2011 Mathematical Methods (CAS) Exam Assessment Report Exam 2

7/27/2019 2011 Mathematical Methods (CAS) Exam Assessment Report Exam 2

1/9

Mathematical

20Asses

Rep

2011

GENERThere were 1across the wopportunity

In the multipstudents obtanormal, a nuThis occurrethe incorrect

,(~ u N X area, not b.

In Section 2hybrid functistudents sho

from their tecareless mistoften only thHowever, if

Correct matsuch as dt inyear, with x Questions 2d Answers mustated. A nuas their finalrounding err gave exact a Students sho

ans giv giv

SPECIFSectionThe table bshading.

Question

1

2

3

4

567

ethods (CAS) 2

1mentort

L CO5 983 studenole spectruor students t

le-choice secined the corr ber of studein multiple-

area was typ)2 , Pr ( X > a

it was pleasion at the star ld be familia

hnology. Thakes with thee equation tot is a show t

ematical notaQuestion 1a.,ften being u, 4bi. and 4c.

t be given to ber of studeanswer. Thisr in Questioswers when

uld always cawered all par en coordinateen the answe

C INFO low indicate

% A

84

56

5

52

1133

GA 3 Exam

MENTSs who sat theof availabledemonstrate

ion, Questioct answers f nts incorrectlchoice Questid into the tec

) = b, Pr ( X

g to see a nuof the questi

r with a varie

s occurred inir algebra, es be solved follhat question,

tion must bedx in Questied when t , m

the required ts gave the eoccurred in1di.; some sn approxima

refully read ats of a questios when requito the requir

RMATI

s the percent

B % C

2 7

2 20

4 16

3 20

2 34 169 68

VICTORIAN C

athema

Mathematicaarks. Respo

what they kn

1 was answer Questions 1gave the rec

ons 2 and 17.hnology whe a) = b1 .

ber of studeon. Technoloy of methods

Questions 1d ecially in Quowed by thesuch as Ques

used. Severaln 1d. and dyor y were req

ccuracy. For act answer uestions 1di.udents wrotete answer wa

nd check quened d accuracy.

N

age of stude

% D

4

1

9

2

718

18

URRICULUM

tical Met

l Methods (Cnses showed ew.

red correctly3, 14, 16, 17,iprocal of theAnother coworking wi

hen a is foun

ts using techy enables st

. Many stude

ii., 3ci., 4bi. aestions 3ci. answer is req tions 1b., 4a.

students leftin Questionsuired. Anothe

numerical rehen it was re

, 4b. and 4d. itheir answersrequired.

tions to mak

ts who chos

E% N

Answ3 0

11 0

5 1

3 0

13 08 0

2 0

ND ASSESSM

ods (CA

AS) Examinahat the paper

by the majori19, 21 and 2gradient of tmon error och the normald by using th

ology efficiedents to appr ts, however,

nd 4di. Othed 4bi. For quired. Studentand 4c., the r

ut the variabaii.2cii. Va

r problem wa

ponses, an equired, but thn particular.to the incorr

sure they ha

each option

r

mtangent

d dx

loge

NT AUTHORI

S) GA 3:

tion 2 in 201was accessib

ty of students. When findie tangent, no

curred in Quedistribution;inverse nor

ntly in Questach questio

transcribed t

s chose not testions worts are not requlevant steps

le they were iiable namess the omissio

act answer isen wrote dowhere appearect number of

ve:

. The correc

Co

4 02

2 ,

f ( x) 2 f 2 f (

Y 2012

Examin

. Students acle and that it

. Less than 5ng the gradiet the negativestions 12 and or example,al and using

on 2 by defis in differenteir equations

use technolomore than oired to showmust be clearl

ntegrating wiere also a pof brackets,

required unlen an approxid to be a comdecimal plac

answer is in

ments

normal

12

x) x)

f ( x) f ( x)

tion 2

ieved scoresrovided an

% of thet of thereciprocal.13, where

b1 as the

ing theways and incorrectly

gy and madee mark,ll the steps.y shown.

th respect to,oblem thisespecially in

ss otherwiseate answer

mones, others

dicated by

12

1


2/9

Mathematical

20Asses

Rep

Question

8

91011

12

13

14

15

16

ethods (CAS) E

1mentort

% A

3

772

5

45

30

14

19

am 2 GA 3 Exa

B % C

8 11

7 65 756 7

4 28

8 15

5 2

7 7

7 11

% D

58

75

10

59

8

44

5

41

Published: 3

E % NAnsw

0 1

2 08 04 0

3 0

3 1

8 0

7 0

1 0

April 2012

r

)( 2x xh

43

23

h

Hence, th

3(~ N X 21.34a

(25.79) isthe area.The answ

)Pr( a Z

11.1a

correct to

obtained i

200017351 .

