Mathematics 3A3B Calculator Assumed Examination 2011 PDF

8/3/2019 Mathematics 3A3B Calculator Assumed Examination 2011 PDF

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Copyright Curriculum Council 2011 Ref: 11-134

Western Australian Certificate of Education

Examination, 2011

Question/Answer Booklet

MATHEMATICS3A/3B

Section Two:Calculator-assumed

Student Number: In figures

In words ______________________________________________________________

______________________________________________________________

Time allowed for this sectionReading time before commencing work: ten minutesWorking time for this section: one hundred minutes

Materials required/recommended for this sectionTo be provided by the supervisorThis Question/Answer BookletFormula Sheet (retained from Section One)

To be provided by the candidateStandard 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 CurriculumCouncil for this examination

Important note to candidatesNo other items may be taken into the examination room. It is yourresponsibility to ensure thatyou do not have any unauthorised notes or other items of a non-personal nature in theexamination room. If you have any unauthorised material with you, hand it to the supervisorbefore reading any further.

Please place your student identification label in this box


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MATHEMATICS 3A/3B 2 CALCULATOR-ASSUMED

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Structure of this paper

SectionNumber ofquestionsavailable

Number ofquestions tobe answered

Working time(minutes)

Marksavailable

Percentageof exam

Section One:Calculator-free 7 7 50 40

Section Two:Calculator-assumed 12 12 100 80

Total 120 100

Instructions to candidates

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

2. Write your answers in the spaces provided in this Question/Answer Booklet. Sparepages are included at the end of this booklet. They can be used for planning yourresponses 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 thepage.

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

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


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CALCULATOR-ASSUMED 3 MATHEMATICS 3A/3B

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Section Two: Calculator-assumed (80 Marks)

This section has twelve (12) questions. Answerall questions. Write your answers in the spacesprovided.

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 thenumber of the question(s) that you are continuing to answer at the top of the page.

Working time: 100 minutes.

Question 8 (4 marks)

Given the universal set {2, 3, 4, 5, 6, 7, 8, 9}, with A = {2, 4, 6, 8} and B = {2, 3, 5, 7}:

(a) list the elements of;

(i) AB (1 mark)

(ii) A B (1 mark)

(b) determine;

(i) n (A B) (1 mark)

(ii) n (A B) (1 mark)


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The masses of lettuces sold at a fruit and vegetable shop are normally distributed with a meanmass 600 g and standard deviation 20 g.

(a) If a lettuce is chosen at random, determine the probability that its mass lies between

570 g and 610 g. (1 mark)

(b) Determine the mass exceeded by 7% of the lettuces correct to three (3) significantfigures. (2 marks)

(c) In one day, 1000 lettuces are sold. Estimate how many weigh less than 545 g.(2 marks)


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A man sells his present business in order to start a small car-hire firm. He has $144 000 tospend on vehicles, so he decides to buyx large cars at $18 000 each andy small cars at$12 000 each. He also decides he must start with at least 10 cars altogether, and that he oughtto have at least twice as many small cars as large cars.

(a) Write three inequalities, other than x 0 and y 0 , which satisfy the above conditions,simplifying where possible. (3 marks)

(b) Draw the three inequalities from (a) on the graph and shade the region satisfying these

inequalities.

(4 marks)

(c) What is the maximum number of large cars he can buy, subject to these conditions?

(1 mark)


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A train is travelling at 30 metres per second when the brakes are applied. The velocity of the

train is given by the equation v = 30 0.3t2

where trepresents the time in seconds after thebrakes are applied.The velocity-time graph is shown below.

The area under a velocity-time graph gives the total distance travelled for a particular timeperiod.

(a) Complete the tables below and estimate the distance travelled by the train during thefirst 6 seconds by calculating the mean of the areas of the circumscribed and inscribedrectangles. (The rectangles for the 46 seconds interval are shown on the grid above)

Time (t) 0 2 4 6

Velocity (v) 28.8 19.2

Rectangle 02 24 46 Total

Circumscribed area 50.4

Inscribed area 38.4

Estimated distance travelled: ____________________________________________________________________________________________________________ (5 marks)

(b) The exact distance travelled during the first 6 seconds is 158.4 m.How could you determine a better estimate of the distance travelled by the train duringthe first six seconds than the one determined in (a)? (1 mark)


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Consider the function4 22 8. y x x=

(a) Using calculus techniques, determine the coordinates of the stationary point(s) for thefunction. (3 marks)

(b) Sketch the graph of the function on the interval 2 2x , labelling all intercepts andstationary point(s).

(3 marks)

(c) Determine the coordinates of the point where the tangent to the curve at the positive

x-intercept intersects they-axis. (2 marks)


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The following figures were obtained from the My School website. The second column showsthe ICSEA value for the school (an indication of the socioeconomic circumstances of the schoolpopulation) and the third column shows the mean score for the school on the 2008 NAPLANnumeracy test for Year 9 students in that school.

School ICSEA value (x) Mean NAPLAN score Year 9 Numeracy (y)A 1117 659B 1082 621C 875 480D 955 575E 1129 616F 917 531G 1065 726H 976 581I 902 472J 956 531K 1115 628L 1161 673

(a) Calculate the correlation coefficient rx and the least squares regression line for theabove data. (2 marks)

(b) Calculate the expected mean NAPLAN score for a school with an ICSEA value of 1100.(1 mark)


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(c) The scatter graph and line of regression have been plotted on the graph below. One ofthese twelve schools may be considered to be an outlier. Circle the outlier on thescattergraph below. (1 mark)

(d) Remove this outlier from the given data and calculate the new correlation coefficient and

line of regression. Graph the new line of regression on the scattergraph. (4 marks)

(e) Calculate the new expected mean NAPLAN score for a school with a ICSEA value of1100. (1 mark)

Scattergraph and line of regression for the 2008 NAPLAN numeracy test.

