2011 trial wace exam solutions

PERTH COLLEGE {hJSW6'LS

Trial Western Australian Certificate of Education Examination, 2011

MATHEMATICS 3A/3B Section One: Calculator-free

Student Number: In figures

In words

Time allowed for this section

Question/Answer Booklet

If required by your examination administrator, please place your student identification label in this box

I

Reading time before commencing work: five minutes Working time for this section: fifty minutes

Material required/recommended for this section To be provided by the supervisor This Question/Answer booklet Formula sheet

To be provided by the candidate Standard items: pens, pencils, pencil sharpener, eraser, correction fluid/tape, ruler, highlighters

Special items: nil

Important note to candidates

No other items may be used in this section of the examination. It is your responsibility to ensure that you do not have any unauthorised notes or other items of a non-personal nature in the examination room. If you have any unauthorised material with you, hand it to the supervisor before reading any further.

Structure of this paper

Section

Section One: Calculator-free

Number of questions available

7

Instructions to candidates

Number of questions to be

7

Working time (minutes)

50

Marks available

40

120

1. The rules for the conduct of Western Australian external examinations are detailed in the Year 12 Information Handbook 2011. Sitting this examination implies that you agree to abide by these rules.

2. Write your answers in the spaces provided in this Question/Answer Booklet. Spare pages are included at the end of this booklet. They can be used for planning your responses and/or as additional space if required to continue an answer. • Planning: If you use the spare pages for planning, indicate this clearly at the top of the

page. • Continuing an answer: If you need to use the space to continue an answer, indicate in

the original answer space where the answer is continued, i.e. give the page number. Fill in the number of the question(s) that you are continuing to answer at the top of the page.

3. Show all your working clearly. Your working should be in sufficient detail to allow your answers to be checked readily and for marks to be awarded for reasoning. Incorrect answers given without supporting reasoning cannot be allocated any marks. For any question or part question worth more than two marks, valid working or justification is required to receive full marks. If you repeat an answer to any question, ensure that you cancel the answer you do not wish to have marked.

4. It is recommended that you do not use pencil except in diagrams.

TRIAL EXAMINATION 2011 SECTION ONE

Question 1

3 MATHEMATICS 3A13B CALCULATOR-FREE

(6 marks)

50 households were asked whether they owned a digital television or had a set top box. There were 5 households who had neither. 33 households owned a digital television. I I of the households who owned a digital television also owned a set top box.

a) Complete the Venn diagram below to show the number of households who owned none, one or both of these items. (3 marks)

Digital Television

../

22

Set Top Box ls'O

/

\2

j v

5

/// (-I per~~)

a) Determine the probability that a randomly selected household from this group only owned a set top box. (1 marks)

(Q~\J =v'<2.~x ) -::. I L / P se,=, """p .,/

:50

b) Determine the probability that a randomly selected household from this group owned a set top box, given that they owned exactly one of these two items. (2 marks)

of (o,-",real Se"'- '=f'1o=<. I"~ exGACKJ..] =-e) -

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~ 3Lj-~


Question 2

4 MATHEMATICS 3AJ3B CALCULATOR-FREE

(4 marks)

Billie O'Phyle has a collection of prized rare and antique books. The collection consists of 15 different books altogether, including 4 books by Jane Austen. Unfortunately, Billie only has room on her bookshelf to display 8 books.

a) Write a numerical expression to determine

(i) the number of different ways there are to display the 8 books ifthere are no further

(ii)

restrictions? (1 mark)

\ S X I '+ x \.3 x 1"2...>< \ \ x \ '" X 9 x 8 /

the number of different ways there are to display the 8 books if all of the books by Jane Austen are to be included, and placed together at the left hand end of the bookshelf.

(1 mark)

4-><3X2xl Xllxl'"Ox9';>( <:'5 /

b) 3 of the 15 books are written by the Bronte sisters. Given that the Jane Austen books are together on the left hand end of the bookshelf, what is the probability that none of the books by the Bronte sisters are on the bookshelf? (2 marks)

4- x 3 x 2. x \ x 8X I x (, x ~ -;:/

=!-

C\.. X"3 x :2.x I x II .x. \ " x "'\ '" ~ 33

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Question 3 (4 marks)

Contestants Jack and Jill Palewater, on the game show Rules my Kitchen, are hosting a dinner pru.iy for the other contestants. The project network below shows the required tasks and completion times (in minutes) for their dinner.