The mean

20001735 is

necessary.Area = Ar

curve = e

Area = Ar

2

12

(loge

e x

Option Aconsidere

)( 2x x f

19% of stcontinuou21% of st

Co

33x . The

, 135h .

domain of

)25, , Pr( X correct to tw

obtained if 0.similar erro

r needed to b

20001735

, Pr(

...465 ,2.0

8.2

three decimal

20001735 is us

A similar err

needed to be

reater than 0.

ea of the rect

1

0

2 dxe x

ea under the i

)

dx

s obtained if .

24 for a

dents thoughs everywhere.dents though

mments

turning point

he range of h

1 is 13,43

2.0)a , Po decimal pla2 instead of 0was made in

e greater tha

21

1)a Z

11465.1

places. Opti

d for the area

or was made i

greater than

5. No calcula

ngle Area

nverse functi

only the first

ll x > 0, x f (

t the graph of .t x x f 2)(

is at23

x .

is

13,43

.

.

8.0)( a X ces. Option C.8 is used for Question 13.the mean.

000735

,

, 023.3

n B (2.577) i

instead of

n Question 1

.8 because

tions were

under the

n =

case is

x2)

f was not

for 2 x .

2

,

.


3/9

Mathematical

20Asses

Rep

Question

17

18

19

20

21

22

ethods (CAS) E

1mentort

% A

14

13

7

2

17

15

am 2 GA 3 Exa

B % C

8 17

1 17

5 27

0 12

5 23

5 17

% D

12

52

43

5

19

8

Published: 3

E % NAnsw

9 1

7 1

8 0

1 1

15 1

15 1

April 2012

r

x y 23

mnormal

A similar

9 23 x x

x = 1 or x 2)5( f

This questand local points. Opof the req Area of th

)1( f 21 22

Actual are

791% p

percentagOption C

considere

to the nea

)Pr( QPPr( QP

Pr )Pr(P

Pr )Pr(2 P

Given the

if and onl

logc (a ) l

)(log)(log

ca

a

a

1log a (c)

Alternati

a c x an

So xlo

Each of thusing tech

Co

,23 21x

dxdy

14

, 4 x y

rror was ma

015 w x ,

= 5, f ( x) x5 , 7w oion could be

aximum antion D is theired form.

e rectangles

)3()2( f 543 222

a = 6

0

2 dx x

%100272

. p is closestould be obt

or 91% p

est percentag

Pr )Pr(PP)Pr() Q)Pr()( QQ

)Pr()( QQ ,

stated conditi

if Pr(P ) =21

oga (b) logb

ll

)(log)(log

ab

b

b

1log b (a ) l

ely, letlog e (

1 x log a (c

1

a (c)and si

e options counology.

mments

1 , at x = 4,

,41

x y

e by students

9 23 x xdxd

9 x2 15 x ,25w for

one by consilocal minimnly one that

)5()4( f 9162

72 ,

%26 correct

to 25%.ined if 791

%10091

72

e.

)Q ,)Pr()( QP ,

Pr()Pr( QP

21

)Pr(P

on, P and Q a

.

c)

)(og)(og

bc

c

c

1g c (b)

) x then

).

ilarly for the

ld also have b

4dxdy ,

47

in Question

015 w x

(1) = 7,ne solution.dering graphm turning

specifies a se

)6(

to the neares

2 was

%21 correc

) ,

re independe

other terms.

een tested

3

.

,

t

t

nt


4/9

Mathematical

20Asses

Rep

SectionQuestion 11a.

Marks

%

5

2t dt dV

,

or t 2

5

dt

This type of

attempt to in

1b.Marks

%

2r A , d d

It is pleasingknow the for showed r = 3 1c.

Marks%

Students seerequired at xabove the x- 1di.

