2016 senior external examination mathematics a paper two — … · 2016-12-01 · 2016 senior...
TRANSCRIPT
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?
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(KP)
i. How much interest will he have earned in that time?
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(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.
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(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.
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(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.
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(KP)
ii. the percentage yield.
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(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.
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(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.
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(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?
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(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.
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(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.
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(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.
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(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?
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(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?
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(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.
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(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.
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(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.
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(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.
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(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.)
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(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.
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(KP)
iii. If the pergola needs to be finished by 14 December, determine the date the project should commence.
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(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?
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(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.
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(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)
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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)
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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
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ight enquiries should be made to:
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