2019 senior external examination: mathematics b paper one ...€¦ · 2019 mathematics — uestion...

2019 Senior External Examination

Mathematics BPaper One — Question book

Thursday 24 October 20199 am to 12:10 pm

Time allowed• Perusal time: 10 minutes

• Working time: 3 hours

Examination materials provided• Paper One — Question book

• Paper One — Response book

• Paper One — Resource book

Equipment allowed• QCAA-approved equipment

• ruler graduated in millimetres

• protractor

• graphics calculator

• additional calculator

Equipment not allowed• calculators with computer alegbra system (CAS) functionality

DirectionsDo not write during perusal time.

Paper One has six questions. Attempt all questions.

AssessmentPaper One 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.

Planning space

12019 Mathematics B — Question book — Paper One

Paper One has six questions. Attempt all questions.Each question assesses Knowledge and procedures (KP), Modelling and problem solving (MP), or a combination of both. Communication and justification (CJ) will be assessed by an overall judgment of your responses to all questions.Write your responses in the response book. Show full working to meet the standards for each criterion.

Question 1a. The ages of nine adults in a group are:

57, 26, 19, 25, 24, 28, 36, 25, 72

i. State the mode.

ii. Calculate the mean and standard deviation of the data.

b. Two sprinters have the following training times in seconds for a 100 metre sprint.

Sprinter A 12.3 12.4 12.5 12.3 12.4 12.8 13.1 12.1 12.7 12.5

Sprinter B 12.1 12.0 12.0 14.2 12.1 12.2 12.4 12.1 12.3 12.4

i. Respond to this question on page 22 of your response book. Represent the data as parallel boxplots.

ii. The coach of the two sprinters believes that Sprinter A is a more consistent sprinter.Evaluate the validity of the coach’s opinion by referring to the data sets.

(KP)

(KP)

(MP)

2 2019 Mathematics B — Question book — Paper One

Question 2a. The following equation describes the relationship f (x)

2 2 2, 3 1( )1, 1 6

− + − ≤ ≤= + < <

x x xf xx x

i. Determine the domain and range of f (x) .

ii. Determine if f (x) is both continuous and a function. Give appropriate reasons.

b. The graph of = − +y a x b c where a, b and c are constants is drawn below.

i. Determine the constants a, b and c.

ii. Describe the set of transformations that converts = = − +y x y a x b c into = = − +y x y a x b c.

c. A cyclist travels at a constant speed to a town 40 km away, then on the return trip reduces speed by 4 km/h. If the cyclist had travelled at 16 km/h for the whole trip, the total time taken would have been one hour less.

i. If x km/h is the cyclist’s constant speed, show that 40 40 64

+ =−x x

.

ii. Solve this equation to find x.

(KP)

(MP)

(KP)


Question 3a. Solve cos 0.5 2 2= − − π ≤ ≤ πx x for x in terms of π wherecos 0.5 2 2= − − π ≤ ≤ πx x .

b. The angle of elevation from a point A on the ground to the top of a flagpole mounted on a building is 72° and to the top of the building is 70°. The flagpole is 4 metres tall.

Calculate the height of the building, BC, correct to the nearest metre.

c. A show ferris wheel is 100 metres in diameter and at the lowest point a ferris wheel carriage is 1 metre above ground level. The wheel takes 25 minutes to make one complete rotation without stopping.

Passengers enter the carriages at the bottom of the wheel.

Respond to the following on page 24 of your response book.

i. Develop a mathematical model that determines the height of the bottom of a carriage above the ground at time t minutes. Graph the developed model.

Respond to the following by using a graphics calculator.

ii. The show fireworks display will be visible to a passenger when the bottom of their cabin is at a height of 65 metres or more from ground level. Determine how long passengers will be able to view the fireworks in one full rotation of the wheel. Comment on any assumptions made and their effect.

(KP)

(KP)

(KP)

(MP)


Question 4a. Given that f (x) = 4x –2 and g (x) = 2x2 + 7x :

i. solve g (x) = –6

ii. find the inverse function f –1(x)

iii. determine and simplify the composite function f ° g (x).

b. Solve algebraically the simultaneous equations 23 2 15 and 13 2+

− = + =x yx y .

c. A basketballer throws a ball from a height of 2 metres at the ring 28 metres away, 3 metres above the ground. The ball goes in the ring after reaching a maximum height of 4 metres.

i. Develop a mathematical model of the form 2+ +y = ax bx c given that a, b and c are constants and y is the height of the ball above the ground when its horizontal displacement from the basketballer is x metres.

ii. Use the developed model to determine how far from the basketballer the ball reaches its greatest height.