Marks%

sin x40

0

40

d

This questioantiderivativ

y

0 10

-60

-40

-20

0

20

40

60

(

(0, 0)

ethods (CAS) E

1mentort

0

74V

t 2

50

a

dt

t 3

156075,

question appe

tegrate and s

0

31

r 2 , dt dA

to report thamula for the. There were

0

23

to have tak = 40. Some sxis or drew s

0

2080

was answer e but did not

20 30

20, 60)

(

am 2 GA 3 Exa

1

36075 or V

5 minutes

ared to be un

lved t 2

560

1

8

20 ,dr dt

d d

many studenrea of a circlsome unusua

1

22 5

en more caretudents wrotetraight lines.

1 Av

80

d well. An evaluate it.

40 50 60

0, 0)

(60, 6

3

20t 2

5

dt

familiar to m

5 , getting t

3

56

t dr dA

20

ts were able t. The stepschoices of v

Aver

4 1.3

hen sketchitheir coordin

rage.8

act answer w

x

70 80

(80, 0)

0)

Published: 3

Aver

0.7t 3

156075

ny students a

45 15 .

Aver

1.9

12 r

103 ,

write downust be showriable names

ge

g graphs thisates as (0, 40

as required.

April 2012

ge

nd was not a

ge

hen r = 3

a related rateclearly in a

, such as V fo

year than in) and (0, 80)

any student

swered well.

equation. Ashow that q r area.

revious year r left out (80,

wrote 25.46.

Many studen

number of stestion. Some

s. A sharp poi0). Others ha

Some wrote

ts did not

dents did notstudents

nt wasd the graph

down the

4


5/9

Mathematical

20Asses

Rep

1dii.Marks

%

75sin( x

800

80

Other metho

6876, or left

incorrect ter

did not trans Question 22ai.

Marks%

.0,3(~ N X This questio 2aii.

Marks%

53Pr( Y

or 3Pr( Y

The hybrid f the hybrid fu

4

3 416

dy y

2b.

Marks%

4

0

2

16dy

y

Some studen

4

0

2

16dy

y

ethods (CAS) E

1mentort

0

55

dx + 120 s0

40

s were acce

their answer

inals or 1

ribe their eq

0

23)4 , Pr(3

was answer

0

27

3Pr() Y

)()55

3

y f

nction could nction this w5

001.

(5.025.0 e y

0

32

4

.025.0 ye

ts gave the e

34 and 2.0

4

am 2 GA 3 Exa

1

16 2

n x

40

dx =

table. Many

n exact form.

0 sin x40

0

40

ation correct

1 Av

775) 0.493

d well.

1

7

4Pr()4 Y

4155.0dy

be defined us a sensible a

)4 dy or 4

31

1

18 1

)4( dy y = 4.

act answer, 1

)4(5.0 d ye y

Aver

9 0.8

875 m2 or 0

80

tudents roun

Many wrote

x or forgetti

y from the te

rage.8

8 correct to fo

3

58

)516

4

3 y

ing technolo pproach. So

5

5

25.06

dy

3

4 37

33 correct to

33 . Others w

3 or only

Published: 3

ge

75sin( x

80)dx

ed incorrectl

+ 120 sin

0

40

g to multipl

chnology out

ur decimal pl

Aver

2

25.05

4

edy

y, giving thee students in

)4(5.0 dye y

Aver

1.6

three decima

rked out eac

considered 0

4

April 2012

+ 12080

y to 6876, ro x

0 dx or its e

their answer

ut.

aces

ge

)4(5.0 dy y

second methcorrectly set

r

5

316

dy y .

ge

l places or

integral sepa

y2

16

dy 1.33

6875 to the

nding 6875.4

quivalent for

to Question

4155.0 correc

d. As the res p their integ

))(( dy y f y

rately and di

3. Some fou

nearest squar

9 to 6875.5 t

incorrectly,

di. by 60. So

t to four deci

t of the questials as:

333.4

not add the

d the mean o

metre

en 6875.5 to

using

e students

al places

on applied to

wo answers

f the two

5


6/9

Mathematical

20Asses

Rep

values,34

21

4

3

2

16dy

y

2ci.

Marks%

16

4

0

dy y

Some studen 2cii.

Marks

%0.25e 0.5( y

4

a

Some studen

2d.Marks

%

W ~ Bi 10,

Many stude

104

94

3223

2e.

Marks%

Pr( X 3)

Pr(Machine

This questio probabilities

Question 33ai.

