2013 FURTHER MATHEMATICS - FUNFURTHER! - … FURTHER MATHEMATICS Written examination 2 STUDENT NAME:...

This trial examination produced by Insight Publications is NOT an official VCAA paper for the 2013 Further Mathematics 2 written examination. This examination paper is licensed to be printed, photocopied or placed on the school intranet and used only within the confines of the purchasing school for examining their students. No trial examination or part thereof may be issued or passed on to any other party, including other schools, practising or non-practising teachers, tutors, parents, websites or publishing agencies, without the written consent of Insight Publications.

Copyright © Insight Publications 2013

INSIGHT YEAR 12 Trial Exam Paper

2013 FURTHER MATHEMATICS

Written examination 2 STUDENT NAME:

Reading time: 15 minutes Writing time: 1 hour 30 minutes

QUESTION AND ANSWER BOOK

Structure of book Core Number of

questions Number of questions to be

answered Number of

marks 3 3 15 Module Number of

modules Number of modules to be

answered Number of

marks 6 3 45 Total 60

x Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, one bound reference, one approved graphics calculator or approved CAS calculator or CAS software and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared.

x Students are NOT permitted to bring into the examination room: blank sheets of paper and/or white out liquid/tape.

Materials provided x The Question book of 35 pages, with an answer sheet for the multiple-choice questions. x A separate sheet with miscellaneous formulas. x Working space is provided throughout the question book. Instructions x Remove the formula sheet during reading time. x Write your name in the box provided above on this page. x All written responses must be in English.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

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CONTINUES OVER PAGE

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TURN OVER Copyright © Insight Publications 2013

Section Page Core.....................................................................................................................................3 Module Module 1: Number patterns.........................................................................................12 Module 2: Geometry and trigonometry.......................................................................15 Module 3: Graphs and relations...................................................................................20 Module 4: Business-related mathematics....................................................................24 Module 5: Networks and decision mathematics..........................................................30 Module 6: Matrices......................................................................................................36

Instructions

This examination consists of a core and six modules. Students should answer all questions in the core and then select three modules and answer all questions within the modules selected. You need not give numerical answers as decimals unless instructed to do so. Alternative forms may involve, for example, ʌ, surds or fractions. Diagrams are not to scale unless specified otherwise.

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Core – Question 1 – continued Copyright © Insight Publications 2013

SECTION A Core: Data Analysis Question 1 (6 marks) The weights, in kilograms, of two different breeds of sheep (Merinos and Dorpers) are compared in the back-to-back stemplot shown below.

Merinos Stem Dorpers

0

0

3 1 3 4 4 4

9 6 1 5 6 7 9 9

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

9 9 8 7 7 6 5 5 2 6 6 7 8 9

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

9 8 7 6 3

4 3

4

5

Key: 3|4 represents 34 kg a. The boxplot of the weights of Dorpers is shown on the diagram below. On the same scale draw the boxplot of the weights of the Merinos.

2 marks

0 5 10 15 20 25 30 Sheep weights (kg) 35 40 45

Dorpers

Merinos

5

Core – continued TURN OVER


b. Compare the distribution of the weights of Dorpers with that of the weights of Merinos in terms of shape, centre and spread.

3 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. Showing your working, explain why the Dorper weighing 43 kg is shown as an outlier.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

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6


Question 2 (5 marks) The table below shows the age of sheep, in months, and the average sale price of the sheep.

Age of sheep (months) Average sale price 6 $65

7 $68

8 $78

9 $85

10 $96

11 $98

12 $103

13 $104

14 $105

15 $107

16 $108

17 $110

18 $110

The data has been used to create the scatterplot shown below.

Age (months)

Average sale price ($)

10 14 18

80

100

60 6

7

Core – continued TURN OVER


A regression analysis was performed and the following results were obtained: r = 0.9283 r2 = 0.862 Least squares regression line: Sale price = 49.66 + 3.79 × age a. What feature of the scatterplot suggests that the least squares regression line

would not be the best model to use to predict the sale price of sheep based on their ages?