EI50

H45........--

Dl20

Task Description Task Description A Purchase ingredients H Chill seafood plates B Prepare panna cotta I Preparelnintsalad C Decorate house to theme J Cook lamb D Chill and serve wine K Chill md serve mint salad E Chilf and serve panna cotta L "Plate up" lamb F Prepare seafood plates M Serve md eat main course G Set table (seafood, lru.nb and mint salad)

a) Determine the minimum completion timynd associated critical path.

23"" "", ~ ~ """-S ./

(2 marks)

Cr,bCo.l ,0"",,,,,,", -0 e F J\ ... M /.

b) The rules for the show state that the preparation time from after the purchasing of ingredients to the commencement of serving the main course should be no longer than three hours.

Unfortunately for Jack md Jill, the decorating of the house takes 30 minutes longer than expected. Determine whether or not they will still make the three hour deadline. (2 marks)

POOCh KG, \~C/e= '=c; 3= o'V),;'" b> 1= vy"~~,,cs /' S,,-,:::.z ~ dcadhrc. "'f- 2> h=.!/"s ""'fjOi,es """ ~ Sica/\.- <01- M ""'\:: ""--E. e,.,J/

~ ~us\c. r-->=V.e. ""Ve... 3 h=.,~ deodi,~ . ./

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Question 4

Solve the following equations:

a) 163x- 1 = 322+X

2 '-I{'6X--1 \ ~ 2 S (2 -r::(..-j /

b)

l+(3;i.. -I )~ 5{;;>.f::LJ

\ 2- :x:.. - <..t- '" \'''' + ,., :L / '""t:x: ". I '* X-~;t.. /

ex - 1)2 - 7 = 74

(x.. ., ) 1. ~ '2' \

OC-- \ -=- JB I /

)C - \ ~ -\- 9

x - \ -::. C\, ::::c. - \ -=0 - 9

6

:x: 0::. Iy 'O/' x... -=- - y

c) 3x - 2y = 7 <D 2x + 3y = -4@

0x...-tJ = 1'+ G");.. -\- s,j -:: -\ 2..-

-\?>j "- 2'(,

j-::--:2-

s " \, , ;; '"'" \D

)<,2..

)(3

/

/

3 X - ( 'l.J< - 2.-) -=- ,.

3:x:. +<..t- =- -i-

3:>e =- 3

X'=' 1 /

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MATHEMATICS 3A13B CALCULATOR-FREE

(9 marks)

(3 marks)

(3 marks)

(3 marks)


Question 5


(3 marks)

Prove that the sum of three consecutive numbers will always be a multiple of3.

~ k + 1,-- -+-1 + \<>-j- 2... /

:::3\c+ 3 ./

-::. 3($+$ ./

o \'-J,S\~Le \oJ "3 S=> al'-00J S 0. ,.....,._<\""'1".'<0. ~f- 3,


The derivative of a function f(x) has the equation f' (x) = 3x2 + bx + 2.

Given that the tangent to f(x) at the point (-1, -1) is y = 3x, detennine the equation ofthe functionf(x).

j -:=. :1 Y'- h:::ts '3.rac1(~"" -=1- 3.

,', ('(-I) '" 3

3 "" 3C- ' )"'- T bH+").. 3"'3-10-1-:2-

3 =- 5 -b

,',6=2 /

S=. ;: / ( x) =- 3 X '- -l- )..;:c.. -t<,

?'(x) = X 3-+ :x.."'--t 2 L + c...

S""'o$~~e (-\ J -I) /

-\ -=-(_t'j1-+(_t"j'"-I-('"l.X-,ji'C

-\::.-It\-:&.-+c... -\ ",,"-')..+c..

,/ I',e-=-See next page

5= .?/:L) -:.. ,:)(.3+ Xl.-+ 2::l::.. -+ I /


Question 7

8

The graph below shows the function y = (x - 1)(2x2 + 2)

•• _d ___ • - - "' - d_ -"_ - - •. - - - r - -

y JI\

-1-' - .• , - '" ··1_· ___ .• '" •. - .

