Midterm review Math 3201 Name:_____________

Chapter 1 - Set Theory Part 1: Multiple Choice : 1) U = {hockey, basketball, golf, tennis, volleyball, soccer}. If B = {sports that use a ball}, which element would be in B ?

(A) Basketball (B) Golf (C) Hockey (D) Soccer

2) Which of the following would represent the shaded region?

(A) G V (C) G V

(B) G (D) V 3) Which of the following phrases describes an empty set? (A) Common factors of 4 and 12

(B) Prime numbers that are even (C) Multiples of 3 that are less than 12 (D) Factors of 10 that are divisible by 4 4) Set M consists of the multiples of 4 from 1 to 50. Which represents set notation?

(A) M={1, 2, 3, … , 48, 49, 50} (C) M={𝑚 = 4, 1 ≤ 𝑥 ≤ 50, 𝑥 ∈ 𝑁}

(B) M={𝑚 = 4𝑥|1 ≤ 𝑥 ≤ 50, 𝑥 ∈ 𝑁} (D)M={𝑚 = 4𝑥|1 ≤ 𝑥 ≤ 12, 𝑥 ∈ 𝑁}

5) Consider the following sets: R={0, 1, 2, 3, 4, 5, 6} S={2, 4, 6, 8} T={1, 2, 3, 6} Which of the following statements is true?

(A) R S (B) R T (C) S R (D) T R

6) Set T is students who like tennis, set B is students who like basketball and set S is students who like swimming, which diagram indicates students who like tennis only?

(A) A (B) B (C) C (D) None of these

Use the following information to answer #7, #8 & #9 A = {natural numbers from 1 to 10} B = {factors of 12} C = {multiples of 11 less than 100} 7) What is the union of sets A and B, (𝐴 ∪ 𝐵)?

(A) {1, 2, 3, 4, 5,6, 7,8, 9,10} (C) {1,2,3,4,6, ,12} (B) {1, 2, 3, 4, 5,6, 7,8, 9,10,12} (D){1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

8) What is A intersect B, (𝐴 ∩ 𝐵)?

(A) {12} (C) {1,2,3,4,6} (B) {1, 2, 3, 4, 6, 12} (D) {1, 2, 3, 4, 5,6, 7,8, 9,10,12}

9) Set B and set C are an example of…. … sets

(A) disjoint (B) empty (C) infinite (D) none of these

G V

2

10) What does the ‘3’ represent in this Venn Diagram shown?

(A) The number of people who like handball and hockey. (B) The number of people who like handball and hockey but not wrestling. (C) The number of people who like hockey and wrestling and handball. (D) The number of people who like hockey and wrestling but not handball.

Part 2- Questions: 11. A survey was conducted to determine where people buy coffee. 82 people buy coffee at Tim Hortons. 65 people buy coffee at StarBucks. 17 people did not buy coffee at all.

If 130 people were surveyed, how many people buy coffee from ONLY Tim Hortons? 12. There are 36 students who study science.

14 study physics 18 study chemistry 24 study biology 5 study physics and chemistry 8 study physics and biology 10 study biology and chemistry 3 study all three subjects

Use a Venn diagram to answer the following questions: (i) Determine the number of students who study physics and biology only. (ii) Determine the number of students who study at least two subjects. (iii) Determine the number of students who study biology only.

13. A grade three teacher asked her class of 28 students about the type of pet they owned. The results were as follows:

28 children have a dog, a cat or a bird 13 children have a dog 13 children have a cat 13 children have a bird 4 children have a dog and cat only 3 children have a dog and bird only 2 children have a cat and bird only

A) Algebraically, determine how many children have a dog, cat and bird?

B) How many children have only one pet?

3

14. In a school of 120 students 5 students took English, Physics, and Chemistry 15 students took Physics and English 8 students took Physics and Chemistry 10 students took English and Chemistry 99 students took English or Chemistry 45 took Chemistry 30 students took Physics James summarized the data using the Venn Diagram shown below:

39 2

22

15

10 85

19

English Physics

Chemistry

U

Identify the regions in James’ Venn diagram that have errors and describe the errors that James made. Provide a correct Venn diagram with the correct entries.