1 mark _______________________________________________________________

b. A residual plot is to be constructed to further investigate the linearity of the scatterplot above. Calculate the residual value for sheep aged 12 months.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. A 1x

transformation could be used to linearise the data. Name one other type of

transformation that may linearise the data. 1 mark

_______________________________________________________________

d. Apply a 1x

transformation to the data and find a regression model for the

relationship between age and sale price. 1 mark

_______________________________________________________________

e. Use the 1x

model you obtained in part d to predict the sale price of a

24-month-old sheep. (Give your answer to the nearest dollar.) 1 mark

_______________________________________________________________

_______________________________________________________________

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Question 3 (4 marks) The rainfall on Doreen’s Dorper Sheep Farm was recorded in each month of the 2012 year and is shown in the table below.

Month Rainfall (mm) January 10

February 34

March 45

April 13

May 45

June 56

July 67

August 75

September 45

October 86

November 45

December 34

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Core – Question 3 – continued TURN OVER


Seasonal indices for the monthly rainfall has been calculated using 100 years of historical data from rainfall records at Doreen’s Farm. Most of the seasonal indices are shown in the table below.

Month Seasonal index January 0.80

February 0.83

March 0.85

April 0.92

May 0.98

June

July 1.06

August 1.10

September 1.15

October 1.15

November 1.12

December 1.00

a. Calculate the seasonal index for June.

1 mark _______________________________________________________________

_______________________________________________________________

10


Most of the raw data for 2012 has been deseasonalised. See the table below.

Month Rainfall (mm) Deseasonalised rainfall January 10 12.50

February 34 40.96

March 45 52.94

April 13 14.13

May 45 45.92

June 56 53.85

July 67

August 75 68.18

September 45 39.13

October 86 74.78

November 45 40.18

December 34 34.00

b. Calculate the deseasonalised rainfall for July. (Round off your answer to 2

decimal places.) 1 mark

_______________________________________________________________

_______________________________________________________________

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END OF CORE Copyright © Insight Publications 2013

c. The deseasonalised rainfall data has been used to calculate the least squares regression equation. When calculating the equation, the numbers 1 – 12 were substituted for the months of January 2012 through to December 2012. The equation was found to be:

Deseasonalised rainfall = 31.72 + 2.04 (month)

Use the equation and the seasonal indices to predict the rainfall for September 2013. (Round off your answer to the nearest whole mm.)

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

12

Module 1: Number patterns – continued Copyright © Insight Publications 2013

SECTION B Module 1: Number patterns Question 1 (3 marks) Henrietta has started raising hens to produce her own eggs. She starts her flock with five Bantam hens and she feeds them 500 g of golden yolk pellets each week. She knows that for each extra Isa Brown hen she keeps, she will need to feed the flock at least an extra 110 g of golden yolk pellets each week. a. If Henrietta increases her original flock to eight hens by adding three Isa Browns, how much will she need to feed the flock each week?

1 mark _______________________________________________________________

_______________________________________________________________

b. Henrietta decides she can afford to purchase 2000 g of golden yolk pellets each week. If she keeps the five Bantams and purchases more Isa Browns, how many hens in total can she afford to feed?

2 marks _______________________________________________________________

_______________________________________________________________

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Module 1: Number patterns – continued TURN OVER


Question 2 (7 marks) After a year, Henrietta decides that the egg production is not working out as well as she had hoped. Instead, Henrietta decides she will breed the hens to increase her flock numbers. In the first year, she starts with 100 hens and each subsequent year Henrietta plans to double the number of hens through breeding, plus she will purchase fifty more hens. a. A difference equation that describes this situation can be written as tn = atn–1 + b, where t1 = c.

Write down the values of a, b and c. 2 marks

_______________________________________________________________

b. Determine the first 5 terms of the sequence generated by this difference equation. Show that the sequence is neither arithmetic nor geometric.

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. Using the formula 1

11

( 1)( 1)

nn

nb at a t

a

��

��

, show that the solution to the difference

equation is given by tn = 150 × 2n–1 Ȃ 50. 2 marks

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

d. Using the expression from part c, or otherwise, determine the 7th term of the sequence.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

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END OF MODULE 1 Copyright © Insight Publications 2013

Question 3 (5 marks) Henrietta is doing so well with her chicken farm that she decides to expand into duck farming. In her first year of duck farming she starts with 200 ducks. Her duck population increases according to a geometric sequence. Duck numbers at the start of each of the first 3 years are shown in the table below. Ducks are sold at market before they reach 1 year of age, so each year the ducks on Henrietta’s farm are new ducks.

Year 1 2 3 Number of ducks 200 300 450

a. Show that the common ratio is 1.5.

