mathematics 504 cst - wordpress.com · which of the following graphs has a path that starts at p,...

Mathematics 504 CST

Mid-Year Examination

January 2011

Question/Answer Booklet

Student's Name

Group Date

563-504

FOR TEACHER USE ONLY

Part A ____________ /24

Part B ___________ /16

Part C ___________ /60

Total __________ /100


Instructions

1. Provide all the required information in the spaces in this booklet. 2. There are 16 questions in this booklet divided into three parts:

A, B and C.

3. Part A contains 6 multiple choice questions worth 4 marks each. Part B contains 4 short answer questions worth 4 marks each. Part C contains 6 application questions worth 10 marks each.

4. Answer the questions in Part A by darkening the letter that corresponds to the answer chosen. Answer the questions in Part B and C in the space provided.

5. For Part C, you must show all your work to justify your answer. The steps in your procedure must be organized and clearly presented.

6. You are permitted to use graph paper, a ruler, a compass, a set square,

a protractor and a calculator.

7. You may use a calculator with or without graphic display (you must indicate the sequence of operations involved, but you do not have to rewrite all the detailed calculations performed with the calculator).

8. You may refer to the memory aid you prepared on your own before the

examination. The memory aid consists of one letter-sized sheet of

paper (8½ 11). Both sides of the sheet may be used. Any mechanical reproduction of this memory aid is forbidden. All other reference materials are forbidden.

9. The length of this examination is 3 hours. Note: Figures are not necessarily drawn to scale.


The following are the evaluation criteria for the different competencies required to

complete the tasks in this booklet.

Evaluation Criteria

Competency 2: Uses Mathematical Reasoning

Cr 1 - Formulation of a conjecture appropriate to the situation

Cr 2 - Correct application of the concepts and processes appropriate to the situation

Cr 3 - Proper implementation of mathematical reasoning suited to the situation

Cr 4 - Proper organization of the steps in a proof suited to the situation

Cr 5 - Correct justification of the steps in a proof suited to the situation


SECTION A

Questions 1 to 6 On page 8 of this booklet, darken the letter of the answer chosen.

A school student council has raised money to hold a Ski Day this year. Students will be able to spend the day skiing or skating. The amount of money raised will cover the expenses for no more than one hundred students. Since skiing is more expensive, there must be more students skating than skiing.

Let x: number of students who ski y: number of students who skate

Which of the following systems of constraints could describe this situation? A)

yx

yx

y

x

100

0

0

C)

xy

yx

y

x

100

0

0

B)

yx

yx

y

x

100

0

0

D)

xy

yx

y

x

100

0

0

1


The system of inequalities below represents the constraints associated with an optimization solution.

Which of the polygons of constraints represents the solutions for this system of inequalities?

2


The solutions for a system of inequalities are represented in the Cartesian plane below. Points P, Q, R and S are shown on the graph.

How many of these four points represent solutions for this system of inequalities? A) 1 B) 2 C) 3 D) 4 . Consider the graph below.

Which of the following paths represents a circuit that passes through all the vertices of this graph? A) P, Q, R, S C) P, R, S, Q, P B) P, Q, R, S, P D) P, R, S, Q, R

3

4


Consider the following directed graph.

Which of the following statements is true? A) There is a circuit that begins at P and that passes through Q. B) There is a circuit that begins at T and that passes through R. C) There is a path that begins at R, ends at S and passes through each vertex of

the graph only once. D) There is a path that begins at T, ends at P and passes through each vertex of the

graph only once. Which of the following graphs has a path that starts at P, ends at S and travels over each edge of the graph only once?

5

6


/4

Mathematics 563-504 CST Result

Competency 2- Essential Knowledge /24

Part A - Multiple-Choice Answer Sheet Darken the letter that corresponds to the answer you have chosen. (4 marks each)

SECTION B

Questions 7 to 10 Write the answers in the appropriate space in the Answer Booklet.

The system of inequalities and the polygon of constraints below are associated with an optimization situation. Each side of the polygon and its corresponding inequality are identified by the same number.