Chapter 2 Counting Methods Part 1. Multiple Choice

1. Samantha is choosing an outfit to wear to the dance. She has 3 different tops, 4 different pants and 2

different pairs of shoes to choose from. How many different outfits could she make from this selection of

clothes?

A) 9 B) 12 C) 14 D) 24

2. Simplify: 8!

2!5!

A) 0.0111 B) 168 C) 336 D) 2,419,200

3. Consider the word CAT. In how many different ways can the letters be arranged?

A) 1 B) 3 C) 4 D) 6

Answers: 1C 2D 3D 4D 5D 6B 7B 8C 9A 10B 11. 48 12. (i)5 (ii)17 (iii) 9 13. A)1 B)18 14.Error occurs where subjects overlap

4 6

9

2

5 73

Physics Chemistry

Biology

U

44 12

32

10

5 35

9

English Physics

Chemistry

U

4

4. Simplify: ( 2)!

!

n

n

A) 2

1

3 2n n B) 2! C) 2 3n D) 2

3 2n n

5. What restriction must be applied to the variable in #4 above?

A) 2n B) 2n C) 0n D) 0n

6. In the grid below, a person must travel from A to B by only heading East (E) or South (S). Under these

rules, which represents the total number of possible routes that can be taken to get from A to B ?

A) 5!

3!2! B)

6!

3!2!

C) 5! D) 6!

7. A briefcase lock opens with the correct four-digit code. If the digits 0 to 9 are allowed and a digit cannot

be repeated, how many different four-digit codes are possible?

A) 24 B) 34 C) 5040 D) 3024

8. How many ways can 5 friends stand in a row for a photograph if Alex and Andrea always stand together?

A) 120 B) 48 C) 240 D) 60

9. How many different arrangements can be made using all the letters in NEWFOUNDLAND?

A) 362,880 B) 479,001,600 C) 39,916,800 D) 665,280

10. How many different combinations of 3 letters can be made using all 26 letters in the alphabet?

A) 15,600 B) 2600 C) 23 D) 69

11. How many different ways can the letters of SASKATOON be arranged if you must start with a T and end

with a K ?

A) 630 B) 1260 C) 5040 D) 24

12. The student council has 10 members, 6 girls and 4 boys. A dance committee is to be formed consisting of

exactly two girls and two boys. Which calculation could be used to determine the number of different

ways this committee could be formed?

A) 6 2 4 2C C B) 6 2 4 2P P C) 10 4C D) 10 4P

13. Solve for n:

1 !12

2 !

n

n

A) 11 B) 13 C) 12 D) -13

14. Calculate: 6

2

A) 3 B) 15 C) 30 D) 24

15. How many different 3 topping pizzas can you create if you have 8 toppings to choose from?

A) 56 B) 24 C) 336 D) 6720

Part 2 Questions:

16. Six different scholarships will be awarded at the graduation ceremonies in November.

If the graduating class has 30 students determine the number of ways to award the scholarships if,

A) there are no conditions on who can win the scholarships

B) no student can win more than one scholarship

C) Clara must be awarded the “Most Improved Student” scholarship and Craig must receive the

scholarship for highest average.

A

B

N

S

EW

5

17. Four students are to be chosen from a group of 10 to fill the positions of president, vice-president,

treasurer and secretary. In how many ways can this be accomplished?

18. Algebraically solve for n and verify your answer: 212

nP

19. A group of five Art club students are to be selected for a field trip to the Rooms. If there are 5 boys and 6

girls in the club, how many ways can the leader select the five students if there must be at most 2 boys?

20. You need to create a password that must be at least 4 characters but no more than 6 characters long. The

password may contain lower case letters as well as the digits 1 to 9. No repeated characters are allowed.

How many different passwords are possible?

Chapter 3: Probability Part 1 Multiple Choice

1. Given the following probabilities, which event is most likely to occur?

(A) P(A) = 0.28 (B) P(B) = (C) P(C) = 0.27 (D) P(D) =

2. Raymond has 12 coins in his pocket, and 9 of these coins are quarters. He reaches into his pocket

and pulls out a coin at random. Determine the odds against the coin being a quarter.

(A) 1:4 (B) 1:3 (C) 3:4 (D) 3:1

3. Julie draws a card at random from a standard deck of 52 playing cards. Determine the odds in

favour of the card being a heart.