1 mark _______________________________________________________________

_______________________________________________________________

b. During which year of duck farming would the number of ducks first exceed 2000?

1 mark _______________________________________________________________

_______________________________________________________________

c. How many ducks would Henrietta have sent to market after 10 years of operating her duck farm? Express your answer to the nearest whole duck.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

d. A difference equation that generates this geometric sequence is of the form

dn+1 = adn + b, where d1 = 200. Find the values of a and b.

2 marks _______________________________________________________________

15

Module 2: Geometry and trigonometry – continued TURN OVER


Module 2: Geometry and trigonometry Question 1 (2 marks) Bill has just built new cattle yards with a loading ramp. The cattle yards are on level ground and the cattle walk up the ramp from point A onto the truck at point B. The sloping edge (AB) is 3 m in length and the ramp makes an angle of 20° with the ground. The diagram below is a cross-sectional view of the ramp. a. How high above the ground are the cattle when they walk onto the truck? (Give your answer correct to 1 decimal place.)

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

b. What is the horizontal distance that the ramp extends into the yards? (Give your answer correct to 1 decimal place.)

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

A

B

20°

16

Module 2: Geometry and trigonometry – continued Copyright © Insight Publications 2013

Question 2 (5 marks) One of the holding yards of Bill’s cattle yards is in the shape of a triangle, as shown below. The angles at vertices A and C are 30° and 47°, respectively. The AB side of the holding yard is 5 m. a. Find the angle at vertex B.

1 mark _______________________________________________________________

_______________________________________________________________

b. Find the length of BC, to the nearest centimetre.

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. Bill is trying to work out how many steers he can fit into the holding yard, so he measures length AC and finds it to be 8.45 m. If a steer requires at least 1.5 m2 of space when in a small yard, how many steers can Bill fit into the triangular holding yard?

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

A

B

C30° 47°

5 m

17

Module 2: Geometry and trigonometry – continued TURN OVER


Question 3 (4 marks) George is planning to build some new fences on one section of his property. He starts by putting some pegs into the ground. He puts in the first peg at point A. He then sets out from point A on a bearing of 035°T for 100 m and places a peg at point B. He then walks on a bearing of 120°T for 120 m and puts a peg in at point C. a. Find the distance from C to A (correct to 2 decimal places).

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

b. What is the bearing of A from C? (Give your answer correct to the nearest degree.)

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

A 35°

B N

100 m

C

120 m

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Module 2: Geometry and trigonometry – Question 4 – continued Copyright © Insight Publications 2013

Question 4 (4 marks) George now plans to build a shed in his newly fenced paddock. The shed will be in the shape of a rectangular prism with a triangular prism on top. The ends of the roof section are in the shape of isosceles triangles with slant edges of 6 m.

a. Find the surface area of the shed, excluding the rectangular base.

1 mark ________________________________________________________________

________________________________________________________________

________________________________________________________________

b. Find the volume of the building.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. A scale model of the shed is built. For the scale model, the 30 m length is represented by a 30 cm length and other side lengths are scaled down accordingly. State the size ratio of the shed to the model.

1 mark _______________________________________________________________

_______________________________________________________________

30 m 10 m

6 m

6 m 6 m

19

END OF MODULE 2 TURN OVER


d. Using your answers to parts a and c or another method, find the area of the model, excluding the base.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

20

Module 3: Graphs and relations – continued Copyright © Insight Publications 2013

Module 3: Graphs and relations Question 1 (4 marks) Stacey has decided to make surfboards in her backyard for some extra income. It costs her $1500 at the start to purchase some equipment she will need and, on top of that, it will cost her $500 in raw materials to make each board. Stacey plans to sell each surfboard for $900. She will make boards only to order, so all boards that Stacey makes she will sell. a. Write a revenue equation for Stacey’s surfboard-making business, where

R = revenue and n represents the number of boards she sells. 1 mark

_______________________________________________________________

b. Write a cost equation for Stacey’s surfboard-making business, where C = cost and n represents the number of boards she makes.

1 mark _______________________________________________________________

c. Write a profit equation (where P = profit) in terms of n.