What are the coordinates of vertex P of this polygon of constraints? The coordinates of vertex P of this polygon of constraints are P (_____, _______).

7

1. [A] [B] [C] [D]

2. [A] [B] [C] [D]

3. [A] [B] [C] [D]

4. [A] [B] [C] [D]

5. [A] [B] [C] [D]

6. [A] [B] [C] [D]


/4

/4

A company manufactures chairs and tables. Different constraints limit the number of chairs and tables that can be manufactured. Polygon of constraints ABCD below is associated with this situation.

x: number of chairs manufactured

y: number of tables manufactured The company earns revenue of $150 per chair sold and $425 per table sold. How many chairs and tables must the company sell to maximize its revenue? The company must sell _________ chairs and __________ tables. Sandy provides a limousine service between the airport and 5 luxury hotels. In the graph below, one of the vertices represents the airport and the other vertices represent hotels A, B, C, D and E. The edges represent the different routes Sandy can take. The value on each edge represents the number of minutes required to cover the corresponding route.

From the airport, Sandy drives guests to the 5 hotels and then returns to the airport. What is the minimum amount of time, in minutes, that Sandy needs to drive the guests to their hotel and return to the airport? The minimum amount of time is ___________ minutes.

8

9


/4

Illustrated below is the layout of a sports centre consisting of 7 soccer fields.

Field A Field B Field C

Field

D Field G Field F Field E

Jerry is organizing a soccer tournament. He wants to draw a graph corresponding to this sports centre, where the vertices represent the soccer fields and the edges represent the opening between the fields. Draw a graph representing this situation.

Part B

Result

/16

10


SECTION C

Questions 11 to 16 Show all your work as well as your answer. The work shown is

taken into consideration when marks are awarded. Your written information must be legible, complete, and clearly

stated in correct language so the marker understands exactly what you have done.

A COLOURFUL DISPLAY!

During a visit to Rouyn-Noranda, you hear that the annual fireworks display will be held at the lake in the centre of town. You decide to attend, and you’d like to have the best possible view of the show. You ask some locals where you should set up your lawn chair. However, each person you ask has a different opinion about the best location on the city map that you show them.

Anna says that if you set up at point A: (6, 1), you will have the best view.

Benny insists that the best place is at point B: (3, 3).

Charlie tells you it’s at point C: (7, 5).

Finally, Dennis is certain that it is at point D: (-1, 4). The organizers tell you that some of the fireworks display can be viewed in the region corresponding to the following system of inequalities:

2x – 3y 6

3x + 2y 12 Which people are giving a correct advice?

11


x

y

The people who are giving a correct advice are: _____________________________________.

Uses mathematical reasoning

Observable indicators correspond

to level

Evalu

ati

on

Cri

teri

a

LEVEL 5 4 3 2 1

Cr3 40 32 24 16 8

Cr2 40 32 24 16 8

Cr4 Cr5

20 16 12 8 4


AN INVESTMENT IN STOCKS Joanne is a stockbroker. A client asked her to buy shares in a software company and in a mining company on Monday and Tuesday. Polygon of constraints ABCD below is based on the constraints Joanne faced on Monday.

x: number of software company shares bought each day y: number of mining company shares bought each day On Monday, each software company share cost $40, and each mining company share cost $8. Joanne bought some shares, investing the lowest possible amount of money. On Tuesday, the price of each software company share rose to $50, while the price of each mining company share dropped to $7. As a result, Joanne bought 3 times as many software company shares as she did on Monday and 10 fewer mining company shares than she did on Monday. How much money did Joanne invest to buy shares for this client on Monday and Tuesday?

