For all Queensland schools

2016 Senior External Examination

Mathematics A Thursday 27 October 2016

Paper Two — Question and response book 1:15 pm to 4:25 pm

Time allowed

• Perusal time: 10 minutes

• Working time: 3 hours

Examination materials provided

• Paper Two — Question and response book

• Paper Two — Resource book

Equipment allowed

• QCAA-approved equipment

• ruler (metric, parallel or rolling)

• protractor

• drawing compass

• set squares

• templates (without formulas)

• non-programmable calculator

• graphing calculator

Not allowed: Calculators with computer algebra system (CAS) functionality.

Directions

Do not write in this book during perusal time.

Paper Two has four extended-response questions. Attempt all questions.

Assessment

Paper Two assesses the following assessment criteria:

• Knowledge and procedures (KP)

• Modelling and problem solving (MP)

• Communication and justification (CJ)

Assessment standards are at the end of this book.

After the examination session

The supervisor will collect this book when you leave.

––

Candidate use

Print your candidate number here

Attach barcode here

Number of books used

1 6

Supervisor use only

QCAA use only

Supervisor’s initials

Marker number

Paper Two has four extended-response questions. Attempt all questions.

Write your responses in the spaces provided. Show full working in all responses. Partial credit can only be awarded if working is shown.

Additional pages for responses are at the back of this book.

Question 1

a. Steven invested $40000 in an account paying 4.8% p.a. interest compounding annually.

i. How much will he have in his account at the end of 15 years?

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

(KP)

i. How much interest will he have earned in that time?

.....................................................................................................

.....................................................................................................

.....................................................................................................

(KP)

b. The purchase price of a car is $15000. A deposit of $2000 is paid and the balance repaid with36 monthly payments of $500.

Calculate the annual simple rate of interest.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(KP)

1

c. A machine that makes boxes costs $45000. Its value depreciates by five cents for every box it makes. Each year it makes 120000 boxes.

Determine the depreciated value of this machine at the end of two years.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(KP)

d. Jayne purchased 5000 $2 shares for $3.40 per share. The company paid a dividend of $0.17 per share.

Find:

i. the percentage dividend.

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

(KP)

ii. the percentage yield.

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

(KP)

2

e. On 1 January 2016 Neville opened two separate accounts. On that day he invested $12500 into the first account which is paying 6% p.a. compounding monthly. He deposited a different amount into the second account which is paying 6.8% p.a. compounding monthly. Neville has estimated that the total of the amounts in both accounts on 1 January 2025 will be $60000, but not $30000 in each account.

If Neville can only invest in the second account in multiples of $100, determine the sum of money that he originally invested in this account to ensure the amount he wants to collect in 2025.

Identify one limitation of the mathematical model.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(MP)

3

Question 2

a. Two buildings on level ground, 120 metres apart, are joined by a cable. It is attached between the rooftops of the two buildings as shown in the diagram below.

Determine the length of the cable.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(KP)

b. The angle of elevation from a worm on the ground to a kookaburra’s feet in a tree is 60°. The kookaburra flies 30 metres to catch the worm as shown in the diagram below.

How high were the kookaburra’s feet above the ground?

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(KP)

120 m

110 m

Not to scale

20 m

Not to scale30m

4

c. Bread can be baked as either a standard sandwich loaf with square ends, or a tank loaf with round ends. They can be approximated by geometric solids as shown.

Loaves of bread with less surface area stay fresher for longer.

Determine which shape loaf of bread will stay freshest the longest. Justify your decision with mathematical working.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(KP)

32 cm

Sandwich loaf Tank loaf

32 cm

12 cm14 cm

Not to scale

5

d. A plane flying at an altitude of 10000 m reached a point 100 km horizontally from the end of the runway at an airport. The plane then begins its descent at an angle of depression of 7° until it reaches an altitude of 6500 m.

Air Traffic Control then instructs the pilot to fly level for 25 km before commencing the final descent to the runway.

Determine the angle of depression the plane flies for the final descent.

Identify one limitation of the mathematical model.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(MP)

6

e. Pontianak has a longitude of 109° E, and Jarvis Island has a longitude of 160° W.

