2019 Senior External Examination: Mathematics A Paper Two ...

2019 Senior External Examination

Mathematics APaper Two — Question and response book

Monday 28 October 20191:15 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.

DirectionsDo not write during perusal time.

Paper Two has four extended-response questions.

Attempt all questions.

AssessmentPaper 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 sessionThe supervisor will collect this book when you leave.

Candidate usePrint your candidate number here

Attach barcode here

Number of books used

Supervisor use only

Supervisor’s initials

QCAA use only

Marker number

1 9 – –

Planning space

12019 Mathematics A — Question and response book — Paper Two

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 be awarded only if working is shown.Additional pages for responses are at the back of this book.

Question 1a. Billy bought a home entertainment unit on a payment plan. The cash price of the unit was $3500.

Billy had to pay 12% deposit and repay the balance at 9% p.a. flat interest in equal monthly instalments for 2 years.

Calculate the amount of each monthly repayment.

(KP)

b. A car is purchased for $24 000. Use the diminishing value formula to calculate the salvage value of the car after 4 years at a depreciation rate of 15% p.a.

(KP)

Question 1 continues over next page ▬►

2 2019 Mathematics A — Question and response book — Paper Two

c. Steven and Michelle borrow $225 000 over 20 years at 4% p.a. reducible rate to purchase a house.

They make the required repayment each month, and after 7 years the balance of the loan has reduced to $195 000.

The interest rate then rises to 5% p.a.

Using the repayment table schedule below, determine whether they will need to increase the size of their monthly repayment and, if so, by how much.

Fully justify your decision with mathematical reasoning.

State one strength and one limitation for this situation.

Monthly repayment per $1000

Years 10 11 12 13 14 15 16 17 18 19 20

4% $12.67 $11.96 $11.38 $10.90 $10.49 $10.14 $9.85 $9.59 $9.36 $9.17 $9.00

5% $13.22 $12.52 $11.95 $11.48 $11.08 $10.75 $10.46 $10.21 $10.00 $9.81 $9.65

(MP)

d. A company’s prospectus predicts that the percentage dividend for the coming year will be 6.25%. Each share has a face value of $16.40.

Calculate the dividend paid if this percentage dividend is paid.

(KP)


Question 2a. Below are the final heights recorded for two athletes in eight high-jump competitions.

Carl 1.80 m 1.80 m 1.90 m 1.90 m 1.70 m 1.80 m 2.00 m 1.60 m

Richard 2.00 m 1.90 m 1.70 m 2.10 m 2.10 m 1.80 m 1.90 m 2.10 m

i. Calculate the mean height for Carl and the mean height for Richard.

Carl: Richard:

ii. Calculate the interquartile range for Carl and the interquartile range for Richard.

Carl:

Richard:

iii. Calculate the sample standard deviation for each student’s heights.

Carl: Richard:

iv. Determine which student is the most consistent and give a reason for your choice.

v. List the five-number summary for Carl and the five-number summary for Richard.

Carl:

Richard:

vi. Draw adjacent boxplots to represent this data on the scale below.

Jump heights


(KP)

(KP)

(KP)

(KP)

(KP)

(KP)


b. The following data represents the number of goals scored in 16 soccer games.

0 1 0 2 3 0 4 3 3 1 3 0 0 2 4 4

i. Complete the frequency table.

Goals scored Tally Frequency

ii. Draw a frequency histogram to represent this data using the axes below.

iii. Find the mean, median, mode and range for this data.

Mean:

Median:

Mode:

Range:

iv. Describe this data using two words, one from each list below.

List 1: continuous, discrete List 1 word:

List 2: categorical, ordinal List 2 word:(KP)

(KP)

(KP)

(KP)

Spare axes are provided on page 16.


c. A class of 30 students sat for an Algebra test and a Geometry test. The results are displayed in a scatterplot with a line of best fit below.

i. How many students scored less than 30 on both the Algebra and Geometry tests?

ii. Describe the correlation between the Algebra and Geometry test results.

iii. Explain why the following statement is incorrect.

In this class all students who are near the top in Algebra are also near the top in Geometry.

(KP)

(KP)

(MP)


Question 3a. Below is a floor plan for a house.

i. If the length of the southern wall of the kitchen is 3 m, find the scale used for this plan.

(KP)

ii. What are the actual dimensions of the living room?

(KP)


iii. The floor of the bathroom is to be tiled. A 600 mm × 600 mm tile costs $29.80 each. Allowing an extra 10% for wastage, calculate the cost of tiling the bathroom.

iv. Wallpaper is to be used to cover the southern and western walls of the bedroom. The walls are 2.4 m high. A 10 m long wallpaper roll is 0.53 m in width.