1 mark _______________________________________________________________

_______________________________________________________________

d. Use your answer to part c to find the minimum number of surfboards Stacey needs to make (and sell) to start making a profit.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

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Module 3: Graphs and relations – Question 2 – continued TURN OVER


Question 2 (2 marks) Stacey’s friend Willis is also a surfer. He has just completed his chemistry degree and has decided to make surfboard wax to finance his surfing career. He makes two types of wax: Sticky wax and Cling On wax. A block of Sticky wax requires 80 g of beeswax and 40 g of coconut oil. A block of Cling On wax requires 50 g of beeswax and 60 g of coconut oil. Both beeswax and coconut oil are difficult to obtain and the most that Willis can purchase each week is 8000 g of beeswax and 6000 g of coconut oil. Using x to represent the number of Sticky wax blocks Willis can make each week and y to represent the number of Cling On wax blocks he can make each week, two of the inequations that describe the constraints are shown below.

00

xytt

Write the other two constraints.

_____________________________________________________________________

_____________________________________________________________________

Question 3 (9 marks) Jake has just finished his business degree and he also surfs. He decides to make his fortune by starting up a surfboard hire business. Jake has two types of boards for hire: Malibus and short boards. x is the number of times Jake hires out Malibus in a month. y is the number of times Jake hires out short boards in a month. Because Jake also surfs, he opens the shop for only 20 days each month. Board hire is for the whole day only. There are constraints on the number of times that Jake can hire out boards in a month due to the number of Malibus he has, the number of short boards he has and the quality of the surf. The inequalities below define these constraints.

00200100

2 300 (quality of surf constraint)

xyyx

y x

ttdd

� d a. How many Malibus does Jake have available for hire?

1 mark _______________________________________________________________

_______________________________________________________________

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Module 3: Graphs and relations – Question 2 – continued Copyright © Insight Publications 2013

The lines x = 0, y = 0, y = 0 and x = 100 are already sketched on the axes below.

b. On the set of axes above, sketch the line 2 300y x� .

1 mark c. Shade in the feasible region defined by these constraints.

1 mark d. Jake makes a profit of $50 for each Malibu he hires out and $70 for each short board he hires out. Write an objective function for the profit in terms of x and y.

1 mark _______________________________________________________________

e. What is the maximum profit Jake can earn in a month?

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

y

200

100

50 100 x

23



f. How many of each type of board must Jake hire out in the month to earn this maximum profit?

1 mark _______________________________________________________________

_______________________________________________________________

Since finding he is able to hire out all of his short boards each month he decides to buy some more short boards. All of the constraints except the short board constraint ( 200y d ) will remain the same. The objective function also remains the same. g. Find the new short board constraint and, hence, the number of extra short boards Jake should purchase to maximise his profit.

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

h. What will be Jake’s new maximum profit with the extra short boards?

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

24

Module 4: Business-related mathematics – continued Copyright © Insight Publications 2013

Module 4: Business-related mathematics Question 1 (2 marks) Tony sells an investment property that he has owned for 10 years for $845 000 and is charged $21 125 in fees by the real estate agent. a. Calculate the percentage of the selling price that Tony is paying to the real estate agent.

1 mark _______________________________________________________________

_______________________________________________________________

b. Tony makes a capital gain of $450 000 on the property. He must pay Capital Gains Tax on half of the capital gains at the rate of 38%. Calculate the amount of Capital Gains Tax that Tony must pay.

1 mark _______________________________________________________________

_______________________________________________________________

25

Module 4: Business-related mathematics – continued TURN OVER


Question 2 (4 marks) Brenda buys a new entertainment unit from a department store. The entertainment unit is advertised for $2499. Brenda does not have enough cash, so she signs up to a hire purchase agreement. The conditions of the agreement are that she pays a $99 dollar deposit and then monthly repayments of $120 for 2 years. a. Calculate the total amount of interest that Brenda pays.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

b. Calculate the flat interest rate that the store advertises to entice customers to sign up to a hire purchase agreement.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. Calculate the effective interest rate that Brenda pays. Give your answer correct to 1 decimal place.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

d. Explain why the flat interest rate that is advertised may be misleading and why the effective interest rate is a more accurate measure of the interest rate being charged to customers.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

26

Module 4: Business-related mathematics – continued Copyright © Insight Publications 2013

Question 3 (3 marks) Harvey has just inherited $55 000 from his grandparents’ estate. He is trying to decide how to invest the money. a. If Harvey invests the money in a simple interest account at the rate of 4.75% per annum for 5 years, how much will he have in total after 5 years?

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

b. Harvey is also considering a compound interest account that earns 4.25% per annum, compounding monthly. If he invests the $55 000 into this account, how much will he have after 5 years?