Marks%

12)( 2x x f This questio

ethods (CAS) E

1mentort

6133

(5.025.0 ye y

0

47

or m

y f )(

ts left their a

0

664) dy 0.2 or

ts attempted

0

38

932

or 104

ts recognised 6

2. Others di

0

65

.5, Pr(Y )

|longer than

was not ansgreater than

0

55

was answer

am 2 GA 3 Exa

167. . Others

)4 dy .

1 Av

53

21 , the me

swers as 4

1

4 30.25e 0.5( y 4

a

o use the nor

1

10 5

9

32

4 2332

that the distri

not give the

1

222332

, Pr(lo

3) =6093.0

.05.0

ered well. Sin their wor

1 Av

95

d very well;

used 3 from

rage.6

ian is 4

and 4.

Aver

0 0.7) dy 0.3 or

al distributi

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2 1.26

, Pr(W = 4)

bution was bi

answer to the

3

9

ger than 3) =

55

= 0.4103 c

udents whoing.

rage1

owever, a s

Published: 3

he previous

ge

f ( y) dya

n. Others sol

ge

0.1812 corr

nomial and g

required nu

Aver0.6

.05.05.0

rrect to four

sed a tree dia

all number o

April 2012

uestion inste

.7 , a = 5.02

ved 0.25e4

a

ct to four dec

ave the correc

ber of decim

ge

3223 = 0.609

decimal place

gram were m

f students trie

d of 0,

correct to 2 d

.5( y 4) dy 0.7

imal places

t n and p val

l places.

375,

s

re successfu

d to integrate

ecimal places

, for a .

es. Some wr

l. Some stude

.

te

nts gave

6


7/9

Mathematical

20Asses

Rep

3aii.Marks

%

02 x , for

Many studeincomplete r 3bi.

Marks%

0, 1, 2

3bii.Marks

%0, 1

In Questionsa one-to-onenumber of st 3ci.

Marks%

Let 23 y

equivalent fo Most studen

3)(1 xq

notation as

q ( x)3

23

students cha

3cii.Marks

%3 2 x3 = x,

Many stude 3di.

Marks%

3)( 2x xg

B A x 3)(

ethods (CAS) E

1mentort

0

53ll of x, hence

ts found it disponse.

0

91

0

80

3bi. and 3bii.function eveationary poin

0

10

3 x , for the i

rm.

s knew to int

2 x and 1(q

3 3)(1 x

x . A numbe

ging their te

0

18= 1, (1, 1)

ts were able t

0

35

04 c x ,

Ax x 23 3

am 2 GA 3 Exa

1 Av

475512 2 x

ficult to expl

1 Av

9

1 Av

20

. many studenthough earli

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1

14 7

verse rule in

rchange x an

32

3) x x

2 x

and (1q

did it by han

hnology outp

1

22 6

o write down

1

31

1216 c

A x A 323

rage.5

for all of x

in their answ

rage.1

rage.2

ts did not knoer parts of thed and others

Aver

6 1.7

erchange x a

d y. Some did

ere common

23)

3 x x

and had pro

ut q 1 ( x) 2

Aver

0 1.4

a correct equ

3

30

0 for one sol

B , 23A

Published: 3

er. Because

w that a cubiquestion wettempted to

ge

d y and solve

not transcrib

answers. So

ere cited. A

blems with t23 3 x3

2to q

ge

ation. Some s

Aver

1.3

ution, 34

c

,323

c

April 2012

t was translat

c function coe helpful in tse the quadra

for y to obtai

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e students n

few students

e algebra and

1( x) 3 x3

213

tudents did n

ge

or

34

ed 5 units up

ld have no stis regard. Sotic formula.

n 323 y x

expression f

ed to be care

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the negative

.

t find the co

was a comm

ationary poinme students t

, 31 3)( xq

rom the tech

ful with their

e derivative

sign. It was p

rdinates.

on

s if it is to beought the

2 x

or

ology.

mathematical

r wrote

leasing to see

7


8/9

Mathematical

20Asses

Rep

It was pleasi

x xg 3)( 2

x xg 3)(

3dii.Marks

%

g ( x) 3 x2

This questio x3 2 x 2 cx Question 44a.

Marks%

n m2 1 ,

Many studesubstitution.able to do thi 4bi.

Marks%

dLdm

2m3

m4

Most studen

incorrect exp

dLdm

m2 1

forgot to fin 4bii.