(KP)

(KP)

(MP)


Question 5a. Determine 2 35 log

2 1= + =

+x

edy xy e x ydx x

in each of the following.

i. 2 35 log2 1

= + =+

xe

dy xy e x ydx x

ii. 2 35 log2 1

= + =+

xe

dy xy e x ydx x

b. The displacement of a particle from its starting point is modelled by the equation x = t2 – 4t where x is the displacement in metres after t seconds.

i. Determine the average speed of the particle across the interval 3 ≤ t ≤ 6 seconds.

ii. Calculate the velocity of the particle when its displacement is 5 metres from its starting position.

c. A cone has a slant height of 15 cm and a base radius of x cm.

i. Show that the volume of the cone is given by 2 2225

3π −

=x xV .

ii. Using calculus methods, determine the value of x that maximises the volume of the cone.

(KP)

(KP)

(KP)

(MP)


Question 6a. A box has 7 green marbles and 3 black marbles. A marble is randomly selected from the box, its colour noted, and then put back in the box. If X is the random variable that represents the number of black marbles selected in 12 trials, calculate P (X ≤ 4).

b. The lengths of a particular species of fish are normally distributed. When fishing, it is only legal to keep fish between 16 cm and 40 cm. Sampling indicates that 18% of fish are undersized and 12% are oversized. In a sample of 1000 fish, what number are expected to fall between lengths of 25 cm and 35 cm?

End of Paper One

(KP)

(MP)


Asse

ssm

ent s

tand

ards

from

the M

athe

mat

ics B

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 qua

lity o

f a

cand

idat

e’s ac

hiev

emen

t acro

ss

the f

ull ra

nge w

ithin

the c

onte

xts

of ap

plica

tion,

tech

nolog

y and

co

mpl

exity

, and

acro

ss to

pics

, co

nsist

ently

dem

onst

rate

s:•

accu

rate

reca

ll, se

lect

ion

and

use o

f def

initi

ons a

nd ru

les

• ac

cura

te u

se of

tech

nolo

gy•

reca

ll and

sele

ctio

n of

pr

oced

ures

and

thei

r acc

urat

e an

d pr

ofici

ent u

se•

effe

ctive

tran

sfer a

nd

appl

icatio

n of

mat

hem

atica

l pr

oced

ures

.

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

• ac

cura

te u

se of

tech

nolo

gy•

reca

ll and

sele

ctio

n of

pr

oced

ures

and

thei

r acc

urat

e us

e.

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 tec

hnol

ogy

• ac

cura

te re

call,

sele

ctio

n an

d us

e of b

asic

proc

edur

es.

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 t

echn

olog

y•

use o

f bas

ic pr

oced

ures

.

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.


(cont

inue

d)

Crite

rion

AB

CD

E

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 ra

nge w

ithin

each

cont

ext,

and a

cross

topi

cs, g

ener

ally

dem

onst

rate

s mat

hem

atica

l th

inkin

g whi

ch in

clude

s:•

inte

rpre

ting,

clarif

ying a

nd

analy

sing a

rang

e of s

ituat

ions

and i

dent

ifyin

g ass

umpt

ions

and v

ariab

les•

selec

ting a

nd us

ing e

ffecti

ve

strat

egies

• se

lectin

g sui

tabl

e pro

cedu

res

requ

ired t

o solv

e a ra

nge o

f pr

oblem

s … and

som

etim

es

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• su

itabl

e syn

thes

is of

pr

oced

ures

and s

trate

gies t

o so

lve pr

oblem

s•

initi

ative

and i

nsigh

t in

expl

orin

g the

prob

lem•

iden

tifyin

g stre

ngth

s and

lim

itatio

ns of

mod

els.

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 an

d id

entif

ying

assu

mpt

ions

and

varia

bles

• se

lect

ing a

nd u

sing e

ffect

ive

stra

tegie

s•

sele

ctin

g sui

tabl

e pro

cedu

res

requ

ired

to so

lve a

rang

e of

prob

lem

s … and

som

etim

es

dem

onst

rate

s mat

hem

atica

l th

inki

ng w

hich

inclu

des:

• su

itabl

e syn

thes

is of

pr

oced

ures

and

stra

tegie

s.

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

requ

ired

to so

lve

prob

lem

s.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

som

etim

es d

emon

stra

tes

mat

hem

atica

l thi

nkin

g whi

ch

inclu

des:

• fo

llowi

ng b

asic

proc

edur

es

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

prop

ositi

ons

• ju

stifi

catio

n of

pro

cedu

res

• re

cogn

ition

of th

e effe

cts o

f as

sum

ptio

ns•

eval

uatio

n of

the v

alid

ity of

ar

gum

ents

.

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

prop

ositi

ons

• ju

stifi

catio

n of

pro

cedu

res.

The o

vera

ll qu

ality

of a

cand

idat

e’s a

chie

vem

ent

in al

l con

text

s gen

eral

ly de

mon

stra

tes:

• ac

cura

te u

se of

bas

ic m

athe

mat

ical t

erm

s and

sy

mbo

ls•

accu

rate

use

of 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 an

d oc

casio

nal

use o

f mat

hem

atica

l rea

soni

ng.

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

and

mat

hem

atics

.

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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

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2019 senior external examination: mathematics b paper one ...€¦ · 2019 mathematics — uestion...

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