(A) 3:1 (B) 1:3 (C) 1:1 (D) 3:13

4. Tia notices that yogurt is on sale at a local grocery store. The last eight times that yogurt was on

sale it was available only three times. Determine the probability of yogurt being available this

time.

(A) 0.220 (B) 0.375 (C) 0.460 (D) 0.625

5. The weather forecaster says that there is a 30% probability of fog tomorrow. Determine the odds

against fog.

(A) 3:7 (B) 3:10 (C) 7:3 (D) 7:10

6. From a committee of 18 people, 2 of these people are randomly chosen to be president and

secretary. Determine the number of ways in which these 2 people can be chosen for president

and secretary.

(A) 2P2 (B) 2P1 (C) 18P2 (D) 18P16

7. Nine boys and twelve girls have signed up for a trip. Only six students will be selected to go on

the trip. Determine the probability that there will be equal numbers of boys and girls on the trip.

(A) 17.23% (B) 22.61% (C) 27.35% (D) 34.06%

8. Cai tosses four coins. Determine the probability that they all land as tails.

(A) 6.25% (B) 12.50% (C) 18.75% (D) 25.00%

9. Helen is about to draw a card at random from a standard deck of 52 playing cards. Determine the

probability that she will draw a black card or a spade.

(A) 1

4 (B)

1

2 (C)

3

4 (D)

5

6

Answers: 1D 2B 3D 4D 5D 6A 7C 8B 9C 10A 11A 12A 13B 14B 15A 16.A) 729000000 B) 427518000 C) 491400 17. 5040 18. n=4 19. 281 20. 1208887680

6

10. Select the events that are mutually exclusive.

(A) Drawing a red card or drawing a diamond from a standard deck of 52 playing

cards.

(B) Rolling a sum of 8 or rolling an even number with a pair of six-sided dice,

numbered 1 to 6.

(C) Drawing a black card or drawing a Queen from a standard deck of 52 playing

cards.

(D) Drawing a 3 or drawing an even card from a standard deck of 52 playing cards.

11. Sarah draws a card from a well-shuffled standard deck of 52 playing cards. Then she draws

another card from the deck without replacing the first card. Determine the probability that both

cards are NOT face cards.

(A) 1

10 (B)

1

3 (C)

10

17 (D)

12

15

12. Misha draws a card from a well-shuffled standard deck of 52 playing cards. Then he puts the

card back in the deck, shuffles again, and draws another card from the deck. Determine the

probability that both cards are even numbers.

(A) 1

100 (B)

3

45 (C)

6

15 (D)

25

169

13. Select the events that are dependent.

A. Rolling a 2 and rolling a 5 with a pair of six-sided dice, numbered 1 to 6.

B. Drawing an odd card from a standard deck of 52 playing cards, putting it back, and

then drawing another odd card.

C. Drawing a spade from a standard deck of 52 playing cards and then drawing

another spade, without replacing the first card.

D. Rolling an even number and rolling an odd number with a pair of six-sided dice,

numbered 1 to 6.

14. A five-colour spinner is spun, and a die is rolled. Determine the probability of spinning yellow

and rolling a 6.

(A) 2.42% (B) 3.33% (C) 6.13% (D) 7.75%

15. Two cards are drawn, without being replaced, from a standard deck of 52 playing cards.

Determine the probability of drawing a five then drawing a two.

(A) 0.603% (B) 1.227% (C) 1.613% (D) 2.009%

Part 2 Questions:

16. A credit card company randomly generates temporary five-digit pass codes for cardholders.

Meghan is expecting her credit card to arrive in the mail. Determine, to the nearest hundredth of

a percent, the probability that her pass code will consist of five different even digits.

17. From a committee of 18 people, 3 of these people are randomly chosen to be president, vice-

president, and secretary. Determine, to the nearest hundredth of a percent, the probability that

Evan, Elise, and Jaime will be chosen.

18. Sonja has letter tiles that spell MICROWAVE. She has selected four of these tiles at random.

Determine, to the nearest tenth of a percent, the probability that the tiles she selected are two

consonants and two vowels.