Marks%

L6

2

2 This questio by substitutirequired.

4c.Marks

%

Distance fro

ethods (CAS) E

1mentort

g to see a va x4 . This res

k cx x2

0

79

4 x 43

0 ,

was not ansk 3 x 2 4

0

40

(n 1)2

ts did not atteFor show ths; however, s

0

253m

3m2 4,

dLd

s were able t

ression2 m

was often se

the coordina

0

487

km

was done wg their m val

0

78

camp (0, 0)

am 2 GA 3 Exa

riety of methult would occ

hen using te

1

9

32 , solve

ered well.c .

1

15

m2 (m2

mpt this quest questions,ome had prob

1

27 1

0 , m 62

get the deri1

3m2 4

n. There was

tes or substit

1

22 3

ll by studentue into L. Ho

1

8

to river ( x, x

ds being useur if a multip

hnology. Oth

3

9

32

g

any students

3

39

2)2 m2

tion. Some ststudents mustlems with the

3

2 36

, m 0 , n

ative. Some t

m3 6m . So

some poor u

ted their m v

Aver

0 0.8

who were sever, some

3

11

1), 2( x

Published: 3

. A number lication sign

ers found c

Aver0.4

3for k , k =

used x = 1 fr

Aver

1.5

m2 m4 4m

dents knewshow the stealgebra.

Aver

1.6

m2 1 12

,

ranscribed it i

e did it by h

e of variable

lue into L in

ge

ccessful in Qtudents gave

Aver

0.52 21) x

April 2012

f students dias not put be

x(3 x 4) .

ge

2710

m Question

ge

2 4 m4

he distance f s involved a

ge

62

, 12

or

ncorrectly fr

and but then

names with x

tead of n

uestion 4bi.an approxima

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4 2 1 x x

not includetween c and

cii. Others tr

3m2 4

rmula but wd the final a

32

, 12

m their tech

id not answe

s and ms in t

2 1 .

thers were ate answer w

c in the deriin

ed to solve

re unsuccessswer. Many

ology or wro

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e same equat

le to gain aen the exact a

vative,

ul with thetudents were

e it as an

e question.

on. Others

ethod mark nswer was

8


9/9

Mathematical

20Asses

Rep

Time from ri

This questio

working wasequation for

4di.Marks

%

dT dx

x(2 x2 x4

The questiowriting 13 as by hand and

4dii.Marks

%

dT dx

0 , x

Some studen

considering

y x2 1.

4e.Marks

%dT dx

0 wit

This questio 4f.

Marks%

dT

dx0 with

This questio

sign, giving

ethods (CAS) E

1mentort

ver to plant =

was not don

required to gthe time fro

0

422 1) x2 1

1313

was done re3 or not scro

made errors

0

563

2,

32

,

ts gave an ap

he domain re

079

x = 1, k 14

was quite w

0

87

x7

2, k

was not ans

k 5 37

74as

am 2 GA 3 Exa

k 34

( x2 1

e well, and m

ain full mark the river to t

1 Av

58 x or dT

dx2

sonably well.lling across then applyin

1

22 214

roximate ans

striction. So

15 1

ell done whe

1

11

5 37

74. This

ered well. O

heir answer.

) ,4 x

T

any students

. Some weree plant.

rage.6

x3 2 x 13(26 x4

. Some studeget all of thethe chain rul

Aver

2 0.7

wer instead o

e did not find

Aver7 0.4

attempted. S

Aver

0.2

can be obser

f those stude

ome student

Published: 3

2 12 x

k

id not attem

able to use th

4 x 2 1) 1 x 2 1

ts transcribed expression o

e. Some simp

ge

f the exact an

the coordina

ge

ome substitut

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ved from con

ts who did u

substituted

April 2012

2 1 x t it. As this

e distance for

3 x or equival

the equationn the right-halified technol

swer. Others

tes or substit

ed x = 1 into

sideration of

e the correct7

2into

4 2

2 x x

as a show t

mula but the

nt form

incorrectly f nd side of theogy output in

gave two or

ted x 32

i

T .

he family of

x value, most

.

274k

x

at question,

struggled to

om their techequation. Ot

correctly.

ore solutions

nto T or 14

k (

raphs genera

left out the i

clear

find the

ology, ofteners did it

, not

4 x 2 ) , not

ted by k .

equality

9

Documents

2011 Mathematical Methods (CAS) Exam Assessment Report Exam 2