esul

ts o

btai

ned

from

inco

rrec

t wor

king

) unl

ess f

ollo

w th

roug

h is

allo

wed

(see

abb

revi

atio

n FT

bel

ow).

•W

rong

or m

issi

ng u

nits

in a

n an

swer

shou

ld n

ot re

sult

in lo

ss o

f mar

ks u

nles

s the

gui

danc

e in

dica

tes o

ther

wis

e. •

For a

num

eric

al a

nsw

er, a

llow

the A

or B

mar

k if

the

answ

er is

cor

rect

to 3

sign

ifica

nt fi

gure

s (sf

) or w

ould

be

corr

ect t

o 3

sf if

roun

ded

(1 d

ecim

al p

oint

(dp)

fo

r ang

les i

n de

gree

s). A

s sta

ted

abov

e, a

n A

or B

mar

k is

not

giv

en if

a c

orre

ct n

umer

ical

ans

wer

is o

btai

ned

from

inco

rrec

t wor

king

. •

Com

mon

alte

rnat

ive

solu

tions

are

show

n in

the A

nsw

er c

olum

n as

: ‘E

ITH

ER

Sol

utio

n 1

OR

Sol

utio

n 2

OR

Sol

utio

n 3

…’.

Rou

nd b

rack

ets a

ppea

r in

the

Parti

al M

arks

col

umn

arou

nd th

e m

arks

for e

ach

alte

rnat

ive

solu

tion.

•

Squa

re b

rack

ets [

] a

roun

d te

xt sh

ow e

xtra

info

rmat

ion

not n

eede

d fo

r the

mar

k to

be

awar

ded.

•Th

e to

tal n

umbe

r of m

arks

ava

ilabl

e fo

r eac

h qu

estio

n is

show

n at

the

botto

m o

f the

Mar

ks c

olum

n in

bol

d ty

pe.

The

follo

win

g ab

brev

iatio

ns m

ay b

e us

ed in

a m

ark

sche

me.

AG

A

nsw

er g

iven

on

the

ques

tion

pape

r (so

ext

ra c

heck

ing

is n

eede

d to

ens

ure

that

the

deta

iled

wor

king

lead

ing

to th

e re

sult

is v

alid

).C

AO

C

orre

ct a

nsw

er o

nly

(em

phas

isin

g th

at n

o ‘f

ollo

w th

roug

h’ fr

om a

n er

ror i

s allo

wed

).C

WO

C

orre

ct w

orki

ng o

nly

FT

Fo

llow

thro

ugh

afte

r err

or (s

ee M

ark

Sche

me

Not

es fo

r fur

ther

det

ails

). IS

W

Igno

re su

bseq

uent

wor

king

OE

O

r equ

ival

ent f

orm

SC

Sp

ecia

l cas

eSO

I

Seen

or i

mpl

ied


of 10© UCLES 2017

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Part

ial

Mar

ksG

uida

nce

1(a)

0 =

202 –

20s

1M

1Fo

r usi

ng v

2 = u

2 + 2

as w

ith a

= –

10 to

find

s, th

e gr

eate

st h

eigh

t rea

ched

s = 2

0 m1

A1

2

1(b)

0 =

20 –

10t

t = 2

1M

1Fo

r usi

ng v

= u

+ a

t with

v =

0, u

= 2

0 an

d a

= –1

0 to

find

the

time

to th

e gr

eate

st h

eigh

t

Tota

l tim

e 4 s

1A

1

2

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

2(a)

[DF

= 13

50]

Pow

er =

135

0 ×

321

M1

= 43

.2 kW

1A

1

2

2(b)

DF

– 13

50 +

120

0g ×

201 =

0[D

F =

750]

1M

1Fo

r usi

ng N

ewto

n’s 2

nd la

w a

pplie

d to

the

car

dow

n th

e hi

ll (3

term

s)A

llow

use

of θ

= 2

.9°

31 50

0 =

thei

r (75

0) ×

v1

M1

For u

sing

DF

= vP

v =

42 m

s–11

A1

3


of 10© UCLES 2017

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ial

Mar

ksG

uida

nce

3(a)