Both places lie on the Equator.

i. Calculate the shortest distance between these two places, to the nearest km.

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

(KP)

ii. Suzanne, who lives at Pontianak, wants to telephone a friend on Jarvis Island.

If Suzanne calls at 8 am Monday, what is the time and day on Jarvis Island?

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

(KP)

iii. The position of Rabaul is 4° to the south and 48° to the west of Jarvis Island.

What is the latitude and longitude of Rabaul?

.....................................................................................................

.....................................................................................................

.....................................................................................................

.....................................................................................................

(KP)

7

Question 3

a. The square below represents a field with an area of 360000 m2.

The square has been accurately drawn to scale.

Determine, in its simplest ratio form, the scale that has been used.

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

(KP)

Drawn to scale

8

b. Using a scale of 1:10, the box below has been accurately drawn.

This box holds exactly 36 identical cans of soft drink. The cans fit upright into the box with no room to spare and are packed in two layers.

Determine the radius and height of a soft drink can.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(KP)

Drawn to scale

9

c. Two different plans for a gate were drawn as shown below.

Building regulations require bracing to be on an angle of between 37° and 53°.

Determine if each of the gate plans satisfy building regulations.

Justify your decisions with mathematical reasoning.

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

..........................................................................................................

(MP)

2.25 m 2.25 m 1.5 m 1.5 m 1.5 m

Not to scale

Gate A Gate B

1.65 m

10

Question 4

a. The diagram below shows the network of roads that Julie can use to travel between home and school.

The numbers on the roads show the time, in minutes, that it takes Julie to ride along each road.

Determine the shortest time that it will take Julie to ride her bicycle from home to school.

..........................................................................................................

..........................................................................................................

(KP)

b. Determine the slack time (in minutes) for activity D in the following network diagram. (A copy of the network diagram is reproduced at the back of this book if required.)

..........................................................................................................

..........................................................................................................

(KP)

Not to scale

school

home

24

8

723

2

6

2

8

13

3

Start Finish

A B C D G L

I

H

KFE

1 4 1

4

5 2

3 3 2

5

3

J 3

11

c. A table showing the activities for building a pergola is shown below.

i. Draw a project network for building the pergola.

(KP)

ii. Determine the critical path for the project.

......................................................................................................

......................................................................................................

(KP)

iii. If the pergola needs to be finished by 14 December, determine the date the project should commence.

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

......................................................................................................

(KP)

Number Activity Time (days) Prerequisites

A Clear area 1 None

B Lay concrete and let it dry 3 A

C Obtain materials 2 None

D Cut wood to required length 1 C

E Put up wooden structure 3 B, D

F Put up roof and gutters 2 E

G Paint pergola 3 F

H Put furniture in 1 G

I Landscape area 3 F

12

d. Lunch hours at Harry’s Hamburger Shop follow a similar pattern every day between 12 noon and 1 pm.

Three customers arrive together at the beginning of lunch and then a customer arrives every 3 minutes.

There is only one server and it takes 4 minutes to serve each customer.A customer will leave Harry’s Hamburger Shop without being served if the customer sees more than three people already waiting in the queue. The customer being served is not waiting in the queue.

i. Use the graph paper below to graph this scenario for 30 minutes.

(KP)

ii. At what time does the first customer leave without being served?

.....................................................................................................

(KP)

0

Cust

om

ers

Time (min)

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

13

iii. Harry decides to assist with serving and starts to serve at 12:10 pm.

Determine the length of time Harry needs to serve until a customer is served immediately upon entering the shop.

......................................................................................................

......................................................................................................

(MP)

End of Paper Two

0

Cust

om

ers

Time (min)

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

14

Spare graph paper (if required)

0

Cust

om

ers

Time (min)

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

15

Additional page for responses (if required)

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

Question 4b

Question

Start Finish

A B C D G L

I

H

KFE

1 4 1

4

5 2

3 3 2

5

3

J 3

16

Additional page for responses (if required)

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

...............................................................................................................