ICSEA value

Mea

nNAPLAN

score


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(f) Describe the influence of the outlier on the different values for the mean NAPLANscores calculated in (b) and (e). (1 mark)

(g) Comment on the reliability of your prediction from (e). (2 marks)


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The equation of a curve is given by2(9 )(3 ) y x x= .

(a) Use the product rule to determine dydx

. Do not simplify your answer. (2 marks)

(b) Solve the equationdydx

= 0 and state what these values ofx represent in terms of the

given curve. (3 marks)


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A curve has a gradient function ddt

= 60 3at2

, where a is a constant.

Given that the curve has a maximum turning point when t= 2 and passes through the point(1, 62), determine the equation of the curve.


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The graphs of the functions (x) and (x) are given below.

(a) State the solution(s) to f(x) =g(x) . (1 mark)

(b) State the domain of (x) . (2 marks)

(c) State the range of f(x) . (1 mark)

(d) Estimate '(3) by drawing an appropriate tangent to the curve. (2 marks)


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A scientist records the number of trout in a section of a river on ten successive days. His resultsare:

28, 26, 23, 23, 25, 27, 21, 58, 22, 23

(a) Calculate the mean and median for these data. (2 marks)

(b) State which is the more appropriate average for describing the number of fish in thatsection of the river on a typical day. Justify your choice. (2 marks)

(c) Calculate the standard deviation and interquartile range. (2 marks)

(d) Explain the large difference between the magnitudes of the two measures of dispersioncalculated in (c).

(2 marks)


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(e) The scientist completes his daily observations after one year and collates the data in thefollowing table.

Number of trout (x) Frequency

21 25 13926 30 10731 35 84

36 40 3441 45 046 50 051 55 056 60 1

Total 365

Calculate approximate values for the mean and standard deviation correct to two (2)decimal places for this distribution. (2 marks)


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To save money for an overseas holiday, Bruce started an investment account. He placed aninitial deposit of $6000, and then deposited an extra $200 at the end of each month for oneyear.

The table below shows the amount in the account at the beginning of each month (An), theinterest added to the account each month (In), the deposit made at the end of each month (Dn),

and the amount in the account at the end of each month (An+1) for the first six months.

Note: The values in this table have been truncated to two decimal places

(a) What is the monthly interest rate? (1 mark)

(b) Write a recursive rule to determine the amount in the account at the end of each month.(2 marks)

(c) What was the amount in the account at the end of 12 months? Give your answer to thenearest 10 cents. (1 mark)

(d) What was the total amount of interest earned during the year? (1 mark)

Month (n) Amount atbeginning of

month (An)

Interest formonth

(In)

Deposit formonth

(Dn)

Amount atend of month

(An+1)

1 $6000.00 $48.00 $200.00 $6248.00

2 $6248.00 $49.98 $200.00 $6497.98

3 $6497.98 $51.98 $200.00 $6749.96

4 $6749.96 $54.00 $200.00 $7003.96

5 $7003.96 $56.03 $200.00 $7259.99

6 $7259.99 $58.08 $200.00 $7518.07


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Sally wants to prepare and eat her breakfast in the minimum amount of time. The activitiesinvolved, their immediate predecessors and the time taken to complete each activity is shownbelow.

Activity ImmediatePredecessors

Time taken(minutes)

F

P

W

G

D

O

M

B

E

T

C

Fill kettle

Put tea bag in cup

Boil water

Toast crumpets

Pour out cereal

Fetch and pour milk

Make tea

Butter crumpets

Eat cereal

Eat crumpets

Drink tea

F

D

P, W

G

O

E, B

M, T

0.5

0.5

10

7

0.5

0.5

0.5

0.5

3

5

3

(a) Complete the partially drawn project network above. You may assume that Sally iscapable of doing several activities at the same time. (2 marks)

(b) State the critical path and the minimum time needed for Sally to complete her breakfast,again assuming that she can do several activities at the same time. (2 marks)

F

W

G

B C

M

E

TFinish

Start


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End of questions

(c) Activities W and G do not require Sallys attention. However, she must do the otheractivities one at a time.

Sally starts preparing her breakfast at 7 am.

To complete her breakfast in minimum time, Sally must start activities as in the tablebelow.

Start time Activity

0700GF

Toast crumpetsFill kettle

0700.5WD

Boil waterPour out cereal

(i) What is the earliest time that Sally can start eating her cereal? (1 mark)

(ii) What is the earliest time that Sally can start eating her crumpets? (1 mark)

(iii) What is the earliest time that Sally can actually finish her breakfast? (1 mark)


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Additional working space

Question number: _____________________________


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Question number: _____________________________


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Question number: _____________________________


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ACKNOWLEDGEMENTSSection Two:

Question 13 Data source:My School[Table]. (n.d.). Retrieved July, 2011,from www.myschool.edu.au/.

This examination paper apart from any third party copyright material contained in it may be freely copied, or communicatedon an intranet, for non-commercial purposes in educational institutions, provided that it is not changed and that the CurriculumCouncil is acknowledged as the copyright owner. Teachers in schools offering the Western Australian Certificate of Education(WACE) may change the examination paper, provided that the Curriculum Councils moral rights are not infringed.

Copying or communication for any other purpose can be done only within the terms of the Copyright Act or with prior writtenpermission of the Curriculum Council. Copying or communication of any third party copyright material can be done only withinthe terms of the Copyright Act or with permission of the copyright owners.

Published by the Curriculum Council of Western Australia27 Walters Drive

OSBORNE PARK WA 6017

Mathematics 3A3B Calculator Assumed Examination 2011 PDF

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