Con

serv

atio

n of

mom

entu

m4 ×

6 [+

0] =

4 ×

2 +

2v

1M

1Fo

r app

lyin

g co

nser

vatio

n of

mom

entu

m

v =

8 [m

s–1]

1A

1

2

3(b)

2 ×

thei

r (8)

[+ 0

] = 2

v +3

v1

M1

For a

pply

ing

cons

erva

tion

of m

omen

tum

v =

3.2 [

m s–1

]1

A1

2

3(c)

Kin

etic

ene

rgy

(KE)

initi

al =

21 ×

4 ×

62

KE

final

= 21

× 4

× 2

2 + 21

× 5

× 3

.22

1M

1Fo

r use

of 21 ×

m ×

v2 , w

hen

eith

er in

itial

or f

inal

ca

lcul

atio

n co

rrec

t usi

ng th

eir v

alue

from

(b)

Loss

of K

E =

72 –

33.

6 =

38.4

[J]

1A

1A

G

For a

ll co

rrec

t

2


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ial

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uida

nce

4(a)

Cor

rect

forc

e di

agra

m w

ith 3

ext

ra fo

rces

show

n

25

200

RF

1B

1A

ccep

t 200

or 2

0 g fo

r wei

ght

4(b)

For r

esol

ving

forc

es in

the

dire

ctio

n pa

ralle

l to

and/

or p

erpe

ndic

ular

to th

e pl

ane

1M

1

F +

25 co

s 20°

= 2

0 ×

g ×

sin 3

0°1

A1

R +

25 si

n 20°

= 2

0 ×

g ×

cos 3

0°1

A1

[F =

76.

5...]

[R =

164

.65.

..]

μ =

(....)

(....)

their

their 16465

765

1M

1U

se F

= μR

to e

valu

ate μ

=

0.46

51

A1

5


of 10© UCLES 2017

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ial

Mar

ksG

uida

nce

5(a)

2500

– 2

000g

× 0

.1 –

400

= 2

000a

1M

1Fo

r New

ton’

s 2nd

law

a =

0.05

m s–2

1A

1

2500

– T

– 3

00 –

120

0g ×

0.1

= 1

200 ×

0.05

OR

T –

100

– 80

0g ×

0.1

= 8

00 ×

0.0

5

1M

1Fo

r New

ton’

s 2nd

law

to e

ither

car

or t

raile

r to

give

an

equa

tion

for T

.

T =

940 N

1A

1

4

5(b)

–200

0g ×

0.1

– 4

00 =

200

0a[a

= –

1.2]

1M

1Fo

r New

ton’

s 2nd

law

0 =

30 –

1.2

t1

M1

For u

sing

v =

u +

at

t = 2

5 s1

A1

T –

100

– 80

0g ×

0.1

= 8

00 ×

–1.

21

M1

For a

pply

ing

New

ton’

s sec

ond

law

to th

e tra

iler

or to

the

car

T =

–60 N

1

A1

Acc

ept (

thru

st) =

60N

5


of 10© UCLES 2017

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ial

Mar

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uida

nce

6(a)

k =

401

B1

6(b)

t

v

O40 28

414

20

, ,y

xx

y yx

x x x

510

40 682

04

414

1420

for

for

for

2GG

GG

GG

− −

= = =

R TS S S S

V XW W W W

1B

1FT

Cor

rect

for 0

⩽ t ⩽

4

For q

uad

grap

h, m

in a

t t =

1;

FT

on k

1B

1FT

Cor

rect

for 4

⩽ t ⩽

14

For h

oriz

onta

l lin

e at

v =

k;

FT

on k

1B

1FT

Cor

rect

for 1

4 ⩽

t ⩽

20

For l

ine

with

neg

ativ

e gr

adie

nt fr

om (1

4, k

) to

(20,

28)

