camrie teratia as a ee

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© UCLES 2017 [Turn over

Cambridge International AS & A Level

MATHEMATICS 9709/02Paper 2 Pure Mathematics 2 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

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

1(a)

Subs

titut

e x

= –1

and

equ

ate

to z

ero

1M

1

Obt

ain

answ

er a

= 7

1A

1

2

1(b)

Subs

titut

e x

= –3

and

eva

luat

e ex

pres

sion

1M

1

Obt

ain

answ

er 1

81

A1

2

Que

stio

nM

arks

Part

ial

Mar

ksG

uida

nce

2U

se si

n 2θ

= 2

sin θ

cos

θ1

B1

Sim

plify

to o

btai

n fo

rm c

1 sin

2 θ =

c 2 or e

quiv

alen

t1

M1

Find

at l

east

one

val

ue o

f θ fr

om e

quat

ion

of fo

rm si

n θ =

k1

M1

Obt

ain

35.3

° and

144

.7°

1A

1A

nd n

o ot

hers

bet

wee

n 0°

and

180

°.U

nsup

porte

d an

swer

rece

ives

0 m

arks

.

4


of 10© UCLES 2017

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

3(a)

(i)D

raw

two

V-sh

aped

gra

phs,

one

with

ver

tex

on n

egat

ive

x-ax

is, t

he o

ther

w

ith v

erte

x on

pos

itive

x-a

xis

1M

1

Show

gra

phs i

n co

rrec

t rel

atio

n to

eac

h ot

her,

with

line

s (ro

ughl

y) p

aral

lel

as a

ppro

pria

te1

A1

2

3(a)

(ii)

Stat

e co

ordi

nate

s (–2

a, 0

), ( 23

a, 0

), (0

, 4a)

, (0,

3a)

1B

1A

ccep

t val

ues i

nser

ted

at c

orre

ct p

oint

s on

axes

3(b)

Squa

re b

oth

side

s to

obta

in li

near

equ

atio

n or

ineq

ualit

y an

d at

tem

pt

solu

tion

O

R so

lve

linea

r equ

atio

n 2x

+ 4

a =

– (2x

– 3

a)O

R e

quiv

alen

t O

R u

se sy

mm

etry

of d

iagr

am to

det

erm

ine

x-co

ordi

nate

of p

oint

of

inte

rsec

tion

1M

1

Obt

ain

valu

e – 41

a an

d no

oth

er1

A1

Stat

e an

swer

x >

–41a

1A

1

3


of 10© UCLES 2017

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

4(a)

Rec

ogni

se q

uadr

atic

equ

atio

n in

5x a

nd a

ttem

pt so

lutio

n fo

r 5x

1M

1

Obt

ain

5x = 3

onl

y1

A1

Use

loga

rithm

s to

solv

e eq

uatio

n of

the

form

5x =

k w

here

k >

01

M1

Obt

ain

0.68

31

A1

4

4(b)

Use

2 ln

x =

ln x2

1B

1

Use

law

for a

dditi

on o

r sub

tract

ion

of lo

garit

hms

1M

1*

Arr

ange

equ

atio

n no

t inv

olvi

ng lo

garit

hms t

o th

e fo

rm y

= ..

.1

DM

1

Obt

ain

y =

x1

5 2 −1

A1

Or e

quiv

alen

t exp

licit

y =

... fo

rm

4


of 10© UCLES 2017

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

5(a)

Atte

mpt

use

of q

uotie

nt ru

le o

r equ

ival

ent

1M

1

Obt

ain

()

()cos

sin

xx

xx

22

22

22

+

+-

or e

quiv

alen

t1

A1

Equa

te n

umer

ator

to z

ero

and

atte

mpt

rear

rang

emen

t1

M1

Con

firm

giv

en re

sult

tan 2

x =

2x +

41

A1

AG

Sh

owin

g ne

cess

ary

deta

il

4

5(b)

Con

side

r sig

n of

tan 2

x –

2x –

4 fo

r 0.6

and

0.7

or e

quiv

alen

t1

M1

Obt

ain

–2.6

3 an

d 0.

40 o

r equ

ival

ent a

nd ju

stify

con

clus

ion

1A

1

2

5(c)

Use

iter

atio

n pr

oces

s cor

rect

ly a

t lea

st o

nce

1M

1

Obt

ain

final

ans

wer

0.6

941

A1

Show

suffi

cien

t ite

ratio

ns to

5 d

ecim

al p

lace

s to

just

ify a

nsw

er

OR

show

a si

gn c

hang

e in

the

inte

rval

(0.6

935,

0.6

945)

1A

1

3


of 10© UCLES 2017

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

6(a)

Use

pro

duct

rule

to d

iffer

entia

te y

1M

1

Obt

ain

dd ty = 4

et + 4

tet o

r equ

ival

ent

1A

1

Use

dddd

ddxy

tytx

'=

1M

1

Obt

ain

give

n an

swer

(

)dd

exy

t2

1t

=+

cor

rect

ly1

A1

AG

4

6(b)

Subs

titut

e t =

0 to

eva

luat

e de

rivat

ive

and

find

coor

dina

tes o

f poi

nt1

B1

Obt

ain

dd xy = 2

and

coo

rdin

ates

(1, 0

)1

B1

Form

equ

atio

n of

nor

mal

at t

heir

poin

t, us

ing

nega

tive

reci

proc

al o

f the

ir

dd xy

1M

1

Stat

e co

rrec

t equ

atio

n of

nor

mal

y =

–21x

+ 21 o

r equ

ival

ent

1A

1

4


of 10© UCLES 2017

Que

stio

nA

nsw

erM

arks

Part

ial

Mar

ksG

uida

nce

7(a)

Rep

lace

tan2 x

by se

c2 x –

11

B1

Expr

ess c

os2 x

in th

e fo

rm ±21 ±

21 co

s 2x

1M

1

Obt

ain

give

n an

swer

sec2 x

+ 21

cos 2

x – 21 c

orre

ctly

1A

1A

G

Show

ing

nece

ssar

y de

tail

Atte

mpt

inte

grat

ion

of e

xpre

ssio

n1

M1

Obt

ain

tan x

+ 41

sin 2

x – 21x

1A

1

Use

lim

its c

orre

ctly

for i

nteg

ral i

nvol

ving

at l

east

tan x

and

sin 2

x1

M1

Obt

ain 45

81−

π or

exa

ct e

quiv

alen

t1

A1

7

7(b)

Stat

e or

impl

y vo

lum

e is

(

)tan

cos

xx

xd2

r+

y1

B1

Pres

ence

of π

impl

ied

by it

s app

eara

nce

late

r if

not s

how

n in

itial

ly

Atte

mpt

exp

ansi

on a

nd si

mpl

ifica

tion

1M

1

Inte

grat

e to

obt

ain

one

addi

tiona

l ter

m o

f for

m k

cos x

1M

1

Obt

ain

π(45 –

81π)

+ π

(2 –

2

) or e

quiv

alen

t1

A1

4

camrie teratia as a ee

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