1 mark _______________________________________________________________

_______________________________________________________________

c. Harvey is not satisfied with either the simple interest investment or the compound interest investment. He wants to have a total of $80 000 at the end of the 5 years. He decides to invest in a compound interest account that pays 4.75% per annum, compounding monthly. At the end of each month Harvey will also deposit an amount into the account (the same amount each month). How much does Harvey need to deposit each month if the investment is to be worth $80 000 at the end of the 5-year period?

1 mark _______________________________________________________________

_______________________________________________________________

27

Module 4: Business-related mathematics – continued TURN OVER


Question 4 (3 marks) Bridgit has purchased a new home at auction for $655 000. To be able to pay for the house, she takes out a reducing balance loan from the bank for the sum of $520 000. The bank charges interest at the rate of 8.75%, compounding quarterly. Bridgit plans to pay off the loan over a period of 20 years. a. What quarterly instalment must Bridgit pay the bank? Express your answer to the nearest cent.

1 mark _______________________________________________________________

_______________________________________________________________

b. How much interest will Bridgit pay over the period of the loan?

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

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END OF MODULE 4 Copyright © Insight Publications 2013

Question 5 (3 marks) Fred buys a new truck for his transport business. The truck costs him $125 000. He will claim the truck’s annual depreciation as a tax deduction. a. The truck travels 40 000 km each year and Fred calculates that it depreciates by 70 cents per kilometre. By how much does the truck depreciate each year?

1 mark _______________________________________________________________

_______________________________________________________________

b. If Fred calculates the depreciation using flat rate depreciation at the rate of 24% per annum, what will be the truck’s value after 3 years?

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. If Fred calculates the value of the truck using reducing balance depreciation at the rate of 20% per annum, after how many years will the truck’s value be less than $40 000?

1 mark _______________________________________________________________

_______________________________________________________________

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CONTINUES OVER PAGE

30

Module 5: Networks and decision mathematics – Question 1 – continued Copyright © Insight Publications 2013

Module 5: Networks and decision mathematics Question 1 (6 marks) The diagram below represents witches hats (A – H) placed on a school oval. The numbers represent the distances, in metres, between the witches hats. a. What is the shortest distance from A to G?

1 mark _______________________________________________________________

b. The school groundsmen wish to mark lines between some of the witches hats so that every hat can be reached from any other hat by following a path along the painted lines. The groundsmen wish to do this by marking the minimum length of total lines. What is the mathematical term used for the network described?

1 mark _______________________________________________________________

35

C A

B E

H

G

F

D

45

11

25 14

17

11

13 13 9

27

28

38

42

31

Module 5: Networks and decision mathematics – continued TURN OVER


c. Draw the network described in part b on the diagram below. 1 mark

d. What is the total length of the lines marked in part c?

1 mark _______________________________________________________________

e. Describe a Hamilton circuit starting at F, and state its length.

2 marks _______________________________________________________________

_______________________________________________________________

A

F

E

D

C

B

H

G

32

Module 5: Networks and decision mathematics – continued Copyright © Insight Publications 2013

Question 2 (3 marks) At the end of each day the groundsmen must also check that all classrooms are locked and that pathways between the classrooms are free of litter. The network below illustrates the classrooms (A–G) and the walkways between the classrooms. a. Can a groundsman check all the pathways without having to go along any pathway twice? If so, at which classroom should he start and at which classroom should he finish?

1 mark _______________________________________________________________

_______________________________________________________________

b. The groundsmen at this school are lazy and wish to start and finish their pathway checking at the same classroom. Between which two classrooms would a new walkway need to be constructed to make this possible? What is the mathematical term to describe this new classroom checking route?

2 marks _______________________________________________________________

_______________________________________________________________

A

G

F

E

D

C

B

33


CONTINUES OVER PAGE

34

Module 5: Networks and decision mathematics – Question 3 – continued Copyright © Insight Publications 2013

Question 3 (6 marks) The school is building a new trade training centre. The tasks that must be completed (A–K) to build the new centre and the times required to complete most tasks, in weeks, are shown in the network diagram below.

The tasks and some of their earliest start times (EST) are shown in the table below. (None of the tasks’ latest start times, LST, are included.)

Task EST LST A 0

B 0

C 0

D

E

F

G

H 8

I

J

K

A, 3

D, 6

I, 4 E, 3

G, 1 B, 5

C, 4 F, x K, 3

H, 2

J, 3

X, 0

35



a. The time to complete activity F is represented on the network diagram with an x. How many weeks does activity F take?