.. -, -, - - ','

- ... - _ .. -- --,--"" -- T'- - -:--

-l'f


(10 marks)

,--, .... "-,----

-'---1"-- "1--"'---

- I •• , __ , _____ ., .,. _ .• _.

- ~

:2 4 x ~I ~. .~

- - T' - -

--~--/1 _

a)

'-'---, I ;/ 'l-~ ~. ~ ~."

- - ".j. - •• T - -, - 'r - - -"1'- .-' V:" -., - •.. "' "' "' -I •. - - .• - - "1

Use the product rule to detennine the gradient function dY

. You do not need to simplifY your dx

answer. (2 marks)

/ ./ ~ :::- (2:e "+2)( \) +(::c-- \ ')( ~ j d:e

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b) Hence detelmine the equation of the tangent to this function at its x- intercept.

X==-i

::J'(-) -::. (2.{I)'-+ __ )(I)-\-(I-I)(,+-(I))

j'Cl) ::- 4- ./

b,,:P"'''=- ~ "C<b~ j::: s...b5k::'~ ~ (, , '" )

o -::: 4-C I ) ~- c. C-;.-4-

l.\-X -+ c.

/ Ta~'c -eClVO~=-- ~ '" ,+x -'-]- ,/

c) Describe the transformations involved in changing this graph to the graph of y = -(x - 1)(2x2 + 2) + 3

R.e~\ecb=-- ,;.., '=-v-= = ax..is ,/

TVo."51"",=~ <..J.p 3 ,-,..., \-\:::.5 v"

(3 marks)

(2 marks)

d) Draw a suitable line on the graph to solve the equation (x - 1)(2x2 + 2) - 2x = -6. State the solution to this equation. (3 marks)

ex.. -j )( 2:G >-+:) .::;.2 X- r..

G..v a1'" h j -= '2..:;L. - '" p

s:, h .. "".;.- ; s x...-=---\ /

END OF SECTION I


ADDITIONAL WORKING SPACE

Question number: __ _


TRIAL EXAMINATION 2011 SECTION ONE ADDITIONAL WORKING SPACE

Question number:. __ _




Question

1

2

3

4

5

6

7

TOTAL SECTION 1

12

Marks Available

6

4

4

9

3

4

10

80

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[]-I

Your Mark

PERTH COLLEGE

Trial Western Australian Certificate of Education Examination, 2011

MATHEMATICS 3A/38

Section Two: Calculator-assumed


In words

Time allowed for this section

Question/Answer Booklet

If required by your examination administrator, please place your student identification label in this box

[r II

Reading time before commencing work: ten minutes Working time for this section: one hundred minutes

Material required/recommended for this section To be provided by the supervisor This Question/Answer booklet Formula sheet (retained from Section One)

To be provided by the candidate Standard items: pens, pencils, pencil sharpener, eraser, correction fluid/tape, ruler, highlighters

Special items: drawing instruments, templates, notes on two unfolded sheets of A4 paper, and up to three calculators satisfying the conditions set by the Curriculum Council for this examination.

Important note to candidates

No other items may be taken into the examination room. It is your responsibility to ensure that you do not have any unauthorised notes or other items of a non-personal nature in the examination room. If you have any unauthorised material with you, hand it to the supervisor before reading any further.

Structure of this paper

Section

Section Two: Calculator-assumed

Number of questions available

12

Instructions to candidates

Number of questions to be answered

12

Working time (minutes)

100

Marks available

80

120

1. The rules for the conduct of Western Australian external examinations are detailed in the Year 12 Information Handbook 2010. Sitting this examination implies that you agree to abide by these rules.

2. Write your answers in the spaces provided in this Question/Answer Booklet. Spare pages are included at the end of this booklet. They can be used for planning your responses and/or as additional space if required to continue an answer. • Planning: If you use the spare pages for planning, indicate this clearly at the top of the

page. • Continuing an answer: If you need to use the space to continue an answer, indicate in

the original answer space where the answer is continued, i.e. give the page number. Fill in the number of the question(s) that you are continuing to answer at the top of the page.