Question

17

Ass

essm

ent

stan

dar

ds

fro

m t

he

Mat

hem

atic

s A

Sen

ior

Ext

ern

al S

ylla

bu

s 20

06

Crit

erio

nA

BC

DE

Kno

wle

dge

and

proc

edur

es(K

P)

The

over

all q

ualit

y of

a c

andi

date

’s

achi

evem

ent a

cros

s th

e fu

ll ra

nge

with

in th

e co

ntex

ts o

f app

licat

ion,

te

chno

logy

and

com

plex

ity, a

nd

acro

ss to

pics

, con

sist

ently

de

mon

stra

tes:

•a

ccur

ate

reca

ll, s

elec

tion

and

use

of

defin

ition

s an

d ru

les

•use

of t

echn

olog

y

•rec

all a

nd s

elec

tion

of p

roce

dure

s,

and

thei

r acc

urat

e an

d pr

ofic

ient

use

.

The

over

all q

ualit

y of

a c

andi

date

’s

achi

evem

ent a

cros

s a

rang

e w

ithin

the

cont

exts

of a

pplic

atio

n, te

chno

logy

an

d co

mpl

exity

, and

acr

oss

topi

cs,

gene

rally

dem

onst

rate

s:•a

ccur

ate

reca

ll, s

elec

tion

and

use

of

defin

ition

s an

d ru

les

•use

of t

echn

olog

y

•rec

all a

nd s

elec

tion

of p

roce

dure

s,

and

thei

r acc

urat

e us

e.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent i

n th

e co

ntex

ts o

f app

licat

ion,

te

chno

logy

and

com

plex

ity,

gene

rally

dem

onst

rate

s:•a

ccur

ate

reca

ll an

d us

e of

ba

sic

defin

ition

s an

d ru

les

•use

of s

ome

tech

nolo

gy

•acc

urat

e us

e of

bas

ic

proc

edur

es.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent i

n th

e co

ntex

ts o

f app

licat

ion,

te

chno

logy

and

com

plex

ity,

som

etim

es d

emon

stra

tes:

•acc

urat

e re

call

and

use

of

som

e de

finiti

ons

and

rule

s

•use

of s

ome

tech

nolo

gy.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent

rare

ly d

emon

stra

tes

know

ledg

e an

d us

e of

pr

oced

ures

.

Mod

ellin

g an

d pr

oble

m s

olvi

ng(M

P)

The

over

all q

ualit

y of

a c

andi

date

’s

achi

evem

ent a

cros

s th

e fu

ll ra

nge

with

in e

ach

cont

ext,

and

acro

ss to

pics

ge

nera

lly d

emon

stra

tes

mat

hem

atic

al th

inki

ng w

hich

incl

udes

:•i

nter

pret

ing,

cla

rifyi

ng a

nd a

naly

sing

a

rang

e of

situ

atio

ns, a

nd id

entif

ying

va

riabl

es

•sel

ectin

g an

d us

ing

effe

ctiv

e st

rate

gies

•inf

orm

ed d

ecis

ion

mak

ing

… a

nd s

omet

imes

dem

onst

rate

s m

athe

mat

ical

thin

king

whi

ch in

clud

es:

•sel

ectin

g an

d us

ing

proc

edur

es to

so

lve

a w

ide

rang

e of

pro

blem

s

•ini

tiativ

e in

exp

lorin

g th

e pr

oble

m

•rec

ogni

sing

stre

ngth

s an

d lim

itatio

ns

of m

odel

s.