Calculate the minimum number of rolls of wallpaper needed to cover both the southern and western walls of the bedroom.


(KP)

(KP)


v. Determine the amount of concrete needed for the footings of this house if they are 350 mm wide and 200 mm deep.

(KP)

Note: This is a duplicate image from page 6.


vi. Calculate the amount of concrete needed for the slab of this house if it is to be 10 cm deep.

vii. The builder has estimated the concrete for the slab and footings will cost $2300.

Concrete costs $300 / m3. Make an informed decision as to whether the builder has allowed sufficient money to pay for the concrete.

(KP)

(MP)



b. After pegging out a rectangular shed, a builder measures one pair of opposite sides of the shed to be 4.8 m and the other pair of opposite sides to be 3.6 m.

Use diagrams and mathematical reasoning to explain how the builder can be sure that the shed is rectangular.

(MP)


Question 4a. The following network shows the completion time for a project in days.

i. Find the minimum time to complete the project.

ii. State the critical path.

b.

The above network shows connecting paths for a cable network. It costs $65 per metre to lay the cable. Use a minimum spanning tree to determine the least cost to connect this network.


(KP)

(KP)

(KP)

all distances are in metres

not to scale

not to scale


c. The graph below illustrates the waiting time in a queue across a 60-minute period.

i. How many customers were waiting to be served 19 minutes after the start of the 60-minute period?

(KP)

ii. What is the longest time a customer has to queue before being served?

(KP)

d. Spectators arrive at the rate of 60 per minute at a football game. If it takes 20 seconds to check each person through security, how many service points are needed if queues are not to grow?

(KP)


e. The table below shows the distances in metres between exhibits at a garden show.

A B C D E F GA – 50 70 85 – – –B – – 30 – 70 – –C – – – 50 65 – –D – – – – 45 90 60E – – – – – 40 –F – – – – – – 80

i. Construct a network diagram to represent the information displayed in the table.

(KP)

ii. All exhibits are to be visited only once. This means that visitors enter the show and progress through the exhibits to the exit without revisiting exhibits. Determine if the minimal spanning tree will identify the route that satisfies these conditions. Fully justify your decision with mathematical reasoning and a network diagram.

(MP)


f. A drive-through fast food outlet has two service lanes but only one operator. It takes one minute for the operator to take the order and then a further two minutes to give the customer the order at the service window.

Up to five cars can fit between the order window and the service window in each service lane. When the outlet opens at 6:00 am there are three cars waiting in each service lane to order.

For the first ten minutes after opening, a car arrives every minute. Following this, a car arrives every two minutes.

i. Respond to this question on page 15. A spare grid is provided on page 17. Graph a scenario for the first 30 minutes the outlet opens.

ii. Determine the first time the queue becomes empty.

iii. Calculate the average waiting time between 6:00 am and 6:30 am.

End of Paper Two(KP)

(KP)

(KP)


Question 4f i.


Spare axes (if required)


Spare grid (if required)


Question

Additional page for responses (if required)


Question



Question


Asse

ssm

ent s

tand

ards

from

the M

athe

mat

ics A

Sen

ior E

xter

nal S

yllab

us 200

6

Crite

rion

AB

CD

E

Know

ledg

e and

pr

oced

ures

(KP)

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

acro

ss th

e ful

l ran

ge w

ithin

th

e con

text

s of a

pplic

atio

n,

tech

nolo

gy an

d co

mpl

exity

, an

d ac

ross

topi

cs, c

onsis

tent

ly de

mon

stra

tes:

• ac

cura

te re

call,

sele

ctio

n an

d us

e of d

efin

ition

s and

rule

s•

use o

f tec

hnol

ogy

• re

call a

nd se

lect

ion

of

proc

edur

es, a

nd th

eir

accu

rate

and

prof

icien

t use

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent a

cros

s a r

ange

with

in th

e con

text

s of

appl

icatio

n, te

chno

logy

and

com

plex

ity, a

nd ac

ross

topi

cs,

gene

rally

dem

onst

rate

s:•

accu

rate

reca

ll, se

lect

ion

and

use o

f def

initi

ons a

nd ru

les

• us

e of t

echn

olog

y•

reca

ll and

sele

ctio

n of

pr

oced

ures

, and

thei

r ac

cura

te u

se.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent i

n th

e con

text

s of a

pplic

atio

n,

tech

nolo

gy an

d co

mpl

exity

, ge

nera

lly d

emon

stra

tes:

• ac

cura

te re

call a

nd u

se of

ba

sic d

efin

ition

s and

rule

s•

use o

f som

e tec

hnol

ogy

• ac

cura

te u

se of

bas

ic pr

oced

ures

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent i

n th

e con

text

s of a

pplic

atio

n,

tech

nolo

gy an

d co

mpl

exity

, so

met

imes

dem

onst

rate

s:•

accu

rate

reca

ll and

use

of

som

e def

initi

ons a

nd ru

les

• us

e of s

ome t

echn

olog

y.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent r

arel

y de

mon

stra

tes k

nowl

edge

and

use o

f pro

cedu

res.