3. Show all your working clearly. Your working should be in sufficient detail to allow your answers to be checked readily and for marks to be awarded for reasoning. Incorrect answers given without supporting reasoning cannot be allocated any marks. For any question or part question worth more than two marks, valid working or justification is required to receive full marks. If you repeat an answer to any question, ensure that you cancel the answer you do not wish to have marked.

4. It is recommended that you do not use pencil except in diagrams.

TRIAL EXAMINATION 2011 SECTION TWO

Section Two: Calculator-assumed (80 Marks)

3 MATHEMATICS 3A1B CALCULATOR ASSUMED

This section has twelve (12) questions. Answer all questions. Write your answers in the spaces provided.

Working time for this section is 100 minutes.


The two valiables hand ware inversely proportional to one another.

a) Circle each of the equations below that reflect this relationship, where k is a constant.

b)

h+w=k w=hk

/ Gi~ ~=w

k

(2 marks)

/ /~-~

/' !=0 ~=k .~ w

Detelmine w when h = 9.7 given that w = 38.8 when h = 12.5. (2 marks)

\.(c::.wh

k""-3'3''6 x\2·.5

\,( -::. 4- 3:S- /"

W:::: \« h

W::. L\_8" ~ --'1 '-==\-

\rJ-:: 50 /

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Question 9


(6 marks)

A quality control officer at a soft-drink bottling plant uses systematic sampling to select close to 0.2% of filled bottles for checking.

a) Describe a practical way that the systematic sampling might be carried out. (2 marks)

o ' d.;. :::--.L 5QQ

Pic-I-<.. ~ pr S 'c; 'o=>""t;<e. O'c; ro~c::Jo-v-, ~t""V"'1 G:;::t"lC?:- (,Y"vd ~c.\.:,1.:-,)--' t,'IC

QV"""Id I~ rer.....-ro.Je. <2.vC ':j ';-;)0 0-> 6~ Qr~er i=hab, ~

On a particular day, the content of bottles of soft-drink was observed to be nOlmally distributed with a mean of382.3 mL and a standard deviation of2.9 mL.

b) How many bottles in a sample of 500 would be expected to contain less than the stated content of

c)

375mL? (2 marks)

X ~ N ( 3'3';?' . 3" 2' 9 ~)

r(x<3-=r:r}='Cl'OClS'1 /

5= x 0-0-=59~' 3 I:= ... ~s /'

Calculate the 95th percentile for this distribution.

f(x< k .. ) =:"O'''''~

\<.. -::0 3~1" I YVI L

/ /

See next page

(2 marks)




The length of time between calls of the grey tree frog is known to vary with the ambient temperature. Data for temperatures from 11°C to 33°C are shown in the table below.

Temperature x 11 12 14 15 17 IS 20 21 23 24 26 30 32 33

(OC) Time between y 4.3 5.1 5.9 6 7.1 6.9 5.S 6.1 5.2 5.2 4.4 3.5 2.5 2.4 calls (seconds) I

a) Use your calculator to graph the data and comment on any trends you notice. (2 marks)

lire ta~ \:::e!CWCE/' ca \ \~

r;",....... \ \ ' C. lao \ ::;-' c. arc!~.J"'

o,f'f<eX)/S ~ \....,&-eC<sz. r:;" be>~t"<3/a"'''/''''S

c.lzcr«:Cle"c. \,;"eev l '1 ~.,..., \ 4-'.::.. b 33-c... ij ...;

b) It has been noticed that there is a different pattern for temperatures below 16°C and above 16°C. By careful selection of data from the above table, calculate a linear regression model of the form y = 111X + c suitable for temperatures above 16°C. (2 marks)

Use.. \<:> ck::vco r',/ \~" \,;:.;;te ~ -=- - 0 ' ?-'4 :2 .::c. -\""':2'0 L\- \

c) Predict the value of Time between calls when the temperature is 2SoC. Comment on the validity of your prediction. (2 marks)

A ~ -::: - '0' 2Y "-(2b) -I-I]' Q(.\.I Sv!:> i~,= CalCA.tk,-'$Y 3, 810 .