19. Homer hosts a morning radio show in Halifax. To advertise his show, he is holding a contest at a

local mall. He spells out NOVA SCOTIA with letter tiles. Then he turns the tiles face down and

mixes them up. He asks Marie to arrange the tiles in a row and turn them face up. If the row of

tiles spells NOVA SCOTIA, Marie will win a new car. Determine the probability that Marie will

win the car. Show your work.

7

20. 8 friends are lining up to get in to see the Hunger Games: Catching Fire movie, including Bob

and Sam.

A) What is the probability that Bob and Sam will be next to each other.

B) What is the probability that they will not be next to each other?

21. The probability that Haley will exercise on Sunday is 0.6. The probability that she will go

shopping on Sunday is 0.5. The probability that she will do both is 0.3. Determine the probability

that Haley will do at least one of these activities on Sunday

22. The probability that a plane will leave Toronto on time is 0.90. The probability that a plane will

leave Toronto on time and arrive in Saskatoon on time is 0.53. Determine the probability that a

plane will arrive in Saskatoon on time, given that it left Toronto on time. Show your work.

23. Bonita has six identical red marbles and ten identical blue marbles in a paper bag. She pulls out

one marble at random and then another marble, without replacing the first marble. Determine the

probability that she pulls out a pair of red marbles.

24. A five-colour spinner is spun, and a four-sided die is rolled. Determine the probability of

spinning orange and rolling a 1.

Chapter 8: Sinusoidal Functions

1. What is 315° written as an exact radian measure?

(A) 5π

3 (B)

11π

6 (C)

7

4

(D)

9

4

2. What is the value of 5

6

in degrees?

(A) 120° (B) 150° (C) 210° (D) 240°

3. Which graph represents a function that is periodic and sinusoidal?

(A) (B)

(C) (D)

Answers: 1A 2B 3B 4B 5C 6C 7D 8A 9C 10D 11C 12D 13C 14B 15A 16. 0.12% 17. 0.12% 18. 47.6% 19. 1

907200

20. (A) 25% (B) 75% 21. 0.8 22. 0.59 23. 0.125 24. 0.05

8

x-180 -120 -60 60 120 180

y

-8

-6

-4

-2

2

4

4. Given the graph:

Which represents the period?

(A) 360° (B) 270° (C) 225° (D) 180°

5. What is the equation of the midline of the function graphed below?

(A) y = –7

(B) y = –3

(C) y = –2

(D) y = 0

6. The midline of a sinusoidal graph is at y = –1 and the minimum value of the

sinusoidal function occurs at –6. Which represents the maximum value of the

sinusoidal function?

(A) 10 (B) 6 (C) 5 (D) 4

7. The graph below shows Jane’s height on a Ferris wheel over a period of time. What is the

amplitude of the sinusoidal function that models the Ferris wheel?

(A) 3

(B) 6

(C) 10

(D) 12

x- 90 ° 90 ° 180 ° 270 ° 360 °

y

- 4

- 2

2

4

6

8

10

Time (sec )4 8 12 16

Height (m)

4

8

12

16

9

8. Which represents the equation of the midline for the sinusoidal function

𝑓(𝑥) = 4𝑐𝑜𝑠 (1

2(𝑥 − 3)) + 5 ?

(A) 𝑦 =1

2 (B) y = 3 (C) y = 4 (D) y = 5

9. What are the amplitude and maximum value for the function below?

( ) 2sin3 60 1f x x

Amplitude Maximum Value

(A) 2 3

(B) 2 4

(C) 3 3

(D) 3 4

10. The graph of which function has a period of 180°?

(A) 12

3cos 1y x (B) 3cos 180 1y x

(C) 4cos 180 1y x (D) 4cos2 1y x

11. Given the equation y = asin(b(x + c)) + d, which value of b would produce a graph

with a period of 120°?

(A) 1

4 (B)

1

3 (C) 3 (D) 4

12. What is the range of the function 4cos 2 y x ?

(A) {y|−2 ≤ y ≤ 4, y ∈ R} (B) {y|2 ≤ y ≤ 4, y ∈ R}

(C) {y|−2 ≤ y ≤ 6, y ∈ R} (D) y R

13. The graph of the function 4cos3y x has its amplitude doubled and its period halved. Which

represents the new function?