Mod

ellin

g and

pr

oble

m so

lving

(MP)

The o

vera

ll qua

lity o

f a

cand

idat

e’s ac

hiev

emen

t acro

ss

the f

ull r

ange

with

in ea

ch co

ntex

t, an

d acro

ss to

pics

gene

rally

de

mon

stra

tes m

athe

mat

ical

thin

king w

hich

inclu

des:

• in

terp

retin

g, cl

arify

ing

and

anal

ysin

g a ra

nge o

f sit

uatio

ns, a

nd id

entif

ying

varia

bles

• se

lect

ing a

nd u

sing e

ffect

ive

stra

tegie

s•

info

rmed

dec

ision

mak

ing

… and

som

etim

es

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• se

lect

ing a

nd u

sing

proc

edur

es to

solve

a wi

de

rang

e of p

robl

ems

• in

itiat

ive in

expl

orin

g the

pr

oble

m•

reco

gnisi

ng st

reng

ths a

nd

limita

tions

of m

odel

s.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent a

cros

s a r

ange

with

in ea

ch co

ntex

t, an

d ac

ross

topi

cs, g

ener

ally

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• in

terp

retin

g, cl

arify

ing

and

anal

ysin

g a ra

nge o

f sit

uatio

ns, a

nd id

entif

ying

varia

bles

• se

lect

ing a

nd u

sing s

trate

gies

… and

som

etim

es

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• se

lect

ing a

nd u

sing

proc

edur

es re

quire

d to

solve

a r

ange

of p

robl

ems

• in

form

ed d

ecisi

on m

akin

g.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• in

terp

retin

g and

clar

ifyin

g a

rang

e of s

ituat

ions

• se

lect

ing s

trate

gies a

nd/o

r pr

oced

ures

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des

follo

wing

bas

ic pr

oced

ures

and/

or u

sing s

trate

gies.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent r

arel

y de

mon

stra

tes m

athe

mat

ical

thin

king

whi

ch in

clude

s fo

llowi

ng b

asic

proc

edur

esan

d/or

usin

g stra

tegie

s.


Asse

ssm

ent s

tand

ards

from

the M

athe

mat

ics A

Sen

ior E

xter

nal S

yllab

us 200

6

Crite

rion

AB

CD

E

Know

ledg

e and

pr

oced

ures

(KP)

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

acro

ss th

e ful

l ran

ge w

ithin

th

e con

text

s of a

pplic

atio

n,

tech

nolo

gy an

d co

mpl

exity

, an

d ac

ross

topi

cs, c

onsis

tent

ly de

mon

stra

tes:

• ac

cura

te re

call,

sele

ctio

n an

d us

e of d

efin

ition

s and

rule

s•

use o

f tec

hnol

ogy

• re

call a

nd se

lect

ion

of

proc

edur

es, a

nd th

eir

accu

rate

and

prof

icien

t use

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent a

cros

s a r

ange

with

in th

e con

text

s of

appl

icatio

n, te

chno

logy

and

com

plex

ity, a

nd ac

ross

topi

cs,

gene

rally

dem

onst

rate

s:•

accu

rate

reca

ll, se

lect

ion

and

use o

f def

initi

ons a

nd ru

les

• us

e of t

echn

olog

y•

reca

ll and

sele

ctio

n of

pr

oced

ures

, and

thei

r ac

cura

te u

se.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent i

n th

e con

text

s of a

pplic

atio

n,

tech

nolo

gy an

d co

mpl

exity

, ge

nera

lly d

emon

stra

tes:

• ac

cura

te re

call a

nd u

se of

ba

sic d

efin

ition

s and

rule

s•

use o

f som

e tec

hnol

ogy

• ac

cura

te u

se of

bas

ic pr

oced

ures

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent i

n th

e con

text

s of a

pplic

atio

n,

tech

nolo

gy an

d co

mpl

exity

, so

met

imes

dem

onst

rate

s:•

accu

rate

reca

ll and

use

of

som

e def

initi

ons a

nd ru

les

• us

e of s

ome t

echn

olog

y.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent r

arel

y de

mon

stra

tes k

nowl

edge

and

use o

f pro

cedu

res.