-:::. -3' 'CS=t- /

\r\~.:2.r (::h IC::I\\,'v:] c,,,,d 5Z\;;;"-~'J t'CC.lQ 1.:.:1 't :J

/ \,\V'\Co.. r Gv~......-e.lC~b~ / f::>-...:::Yq:-rr:..- vc;.'".{, ob''':::'

d) Calculate the residual time for a temperature of 20°C using the regression model from part (b).

- a - 2c{ :L (2,.",,) + \ 2. . Cl '>-\ "- <S' 1 "1(0 ~. (2 marks)

5"8 -G'I'11.., '" -0' ::0"'1(, ::::: -0"+ S::"c~~J~/

e) Comment on the statement "Due to the strong association evident between temperature and time, we can be certain that high temperatures cause grey tree frogs to have shOlter time intervals between calls." (1 mark)

I~_ ·~a~~'er>~c,r'""Ic., ~~ fo\~~c Ir-. \.:;:''"-a~ l..rJC2- GO"-'rr=;:)~ \~r=.'J Cau~('j

( ~ . / j'""'\(".!!""""'" G:::::,..r/r..:.! 10 ~\~, '" V""

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Brendon Ball, the bulldog breeder, uses two types of food, Bully Beefies and Canine Chewies, to feed his prize animals. Each 50g serve of Bully Beefies contains 12g of protein, 2g of fats and 36g of carbohydrates, while a 50g serve of Canine Chewies contains 8g of protein, 6g of fats and 36g of carbohydrates. A bulldog's minimal daily nutritional requirements are 96g of protein, 24g offats and 360g of carbohydrates.

Brendon gives b 50g serves of Bully Beefies, and c 50g serves of Canine Chewies each day to each bulldog. Two of the three constraints are shown on the graph below.

a)

c

12 k /; /"

1/' /

.'

/ / / / '/1 10 l : : ~ /" . /" 7 0 /' . ./ /'

. - ~7"-'- "' - ,

81 ~ ,

/'~/

61"~ J'

, 4 1----:

'. ,

__ , __ ....J,,_""_, y o'

\.- : -l- --. ---- ---;/ --- -, . /" .

2,j1----'-----'--+--'-------t---;--;--~" "<I: 7/ I /

2 4 6

Determine the inequality for the third constraint.

8 10

. -",. , .. ,.

, /

"

PI" 1 re"'-"'d ,:V "" e~,z J \"2 b .. + '3 c. ::::- "".,,, / 26"T <0 C ~ ;2.4-

~\;cd \V\cq"-Jo·ll~ 2,(,b·-\- 3Gc ::; 3bQ IokC ~., 10

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///

12 b

(2 marks)



b)

c)

Graph this constraint on the graph on the previous page, and shade the feasible region.

II:'c 0/ ye3l:O~ ./' (2 marks)

If each serve of Bully Beefies costs $1.60, and each serve of Canine Chewies costs $1.20, determine the number of serves of each food that should be given to minimise the daily cost of feeding each bulldog. State this daily cost.

(6, c) Co, ':l-)

(Il.,~ ) ("1,1)

(4-, b)

1-('6 -t\. 2c.

~ 1 '1-' '1-'"

~\"I')..Q

'$ \ 5 . 6<>

'* 13 1,,<>

/

4- 5e<ves.qt- 8 ... )\j -&=/:,.8S C\~d (0

MI"i";~"'" Goslc =.? ~\3'6"" /

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(3 marks)

sa-ves or Cc:I'-'I,.,e c.hee,.>,eS/


Question 12

Consider the following sets:

8

U = {the counting numbers less than or equal to IO}

F = {the set of Fibonacci numbers}

P = {the set of Prime numbers}

T = {I, 3, 6, 1O}

Use this to state the following:

a) n(PUT) -:r/

b) F L 4-1 b / -=J I 9 I 1'0 1 ./

c) n (F n T) 3 /

d) F uP ~ '+1(,/;'11''O~ /

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MATHEMATICS 3AJB CALCULATOR ASSUMED

(4 marks)

(1 mark)

(1 mark)

(1 mark)

(1 mark)