1 mark ________________________________________________________________

________________________________________________________________

b. Complete the table opposite for earliest start and latest start times.

1 mark c. How much float (or slack) time does activity D have?

1 mark _______________________________________________________________

d. What is the minimum number of weeks in which the trade training centre can be completed?

1 mark _______________________________________________________________

_______________________________________________________________

e. Activities B and C can each be completed 3 weeks quicker than originally planned. However, for each activity, each week of time reduction will cost the school an extra $10 000. What is the new minimum completion time and what is the minimum extra money that needs to be spent to achieve this?

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

36

Module 6: Matrices – continued Copyright © Insight Publications 2013

Module 6: Matrices Question 1 (3 marks) The price for cattle, sheep, pigs and goats on a particular day at a market are shown on matrix P below.

550 cattle85 sheep

180 pigs120 goats

P

ª º« »« » « »« »¬ ¼

The number of cattle, sheep, pigs and goats are shown in matrix N below.

> @ C S P G

(203) (567) (45) (32)N

a. What is the order of matrix P?

1 mark _______________________________________________________________

b. Which matrix product, PN or NP, gives the total sales of all cattle, sheep, pigs and goats sold at the market? Give reasons for your answer.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. Calculate the total sales of all cattle, sheep, pigs and goats sold at the market.

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

37

Module 6: Matrices – continued TURN OVER


Question 2 (4 marks) A butcher gets his meat directly from the market and is able to sell direct to the public from one of three outlets. His prices for beef, goat and pork per kilogram are represented by b, g and p, respectively. His sales from his three butchers’ outlets are represented by the three equations below.

50b + 20g + 15p = 1140 25b + 30g + 25p = 985 45b + 25p = 925

These equations can be represented by the matrix equation

50 20 15 114025 30 25 98545 0 25 925

bgp

ª º ª º ª º« » « » « »u « » « » « »« » « » « »¬ ¼ ¬ ¼ ¬ ¼

a. Find the determinant of 50 20 1525 30 2545 0 25

ª º« »« »« »¬ ¼

.

1 mark _______________________________________________________________

b. Find the inverse matrix of 50 20 1525 30 2545 0 25

ª º« »« »« »¬ ¼

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. Find the price per kilogram of beef, goat and pork by solving the matrix equations.

2 marks _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

38

Module 6: Matrices – Question 3 – continued Copyright © Insight Publications 2013

Question 3 (8 marks) The management group of the Safeless supermarket chain was researching the grocery shopping habits of people in the town of Brightbeach. The number of people shopping at each of three supermarkets, Safeless, Myway and Coldstore, in the town of Brightbeach in the first week of the study is given by the initial state matrix S0, where

0

250 375

120

SS M

C

ª º« » « »« »¬ ¼

Shopping habits of customers were tracked by identifying them through credit card numbers and a transition matrix was created to predict shopping habits from one week to the next. The transition matrix T is shown below. This week

0.7 0.0 0.1 0.1 0.9 0.3

0.2 0.1

S M CS

T M Next weekCx

ª º« » « »« »¬ ¼

a. How many people went shopping in the town of Brightbeach in the first week

of the study? 1 mark

_______________________________________________________________

b. In the transition matrix T above, element x = 0.6. What does the matrix tell us

about the number of people shopping in Brightbeach each week? 1 mark

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

c. What proportion of people who shopped at Myway in the first week will shop at Coldstore in the second week?

1 mark _______________________________________________________________

39


d. If S1 = TS0, find the matrix S1. 1 mark

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

e. S1 represents the number of shoppers at each of the three supermarkets in which week of the study?

1 mark _______________________________________________________________

f. How many shoppers in Brightbeach switched from Safeless to Myway from the first week to the second week of the study?

1 mark _______________________________________________________________

g. How many people are expected to shop at Myway in the 6th week of the study?

1 mark _______________________________________________________________

_______________________________________________________________

_______________________________________________________________

_______________________________________________________________

h. Find the number of shoppers at each of the three supermarkets in the long term.

1 mark _______________________________________________________________

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END OF QUESTION AND ANSWER BOOK

2013 FURTHER MATHEMATICS - FUNFURTHER! - … FURTHER MATHEMATICS Written examination 2 STUDENT NAME:...

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