(A) 32

2cosy x (B) 2cos6y x (C) 32

8cosy x (D) 8cos6y x

14. Which equation represents the function graphed below?

(A) 𝑦 = 2sin (𝑥 − 90o)

(B) 𝑦 = 2sin (𝑥 + 90o)

(C) 𝑦 =1

2sin (𝑥 − 90o)

(D) 𝑦 =1

2sin (𝑥 + 90o)

x- 270 ° - 180 ° - 90 ° 90 ° 180 ° 270 ° 360 ° 450 ° 540 °

y

- 3

- 2

- 1

1

2

3

10

15. The graph below represents a sinusoidal function.

Which describes the transformation of the base graph to produce the graph above?

(A) The base graph of sine has been vertically stretched by a factor of 2 and

vertically translated 2 units up.

(B) The base graph of cosine has been vertically stretched by a factor of 2 and

vertically translated 2 units up.

(C) The base graph of sine has been horizontally stretched by a factor of 2 and

vertically translated 2 units up.

(D) The base graph of cosine has been horizontally stretched by a factor of 2 and

vertically translated 2 units up.

16. Use the sinusoidal function shown below to answer the questions that follow.

(i) Determine the amplitude, equation of midline, range and period.

Amplitude:____

Equation of midline:____

Range:________________

Period:

(ii) Use the information from part (i) to determine a function that represents the

graph in the form cosy a b x d .

x- 90 ° 90 ° 180 ° 270 ° 360 °

y

- 4

- 2

2

4

6

8

10

x-90 ° 90 ° 180 ° 270 ° 360 °

y

-5

-4

-3

-2

-1

1

2

11

x- 90 ° 90 ° 180 ° 270 ° 360 ° 450 °

y

- 4

- 2

2

4

6

8

x- 90 ° 90 ° 180 ° 270 ° 360 ° 450 °

y

- 4

- 2

2

4

6

8

17. Sketch a possible graph of a sinusoidal function with the following characteristics:

Domain: {𝑥|0 ≤ 𝑥 ≤ 360°, 𝑥𝜖𝑅}

Range: {𝑦| − 2 ≤ 𝑦 ≤ 6, 𝑦𝜖𝑅}

Period: 120°

y-intercept: 0

18. The height of a chair on a Ferris wheel is described by the function h(t) = 10cos (π

3t) + 12

where h(t) represents the height of the chair in metres and t represents the time in seconds

(a) Determine the height of the axis of rotation above the ground.

(b) Determine the maximum height attained.

(c) Determine the time it takes for one revolution of the ferris wheel.

(d) Determine the height of the chair at 9 seconds.

19. The average monthly rainfall in inches for Honolulu, Hawaii was calculated for 2013. The data

followed a sinusoidal trend and hence a sinusoidal regression analysis was performed to

determine the equation of best fit for the data.

(a) Given that the regression analysis produced the following output, what is the regression

equation for this data?

(b) Using your equation from (a), predict the average monthly rainfall for

February, 2014 (month 14).

17.

Answers: 1C 2B 3D 4D 5C 6D 7B 8D 9A 10D 11C 12C 13D 14A 15B 16(i) Amp = 2 midline y = -1 Range -3 ≤ y ≤ 1 Period 120ᶱ (ii) y = 2 cos(3x) – 1 18.(a) 12 m (b) 22 m (c) 6 sec (d) 2 m 19.(a) y = 1.65sin(0.45x + 1.69) + 2.04 (b) 3.67 inches

SinReg y=a*sin(bx+c)+d a=1.65 b=0.45 c=1.69 d=2.04

12

Chapter 5: Polynomial Functions

1. Which of the following functions represents a polynomial function?

A) f(x) = 3x + 4x2 + 5 B) f(x) = x – 9x2 – 1

C) f(x) = 2x2 + 6x – D) 𝑓(𝑥) =𝑥2

5+ √6𝑥 −

7

𝑥

2. Determine the leading coefficient of the polynomial function: 𝑓(𝑥) = 3𝑥 + 2

A) 1 B) 2 C) 3 D) 5

3. What is the end behavior of the graph of:𝑓(𝑥) = 4𝑥2 − 3𝑥 + 5 ?

A) Q2 to Q1 B) Q3 to Q1 C) Q2 to Q4 D) Q3 to Q4

4. What is the y-intercept of the graph of the function, 𝑓(𝑥) = 5𝑥3 + 2𝑥2 + 𝑥 + 7 ?

A) 2 B) 3 C) 5 D) 7

5. The graph of a third-degree polynomial function is shown.

The range of the function is:

A) {𝑦|𝑦 ∈ 𝑅}

B) {𝑦|𝑦 ≤ −1 𝑎𝑛𝑑 𝑦 ≥ 2, 𝑦 ∈ 𝑅}

C) {𝑦|−1 ≤ 𝑦 ≤ 2, 𝑦 ∈ 𝑅}

D) {𝑦|𝑦 ≤ 8, 𝑦 ∈ 𝑅}

6. What is the range of the function shown?

A) {𝑦|𝑦 ∈ 𝑅}

B) {𝑦|𝑦 ≤ −1, 𝑦 ∈ 𝑅}

C) {𝑦|𝑦 ≥ 1, 𝑦 ∈ 𝑅}

D) {𝑦|𝑦 ≤ 1, 𝑦 ∈ 𝑅}

7. Which represents the equation of the polynomial function:

A) 𝑓(𝑥) = 𝑥3 − 4𝑥2 − 3𝑥 + 19 B) 𝑓(𝑥) = 𝑥3 − 4𝑥2 − 3𝑥 + 18

C) 𝑓(𝑥) = −𝑥3 − 4𝑥2 − 3𝑥 + 19 D) 𝑓(𝑥) = −𝑥3 − 4𝑥2 − 3𝑥 + 18

x-6 - 4 - 2 2 4 6

y

-6

-4

-2

2

4

6

x- 4 - 2 2 4

y

- 10

- 8

- 6

- 4

- 2

2

4

6

8

10

12

14

16

18

20

13

8. A student was given the following information about a polynomial function:

o It is cubic.

o It has a negative leading coefficient.

o It has one x-intercept going through the origin.

o It has exactly two x-intercepts.

Based on the criteria provided, which of the following graphs could represent the

function described?

A) C)

B) D)

9. What is the range of y = 4𝑥 + 5 ?

A) {𝑥|𝑥 ∈ 𝑅} B) {𝑦|𝑦 ≤ 5, 𝑦 ∈ 𝑅}

C) {𝑦|𝑦 ∈ 𝑅} D) {𝑦|𝑦 ≥ 5, 𝑦 ∈ 𝑅}

Constructed Response

10. Determine the following characteristics of each function: (6 points)

Characteristics 𝑓(𝑥) = 3𝑥2 − 6𝑥 + 8 𝑓(𝑥) = −2𝑥3 − 5𝑥 + 3

Number of possible x-intercepts

y-intercept

Domain

Range

Number of possible turning points

End behaviour

11. Sketch a cubic with two turning points,

a constant term of -2 and end behaviour

Q3 to Q1

x

y

x

y

x

y

x

y

y

x

14

12. Determine the following characteristics for the following polynomials:

Characteristics

Degree

Sign of Leading Coefficient

Constant term of function

End behaviour

y-intercept

Domain

Range

10.

Characteristics 𝑓(𝑥) = 3𝑥2 − 6𝑥 + 8 𝑓(𝑥) = −2𝑥3 − 5𝑥 + 3

Number of possible x-intercepts 0, 1 or 2 1, 2 or 3

y-intercept 8 3

Domain x € R x € R

Range y ≥ 5 y € R

Number of possible turning points 1 0 or 2

End behaviour Q2 to Q1 Q3 to Q1

11.

12. Degree: 3 Degree: 1

Sign of L.C.: Positive Sign of L.C.: Negative

Constant Term: –1 Constant Term: 5

End Beh: Q3 to Q1 End Beh: Q2 to Q4

Y – intercept: –1 Y – intercept: 5

Domain: x € R Domain: x € R

Range: y € R Range: y € R

x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10

y

- 10

- 8

- 6

- 4

- 2

2

4

6

8

10

x- 10 - 8 - 6 - 4 - 2 2 4 6 8 10

y

- 10

- 8

- 6

- 4

- 2

2

4

6

8

10

Answers: 1C 2C 3A 4D 5A 6D 7B 8C 9C

x

y