The graph below shows three functions, f(x),g(x)and hex)

The equations of the three functions are y = _1_ -b, y = c x O.Sx + d and y = -..Jx + e - f, x+a

but not necessarily in that order. y

a)

b)

'I '10 I /- __ n., __ ... I--ln-mhJ

l --)1- - - - - - - 1--- ------. -I- c - - - - - -

__________ h __ ---T- ~-----~

[[ r~1 .+uu~u .~ ..••..•.••. li- -- -,-1-- 1 -"-'-' ---: --:~----:--I"~' •

. '--,. r -----.-.-- : ,. "

x -10

p~=·ri ........ 1" -----.h&JK-I-:-~, ~ -. ~~ ~x)- <~-I I . 0-" ·····--~--·--I--·'·--~=-~I

. - '. , I

n.I.· ~ ..... f(~)- .. , .. __ ,_ .... 1 1--- .... -l<-- , I

.... "111--: -~~+--r---'~-~~~_ I

Match each function with its correct equation, determining the values of a, b, c, d, e and f

f(x) = S -.z <::\. 5 =<':- .,.. \ /

(e)<~b,"1 f.,~c~~)

g(x) = - Jx + S - 2. / ( S:j .... l Q rC ~k:. f:-. ..... c. b '0 ..... )

h(x) = _I -3 /

.y ±l !_

(('.eo f'r«.o.l f..,.. t'lc).;:t .-;,' ..... )

State the natural domain and range of g(x).

o""o'V'C\ i.... -:: ? X> :C. >/ -- r J ./ r(c.-,'7C 0;: ~'j; j S - 2J /

, (9 marks) ., ../ v

C=S d=1

",/ ,,//

<£.:::5 ;:.=.2.

/' Co'

/ C\"C l\- \::. ::::...2

(2 marks)

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Question 14 (5 mal'ks)

The histogram below refers to the percentage scores obtained by a group of 49 students on a test.

a)

b)

Test scores of students frequency

15

10

5

- .. ,-

- . -: ' .

percentage score

- --;

_i , I---c----- --- -+------.. -.-.-----

Use the information in the histogram to calculate an estimate for the mean and standard deviation. (2 marks)

me""..., ::::: 12' <0 '+ /. / S~f"'dal'd .J;;A:J\a~~.-.:::r- ~ \4-' 4-2- /. /

The actual mean for the data was 71.5%. Explain why this is different to the estimate you obtained in a) above, (I mark)

~ l>-Y"1 tdp~I""~= ~ \,;'"\ /'~i! ~ ~hcr~ .~ f"l~ be.. ~ C~e.

\::Chalco ~ SGo"",-s, ''-' each I"'lc~o \ O-=r0Td ":" \,.//

!::= ASsv'~d

c) Briefly describe the symmetry of the distribution, and whether you would expect the median to be higher than, or lower than the mean, (2 marks)

I~ d-",<'c.=. ""f'f=rS <:::0.'0 ~ sk:~01 ~;-!:;... / IZK~ ~,o'" "'=> (r-,= h'd~"Y ""-q~ ',,;:-Jl--e. VVlC ... ?"-'"

.-'/

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The daily sales figures for a new clothing store are shown in the table below. The figures in column a are the six-point centred moving averages of the sales data, rounded to the nearest dollar. The regression line of a on t from this data is a = 122.51 + 2871 , with a correlation coefficient of 0.97.

a)

b)

c)

Date t Sales ($) a Thu 05 May 1 3700 Fri 06 May 2 2730 Sat 07 May 3 2850 Mon 09 May 4 4190 3353 Tue 10 May 5 b 3496 Wed 11 May 6 4280 3605 Thu 12 May 7 4530 3723 Fri 13 May 8 3620 3859 Sat 14 May 9 3270 3973 Mon 16 May 10 5180 4083 Tue 17 May 11 2600 4223 Wed 18 May 12 5000 c Thu 19 May 13 5120 Fri 20 May 14 4710 Sat 21 May 15 3630

Find the missing values, band c in the table above. (2 marks)

b::;; \ ~ G J..~_ ~ "" 3\.\.-9(." :. 10::::: \ "\ 5 \ ./ ---(,

C::. 4- 3 4- 3 ("<>--l-0ed '=- .--ea'""S;". ch>! !a) /'