12


Joanne invested $______________ to buy shares for this client on Monday and Tuesday.



to level

Evalu

ati

on

Cri

teri

a

LEVEL 5 4 3 2 1

Cr3 40 32 24 16 8

Cr2 40 32 24 16 8

Cr4 Cr5

20 16 12 8 4


THE MUNICIPAL CAMPGROUND The owner of a campground rents out campsites to residents and nonresidents. The sign on the right shows the daily rental fees for a campsite at this campground in the summer of 2009. Polygon of constraints PQRS below represents the possible combinations of the number of campsites that can be rented out each day at this campground.

where x : number of campsites rented out to residents each day

y : number of campsites rented out to nonresidents each day In the spring of 2010, the camping association demanded that there would be at most 40 campsites rented out to non-residents. In the summer of 2010, the daily rental fees for a campsite will be the same as they were in the summer of 2009. By how much would the owner’s maximum possible daily revenue decrease after having agreed to the camping association’s demands?

13


POSSIBLE COMBINATIONS OF CAMPSITES

IN THE SUMMER 2010

The owner’s maximum possible daily revenue will decrease by _________________ after having agreed to the camping association’s demands.



to level

Evalu

ati

on

Cri

teri

a

LEVEL 5 4 3 2 1

Cr3 40 32 24 16 8

Cr2 40 32 24 16 8

Cr4 Cr5

20 16 12 8 4


A VISIT TO THE MUSEUM A museum has 5 exhibition rooms connected by corridors. Paintings are displayed in each room and in each corridor. Visitors to the museum can begin their tour in any room. In the following graph, the vertices represent the exhibition rooms. The edges represent the existing corridors. The number on each edge indicates the length, in metres, of the corresponding corridor.

Because of the addition of several paintings, the museum owner decides that a new corridor should be built so that visitors can begin and end their tour in the same room, while travelling down each corridor only once. This new corridor will be 22 metres long. Sophie visits the museum specifically to see the paintings displayed in the new corridor. She begins and ends her visit in room A. In metres, what is the minimum distance Sophie must travel during her visit to the museum?

14


The minimum distance Sophie must travel during her visit to the museum is ___________________ m.



to level

Evalu

ati

on

Cri

teri

a

LEVEL 5 4 3 2 1

Cr3 40 32 24 16 8

Cr2 40 32 24 16 8

Cr4

Cr5 20 16 12 8 4


SNOW REMOVAL Marco is responsible for clearing snow from the roads connecting six villages in a region. In each of the following graphs, vertices A, B, C, D, E and F represent the villages in this region. The edges represent the roads that must be cleared of snow. In the graph on the left, the value on each edge indicates the average number of people who use the corresponding road each day. In the graph on the right, the value of each edge indicates the length of the corresponding road in kilometres.

Last year, all the roads were cleared and the cost of snow removal was $1000/km of road. This year, the cost of snow removal will be $1300/km of road. To reduce total snow removal costs, Peter has decided to close as many roads as possible, while making sure that each village can be reached. As requested by the users, the roads used the most be kept open as much as possible. Compared with last year, what is the decrease in total snow removal costs this year?

15


The decrease in total snow removal costs this year is $ ___________________ .



to level

Evalu

ati

on

Cri

teri

a

LEVEL 5 4 3 2 1

Cr3 40 32 24 16 8

Cr2 40 32 24 16 8

Cr4

Cr5 20 16 12 8 4


MULTITASKING AT THE AIRPORT When an airplane lands at the Montreal airport, there are three crews who are very busy to get the plane ready as quickly as possible for its next flight back. The following table shows the different tasks that must be carried out. Some tasks can be carried out at the same time.

Step Time

(minutes) Prerequisite

Task(s)

A. Unload luggage 15 None

B. Deplane passengers 10 None

C. Clean the plane 15 B

D. Fill the fuel tanks 20 None

E. Load the luggage 20 A

F. Board passengers 15 C

G. Warm up the engines 5 D, E and F

Today, there was a technical problem with step E, the loading of the luggage, which took 35 minutes instead of the usual 20. How much longer did the crews take to get the plane ready?

16


The crews took ____________ minutes more to get the plane ready.



to level

Evalu

ati

on

Cri

teri

a

LEVEL 5 4 3 2 1

Cr3 40 32 24 16 8

Cr2 40 32 24 16 8

Cr4

Cr5 20 16 12 8 4

mathematics 504 cst - wordpress.com · which of the following graphs has a path that starts at p,...

Documents