Mod

ellin

g and

pr

oble

m so

lving

(MP)

The o

vera

ll qua

lity o

f a

cand

idat

e’s ac

hiev

emen

t acro

ss

the f

ull r

ange

with

in ea

ch co

ntex

t, an

d acro

ss to

pics

gene

rally

de

mon

stra

tes m

athe

mat

ical

thin

king w

hich

inclu

des:

• in

terp

retin

g, cl

arify

ing

and

anal

ysin

g a ra

nge o

f sit

uatio

ns, a

nd id

entif

ying

varia

bles

• se

lect

ing a

nd u

sing e

ffect

ive

stra

tegie

s•

info

rmed

dec

ision

mak

ing

… and

som

etim

es

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• se

lect

ing a

nd u

sing

proc

edur

es to

solve

a wi

de

rang

e of p

robl

ems

• in

itiat

ive in

expl

orin

g the

pr

oble

m•

reco

gnisi

ng st

reng

ths a

nd

limita

tions

of m

odel

s.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent a

cros

s a r

ange

with

in ea

ch co

ntex

t, an

d ac

ross

topi

cs, g

ener

ally

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• in

terp

retin

g, cl

arify

ing

and

anal

ysin

g a ra

nge o

f sit

uatio

ns, a

nd id

entif

ying

varia

bles

• se

lect

ing a

nd u

sing s

trate

gies

… and

som

etim

es

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• se

lect

ing a

nd u

sing

proc

edur

es re

quire

d to

solve

a r

ange

of p

robl

ems

• in

form

ed d

ecisi

on m

akin

g.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• in

terp

retin

g and

clar

ifyin

g a

rang

e of s

ituat

ions

• se

lect

ing s

trate

gies a

nd/o

r pr

oced

ures

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des

follo

wing

bas

ic pr

oced

ures

and/

or u

sing s

trate

gies.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent r

arel

y de

mon

stra

tes m

athe

mat

ical

thin

king

whi

ch in

clude

s fo

llowi

ng b

asic

proc

edur

esan

d/or

usin

g stra

tegie

s.


(cont

inue

d)

Crite

rion

AB

CD

E

Com

mun

icatio

n an

d ju

stifi

catio

n(C

J)

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

acro

ss th

e ful

l ran

ge w

ithin

ea

ch co

ntex

t con

siste

ntly

dem

onst

rate

s:•

accu

rate

use

of m

athe

mat

ical

term

s and

sym

bols

• ac

cura

te u

se of

lang

uage

• or

gani

satio

n of

info

rmat

ion

into

vario

us fo

rms s

uita

ble f

or

a give

n us

e•

use o

f mat

hem

atica

l re

ason

ing t

o dev

elop

logic

al

argu

men

ts in

supp

ort o

f co

nclu

sions

, res

ults

and/

or

decis

ions

• ju

stifi

catio

n of

pro

cedu

res.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent a

cros

s a r

ange

with

in ea

ch co

ntex

t ge

nera

lly d

emon

stra

tes:

• ac

cura

te u

se of

mat

hem

atica

l te

rms a

nd sy

mbo

ls•

accu

rate

use

of la

ngua

ge•

orga

nisa

tion

of in

form

atio

n in

to va

rious

form

s sui

tabl

e for

a g

iven

use

• us

e of m

athe

mat

ical

reas

onin

g to d

evel

op si

mpl

e lo

gical

argu

men

ts in

supp

ort

of co

nclu

sions

, res

ults

and/

or

decis

ions

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

in so

me c

onte

xts g

ener

ally

dem

onst

rate

s:•

accu

rate

use

of b

asic

mat

hem

atica

l ter

ms a

nd

sym

bols

• ac

cura

te u

se of

bas

ic la

ngua

ge•

orga

nisa

tion

of in

form

atio

n in

to va

rious

form

s•

use o

f som

e mat

hem

atica

l re

ason

ing t

o dev

elop

sim

ple

logic

al ar

gum

ents

.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

som

etim

es d

emon

stra

tes

evid

ence

of th

e use

of th

e bas

ic co

nven

tions

of la

ngua

ge an

d m

athe

mat

ics.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent r

arel

y de

mon

stra

tes u

se of

the b

asic

conv

entio

ns of

lang

uage

or

mat

hem

atics

.

© The State of Queensland (Queensland Curriculum and Assessment Authority) 2019

Copyright enquiries should be made to:

ManagerPublishing UnitEmail: publishing@[email protected]

Queensland Curriculum& Assessment AuthorityPO Box 307, Spring Hill QLD 4004 AustraliaLevel 7, 154 Melbourne Street, South BrisbaneT +61 7 3864 0299

www.qcaa.qld.edu.au

2019 Senior External Examination: Mathematics A Paper Two ...

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