Write down the calculation used to find the six-point centred moving average for Mon 16 May. (1 mark)

( 0' S:;< 4-530 ") -\- 3",?-", -I- 3:2-'1-0 -\- SI i5V"I- 21;,=-+ S= + C to'S"" 5llo.)_ /

<;

Predict, with seasonal adjustment, the likely sales for Mon 23 May, assuming that existing trends continue. (4 marks)

C ClIC<..(alc<Z. t,.J.."""cJa::J teSd'-l;=<ls

'-\- \ "I <:> - 3 3 5::' " ";;;?, ';1· ( ..,//

C\..-d .9Q:;i,~! G::>~~~.-t;:;;;

S 1&0>- 4-<:>'3 6 =' ''''>9 '"'1-) /

%3-=1 +I~ T -:c 96"4- ,/ 0J_

use bre,.-d \ l.-e K:,~ ~ I;;;. =- \ b

0.=-1-).2.' SCI",) + 2'8"1-1 -::::. ,+S 3 ( /

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AdJ'-ISlc: f=r sa,,~ ~ ~\;:; fredcJo~ 4-"631 -t 9-0=1 '" ~ S''t 9'6

'c~-_ / ~;; ~ ....,...61::')-Q



Question16 (7 marks)

The spreadsheet below was used to model the balance of an annuity used to help pay the university fees of a student. $19 000 was deposited in a fixed-interest savings account at the start ofthe year 2011. One year later interest was added and then the student withdrew $4 480. This process was repeated until the balance of the annuity reached zero at the start of the year 2016.

Date Balance Interest Withdrawal 0110112011 19000.00 1092.50 4480.00 0110112012 15612.50 897.72 4480.00 0110112013 12030.22 691.74 4480.00 0110112014 8241.96 473.91 4480.00 0110112015 4235.87 243.56 4479.43 0110112016 0.00

a) How much did the annuity provide in total towards the student's university fees? (1 mark)

t..\.-'-\- 'S\::> X It- + 4-4-"'T 9 . Lf. 3::::' -$?. 2- 399· ,+3 ./

b) What was the annual percentage interest rate? (1 mark)

c)

d)

e)

(\ 0"\'.1. . S'o ~ \9 Cl'Q"O) X. \ = -:::... '5. -=r r to ,../

Explain why the final withdrawal was less than $4 480. (1 mark)

Bz~\.t9Z ~ r=vCY\'~'S. ~\O.-1ce~ pl 0""$ I~Z:/~S~ ortJ o<"V\~!""lc....zd

b=> :$ Li-1.!-'::j-"f'y..3 / 9=> ~e I....-<:l.S v'l='-" e.~e'.-, ""', ~-J 'S). Ll-'+-S'c ,../

Write a recursive rule to generate the balance at the start of each year from 2011 until 2015. ./ ../ (2 marks) t: ... \ '" T,:, X I' 0 s ::r s- - 4-1+"8"<:> T, -= 19 -0"""'"

Suggest two changes to this annuity so that it would provide a larger amount towards the student's university fees. (2 marks)

Iv'u <:2ose l;;hZ:

\ V'"C..f El:Asc, = ~, .. n',,\f"}~"'-.,_,~ \~

r'

'5\oOr'''' ea~ Ii<?.-/"

,,;..,,,=,,,,,\ ~~i'.;:;. j O.-"lV4.....a.\ \-rk;;c:ies.1;;:; 'IC\\;:C a 0 \"'-'\:;..5JC.~;C; (';r,~-~ r:...<"C<~ ,c:/-"",,",Ij

;2.. {\ "",,,,,,IA E'CKJA J

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Question 17

13 MATHEMATICS 3AJB CALCULATOR ASSUMED

(6 marks)

A cylinder of radius r and height h is such that the sum of the radius and height is 42cm.

a) Show that the volume of the cylinder is given by V = 42m,2 _1I:r3 .

l-\-h':::' l.\~ 2.

