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FOR ALL PROBLEMS: You MUST show your work as if a computer was not available.

Write your answers NEATLY, putting a box around all your answers, so that it is clearly indicated.

Part I – Chapters 1 to 5 (/34)

1. [4] Consider the linear programming problem:

Maximize ? = 4? + 5?

Subject to {

? + 2? ≤ 10 −? + ? ≤ 2 4? + ? ≤ 20

?, ? ≥ 0

Graph the feasible set, labelling all lines and corner points. Then find the maximum value ? can obtain.

2. [2] Transcribe the following problem into a linear programming problem by stating the objective equation, along with all constraining inequalities. You do not need to solve it!

A team produces two types of doghouses: large and small. Each large house requires 3

hours to construct, 2 hours to paint and 0.5 hours for testing. Each small house requires 2

hours to construct, 1 hour to paint and 0.5 hours for testing. The large houses earn a profit

of $100, while the small houses earn a profit of $70. There are 22 hours available for

building, 14 hours available for painting and 4.5 hours available for testing. What is the

maximum profit the team can obtain?

3. [2] Express the shaded region in the Venn diagram below as some union, intersection and/or complement of the sets ?, ? and ?.

4. [2] A doghouse manufacturing team has produced 540 doghouses. Of these:

100 houses have air conditioning

160 houses have plumbing

290 houses have beds

60 houses have both air conditioning and plumbing

130 houses have beds, but not air conditioning nor plumbing

130 houses have both plumbing and beds

210 houses have beds, but not air conditioning

Let ?, ?, ? denote the sets of houses with air conditioning, plumbing and beds. Build a 3-circle

Venn diagram and fill in all regions with the number of houses contained within that region.

Then give the number of houses that have none of the three features.

5. [2] Let ? = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set with ? = {1, 2, 3} and ? = {3, 4, 5} and ? =

{2, 4, 6}. Find ?′ ∩ (? ∪ ?).

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6. [2] How many four-letter words exist (including nonsense words) that satisfy all of the following

conditions:

No letter is repeated.

The first letter is a vowel (A, E, I, O or U).

The last letter is not a vowel.

7. [2] An accounting division has ten accountants. Some are to be divided into groups to aide two

clients: Clients A and B. How many ways can they be selected so that Client A receives four

accountants and Client B receives three (some accountants will be left unassigned)?

8. [2] Tom and Jerry are two friends in the same class of nine people (including them). There are six

tables in the classroom, each containing two seats (similar to the classroom used in our lectures, but

with only six tables). How many seating arrangements exist so that Tom and Jerry sit at the same desk?

9. [2] Mr. Tinsley plays nine checkers games against the computer program Chinook, and the sequence

of wins, losses or ties (draws) is recorded. How many sequences are possible if Mr. Tinsley won at

least one game?

10. [2] A teacher has 10 gold stars and 9 smiley face stickers to give to six students. Suppose the

stickers are selected randomly. What is the probability that at least four students will get a smiley face?

11. [2] A teacher assigns their students a book report. There are ten different books to choose from.

There are seven students in the class, and each student selects a book at random. What is the

probability that at least two students will choose the same book?

12. [1] Suppose you are flipping a coin 100 times. Miraculously, the first 99 flips have all been tails.

What is the probability that the final flip, and thus all 100 flips, will be tails?

13. [2] Suppose a table illustrating the number of pink and white cherry blossoms in Victoria and

Saanich is given in the table below. Using the table, find the probability that a sampled pink cherry

blossom came from Victoria.

Pink White Total

Victoria 35 28 63

Saanich 25 20 45

Total 60 48 108

14. [3] A doghouse distributor has both an online and an ‘offline’ retail store. They find that when a

person shops online, they have a 72% probability of purchasing a deluxe model, whereas offline retail

shoppers have a 54% probability of purchasing a deluxe model. 80% of all transactions with the

distributor are online. Build a tree-diagram illustrating this scenario and labelling all events clearly. If

a person did not purchase a deluxe model, what is the probability that it was purchased online?

15. [2] A fair game has two outcomes: win $60 or lose $100. What is the probability of winning?

16. [2] A promotor invites 200 people to a party. If each person has a 10% chance of attending the

event, what is the probability that exactly twelve people will attend the event?

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Part II – Chapters 6 to 9 (/20)

17. [1] After performing the Gauss-Jordan Elimination Method on a system of linear equations (with

variables ?, ?, ?, ? as usual), the resulting augmented matrix remains. Write the solution to the original

system.

[

1 0 −3 0 | 0 0 1 2 0 | 3 0 0 0 1 | −5

]

18. [2] Solve the given system of linear equations using matrix inverses. No credit will be given for any

other method.

{ 4? + 6? = 2 3? + 5? = 4

19. [2] Find the inverse of the matrix below using the Gauss-Jordan Elimination Method. Show all

work.

[ 1 0 2

−4 2 6 0 1 6

]

20. [2] Suppose Lego begins tracking consumers’ preferences between Lego brand toys and their major

competitor – Mega Bloks. They find that each year, 15% of consumers that preferred Lego switch to

preferring Mega Bloks. Also, 20% of consumers that preferred Mega Bloks switch to preferring Lego

each year. If half of the current population prefers Lego, find the proportion that will prefer Lego two

years from now. Use Markov chains in your work.

21. [3] In the scenario above, what proportion will prefer Lego in the long run?

? = ?(1 + ??) ? = ? (1 + ?

? )

??

? = ???? ?? = ??? [ (1 +

? ?)

??

− 1

? ]

??? = 1 − (1 + ?

? )

?

? = ?

? ? = ?? ?? = ??? [

1 − (1 + ? ?)

−??

? ?

]

22. [1] If $1000 is deposited in an account with an annual interest rate of 2%, compounded quarterly,

what will the value of the account be after 5 years? In your work, show which of the above formulas is

used to solve this.

23. [2] Garret opens an RRSP account with an interest rate of 5% compounded monthly. He pays $400

each month into the account. How much will the RRSP be worth in 35 years? Clearly indicate which

formula is being used.

24. [2] The current nominal interest rate on Canada Student Loans is 5.95%, compounded monthly. If a

student takes out a $20,000 student loan, and wishes to make payments each month for exactly 5 years,

how much will each payment be? Clearly indicate which formula is being used.

25. [1] Consider the problem above. How much of the amount paid on the loan be due to interest?

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26. [1] Given the statements below, write (~? ∧ ~?) → ? in conversational English:

? = “The budget is balanced. ”

? = “Taxes will be raised. ”

? = “Programs will be cut. ”

27. [1] Let ? be the statement: “The ferry will depart at 9am.” Let ? be the statement, “The ferry will

arrive at noon.” Write the following statements in logical syntax using ?, ? and connectives.

“The ferry will not depart at 9am, but will arrive at noon anyway.”

28. [2] Construct a truth table for the statement ? → (~? ∨ ?).