[gra

dien

t = –

2];

FT

on k

3

6(c)

a =

10t –

10

1M

1Fo

r atte

mpt

ing

to d

iffer

entia

te to

find

a

1 <

t < 4

1A

1A

llow

⩽ a

t 4 b

ut n

ot a

t 1

2

6(d)

Dis

t = (

)(

)d

dt

tt

tt

t5

105

102

022

24−

−+

yy

+ (4

0 ×

10) +

(0.5

× (4

0 +

28) ×

6)

1M

1Fo

r atte

mpt

ing

to in

tegr

ate

1A

1Fo

r cor

rect

exp

ress

ion

incl

udin

g lim

its a

nd th

e di

stan

ce tr

avel

led

over

full

20 s

1B

1Fo

r cor

rect

inte

grat

ion

tt

tt

55

353

202

353

224

−−

=+

88

BB

+ 40

0 +

204

1A

1FT

FT o

n th

eir k

= 64

4 m1

A1

5


of 10© UCLES 2017

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ial

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uida

nce

7(a)

EIT

HE

R S

olut

ion

1T

= 0.

8a

for A

0.2g

– T

= 0

.2a

fo

r B0.

2g =

(0.2

+ 0

.8)a

fo

r sys

tem

2(M

1M1

M1

for a

pply

ing

New

ton’

s 2nd

law

eith

er to

pa

rticl

e A

or to

par

ticle

B o

r to

the

syst

em

M1

for a

pply

ing

New

ton’

s 2nd

law

to a

seco

nd

parti

cle

(if n

eede

d) a

nd so

lvin

g fo

r a

a =

21

A1

2.5

= 0.

5 ×

2 ×

t21

M1

For u

sing

a c

ompl

ete

met

hod

for f

indi

ng t

such

as

usin

g s =

ut +

21at

2

t = 1

.58 s

1A

1)A

llow

t =

1021

OR

Sol

utio

n 2

0.2 ×

g ×

2.5

OR

21(0

.2 +

0.8

)v2

1(M

1Fo

r fin

ding

pot

entia

l ene

rgy

(PE)

loss

or K

E ga

in

(sys

tem

)

0.2 ×

g ×

2.5

= 21(0

.2 +

0.8

)v2

1M

1Fo

r usi

ng P

E lo

ss =

KE

gain

and

find

ing

v

v2 = 1

01

A1

.t

25

010

21+

=^

h1

M1

For u

sing

s = 21(u

+ v

)t

t = 1

.58 s

1A

1)A

llow

t =

1021

OR

Sol

utio

n 3

T =

0.8a

0.

2g –

T =

0.2

a

→

T =

1.6

N

1(M

1Fo

r app

lyin

g N

ewto

n’s 2

nd la

w to

A a

nd B

and

so

lvin

g fo

r T

T ×

2.5

= 21(0

.8)v

21

M1

For u

sing

wor

k do

ne (W

D) b

y T

= K

E ga

in b

y A,

fin

d v

v2 = 1

01

A1

.t

25

010

21=

+_

i1

M1

For u

sing

s = 21(u

+ v

)t

t = 1

.58 s

1A

1)A

llow

t =

1021

Avai

labl

e m

arks

5


of 10© UCLES 2017

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ial

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uida

nce

7(b)

0.2 ×

g ×

2.5

= 21(0

.8 +

0.2

)v2 +

22

M1A

1M

1 Fo

r app

lyin

g w

ork/

ener

gy to

the

syst

em a

s PE

loss

= K

E ga

in +

WD

aga

inst

resi

stan

ceA

1 Fo

r cor

rect

wor

k/en

ergy

equ

atio

n

For s

olvi

ng fo

r v1

M1

v =

6 (o

r 2.4

5)1

A1

4

Camrie nernaina AS A ee

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Transcript of Camrie nernaina AS A ee