\,.., ,," \\0 02.,0' r / V~ 1\1" • h V-:::.11,"'"C ll-2 -( )

V -::::: 4-'k 7T r""]. -'Uyo 1 /

(2 marks)

b) Use calculus techniques to find the maximum possible volume ofthe cylinder, con-ect to 3 significant figures, (4 marks)

- '34- IT r - 371,. .,. ,/ dv ck

dv'. Q --""""""--~ ~.

",;\t'

(::::0 Or' (-:';1.'1$ /

i . V"VO.l('rv''',-v"\ v",I".......-g ~/' yo::' 28 /

V <J \ '-I V\'"'<Z::: 3 4- '+ S:2. CI""Y\ 3.

-"'.. "3 4- S""""" G-w->::' boo 3,sr/

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Question 18


(8 marks)

An isosceles triangle has a base length that is k times the length of the two congruent sides, as shown in the diagram.

e x x

kx

a) Use the cosine rule to write an expression for cos 8 in telms of k only, where 8 is the angle between

b)

the two congruent sides. (3 marks)

(;::'s e-::; X"'-+:;c." - ( \A.:x:-)''' J '- ~ :x.-x:>C-

Cos e -::. 2.:X;,:J. _ It ." ;{~ :J.

t,~ 2<.

c::'s e "

G::>s <a ::

-:X .. 1. (t. - 1<:.') :1.::(" ,.

2-\'<"'- / ~-.:::.~ 0 r .J..

CoS e::: - Q • S Ie. "'" -j- I

Determine the size of 8, correct to 3 significant figures, when k = 1.5.

Cos e '" (2. - \' S7.) -=:. - Q' 1:2 S 2.

e -: '4 --'1 • ;2..'" /

(l mark)

c) Use your answers to questions a) and b) above to write an expression for the area of an isosceles triangle with a k value of 1.5 in terms of x only, correct to three significant figures.

(2 marks)

.,.. 0/ Aveo. == o· S -::;t::.- X Si", 9-=1-' :L

PI V' e '" -:::.. C). 4-9 (" ::>G '2. /

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Question 19


(6 marks)

A small toy train is able to travel backwards and forwards along a straight track built on level ground. The displacement in metres, of the train relative to point A, is shown on the graph below for the interval o ~ t ~ 20 seconds.

Displacement LUE 15-

1

5

-10

-1 ~

-2

t

[0

a) In what interval of time was the rate of change of displacement constant and what was that rate of change, in centimetres per minute? / (3 marks)

'1 S I::. S :l <:) se~J5 v G-.radi e,.-,"=- ~ 2 V-'-' I 'S /

-:: :<.x 1<:>"" X b~

-.;.. \ 2. 0 <:)"'> G..-n I .... .., i...., /

b) Estimate the area between the time axis and the displacement graph for the interval 0 ~ t ~ 4 seconds using the average area of circumscribed and inscribed rectangles with widths of 1 second .

C\re"....rv\s::.r-.~ "'" Ib -1-15 + 12t--=t::. 5"", / .' (3 marks)

\'rl.5C ..... ""cd "'" \ 5-1-\;' -I- -4 +- <:) ::;;. 3 'to /

{\ • .+Gr "':le- --.:;- :5=+31..\· 4-2 M5/ ~

END OF EXAMINATION


15 MATHEMATICS 3AIB CALCULATOR ASSUMED

d) Hence detennine the value of x for the triangle below, COlTect to the nearest integer. (2 marks)

e x x

Area = 71.435 em2

1.5x

b / 1\'4-35 "': '0'4-9 (,% ,/

\ 4-lt- • Q 2.2. "" :;!L 1-

" . X= I-:;LCn"'I /.

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TRIAL EXAMINATION 2011 SECTION TWO ADDITIONAL WORKING SPACE

Question number:. __ _


TRIAL EXAMINATION 2011 SECTION TWO ADDITIONAL WORKING SPACE

Question number: __ _




Question

8

9

10

11

12

13

14

15

16

17

18

19

TOTAL SECTION 2

20

r---'----,

Marks Available

4

6

9

7

4

11

5

7

7

6

8

6

80

MATHEMATICS 3A1B CALCULATOR ASSUMED

Your Mark

2011 trial wace exam solutions

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