The

over

all q

ualit

y of

a c

andi

date

’s

achi

evem

ent a

cros

s a

rang

e w

ithin

ea

ch c

onte

xt, a

nd a

cros

s to

pics

, ge

nera

lly d

emon

stra

tes

mat

hem

atic

al th

inki

ng w

hich

incl

udes

:•i

nter

pret

ing,

cla

rifyi

ng a

nd a

naly

sing

a

rang

e of

situ

atio

ns, a

nd id

entif

ying

va

riabl

es

•sel

ectin

g an

d us

ing

stra

tegi

es

… a

nd s

omet

imes

dem

onst

rate

sm

athe

mat

ical

thin

king

whi

ch in

clud

es:

•sel

ectin

g an

d us

ing

proc

edur

es

requ

ired

to s

olve

a ra

nge

of

prob

lem

s

•inf

orm

ed d

ecis

ion

mak

ing.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent

dem

onst

rate

s m

athe

mat

ical

th

inki

ng w

hich

incl

udes

:•i

nter

pret

ing

and

clar

ifyin

g a

rang

e of

situ

atio

ns

•sel

ectin

g st

rate

gies

and

/or

proc

edur

es.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent

dem

onst

rate

s m

athe

mat

ical

th

inki

ng w

hich

incl

udes

fo

llow

ing

basi

c pr

oced

ures

and/

or u

sing

stra

tegi

es.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent

rare

ly d

emon

stra

tes

mat

hem

atic

al th

inki

ng w

hich

in

clud

es fo

llow

ing

basi

c pr

oced

ures

and

/or u

sing

st

rate

gies

.

18

(co

nti

nu

ed)

Crit

erio

nA

BC

DE

Com

mun

icat

ion

and

just

ifica

tion

(CJ)

The

over

all q

ualit

y of

a c

andi

date

’s

achi

evem

ent a

cros

s th

e fu

ll ra

nge

with

in e

ach

cont

ext c

onsi

sten

tly

dem

onst

rate

s:•a

ccur

ate

use

of m

athe

mat

ical

term

s an

d sy

mbo

ls

•acc

urat

e us

e of

lang

uage

•org

anis

atio

n of

info

rmat

ion

into

va

rious

form

s su

itabl

e fo

r a g

iven

us

e

•use

of m

athe

mat

ical

reas

onin

g to

de

velo

p lo

gica

l arg

umen

ts in

sup

port

of c

oncl

usio

ns, r

esul

ts a

nd/o

r de

cisi

ons

•jus

tific

atio

n of

pro

cedu

res.

The

over

all q

ualit

y of

a c

andi

date

’s

achi

evem

ent a

cros

s a

rang

e w

ithin

ea

ch c

onte

xt g

ener

ally

de

mon

stra

tes:

•acc

urat

e us

e of

mat

hem

atic

al te

rms

and

sym

bols

•acc

urat

e us

e of

lang

uage

•org

anis

atio

n of

info

rmat

ion

into

va

rious

form

s su

itabl

e fo

r a g

iven

us

e

•use

of m

athe

mat

ical

reas

onin

g to

de

velo

p si

mpl

e lo

gica

l arg

umen

ts in

su

ppor

t of c

oncl

usio

ns, r

esul

ts a

nd/

or d

ecis

ions

.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent i

n so

me

cont

exts

gen

eral

ly

dem

onst

rate

s:•a

ccur

ate

use

of b

asic

m

athe

mat

ical

term

s an

d sy

mbo

ls

•acc

urat

e us

e of

bas

ic

lang

uage

•org

anis

atio

n of

info

rmat

ion

into

var

ious

form

s

•use

of s

ome

mat

hem

atic

al

reas

onin

g to

dev

elop

sim

ple

logi

cal a

rgum

ents

.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent

som

etim

es d

emon

stra

tes

evid

ence

of t

he u

se o

f the

ba

sic

conv

entio

ns o

f la

ngua

ge a

nd m

athe

mat

ics.

The

over

all q

ualit

y of

a

cand

idat

e’s

achi

evem

ent

rare

ly d

emon

stra

tes

use

of

the

basi

c co

nven

tions

of

lang

uage

or m

athe

mat

ics.

19

© Th

Copyr

ManaPublisEmail

Que& APO BoLevel 7T + 61

www.

e State of Queensland (Queensland Curriculum and Assessment Authority) 2016

ight enquiries should be made to:

ger hing Unit

: publishing@qcaa.qld.edu.au

ensland Curriculumssessment Authorityx 307, Spring Hill QLD 4004 Australia

20

, 154 Melbourne Street, South Brisbane 7 3864 0299

qcaa.qld.edu.au