Math 621A - Final Exam Review – January 2016

1. State whether each of the following statements is TRUE or FALSE.

______ (a) According to the Rule of 72, it would take 12.5 years for $400 to double at an interest rate of 5.76 %, compounded monthly. ______ (b) The original amount of money invested in an investment is called the principal. ______ (c) A compound interest loan will accumulate less interest if payments are made more frequently to pay off the loan more quickly. ______ (d) The intersection symbol, , represents all of the elements in two overlapping sets.

______ (e) The symbol can be used to represent an empty set.

______ (f) 0! =1

______ (g) 8P3 = 8 x 7 x 6

______ (h) The probability of tossing a coin and getting three heads in a row is 1

2.

______ (i) The probability of a family having two girls in a row is 1

4 .

______ (j) Multiplying the number of choices for each part of a meal is called the fundamental counting principal. ______ (k) Drawing a face card from a standard deck of 52 playing cards, putting it back, and then drawing another face card are independent events. ______ (l) Winning 5 games and losing 3 games results in odds of winning of 5:3. ______ (m) Theoretical probability means that you determine probability by way of an experiment.

______ (n) The end behaviour of the polynomial function, 2 5 4y x x , extends from Q II to Q I.

______ (o) The following function 446 23 xxxy has a maximum of 5 possible x-intercepts

______ (p) The end behaviour of the exponential function graph, xy 53 , extends from Q II to Q I.

______ (q) The function xy 25.03 , is a increasing exponential function.

______ (r) The function xy log2 , is a decreasing logarithmic function

______ (s) The value of the y-intercept of the following function is xy 25.03 is 3.

______ (t) The amplitude of the function 4 cos 3y x is –4.

______ (u) The period of the function xy cos is 2

______ (v) One degree is a larger unit of measurement than one radian. ______ (w) Two cards are drawn from a deck without replacement. The first card is a multiple of 3 and the second card is a multiple of 5. This is an example of an independent event.

2. For each of the following, place the NUMBER of the correct answer in the space provided.

_____ (a) Determine the future value of a simple interest investment with a 4-year term with a principal of $400 at 1.9% simple interest, paid annually. [1] $407.60 [2] $460.80 [3] $404.00 [4] $430.4 _____ (b) Sokka invested $500 annually for 3 years. At the investment’s maturity, its value was $578. What was the annual simple interest rate? [1] 5.2% [2] 4.4% [3] 6.2% [4] 5.8% _____ (c) How many compounding periods are there for $1000 invested for 6 years at 4.2% compounded semi-annually? [1] 6 [2] 12 [3] 42 [4] 6000

_____ (d) A loan worth $10,000 is to be paid in 5 years. Which compounding period will result in the highest amount of interest? [1] annually [2] quarterly [3] monthly [4] daily _____ (e) A new home is purchased for $250,000. The value of the home appreciates by 4% each year. Which of the following functions models this situation?

[1] A=250000(0.04)n [2] A=250000(1.4)n [3] A=250000(1.04)n [4] A=250000(0.96)n _____ (f) The population of a small town decreases by a rate of 3% every year. What is the value of the decay factor (the number in the bracket) for the exponential function describing this scenario? [1] 0.03 [2] 0.3 [3] 0.97 [4] 1.03

_____ (g) Express 20 19 18 17 in the form Pn r .

[1] 20 3P [2] 20 4P [3] 20 17P [4] 4 20P

_____ (h) Zahra likes to go rock climbing with her friends. In the past, Zahra has climbed to the top of the wall 7 times in 28 attempts. Determine the odds in favor of Zahra climbing to the top. [1] 3 : 1 [2] 1 : 3 [3] 4 : 1 [4] 3 : 11 _____ (i) The odds in favour of Justin passing his drivers test is 8:3. Determine the probability that he will pass his drivers test.

[1] 11

8 [2]

11

3 [3]

3

8 [4]

8

3

_____ (j) Roena is about to draw a card at random from a standard deck of 52 playing cards. Determine The probability that she will draw a heart or a King.

[1] [2] [3] [4]

_____ (k) Determine the degree of the following polynomial function: 862)( 3 xxxf .

[1] 0 [2] 1 [3] 2 [4] 3 _____ (l) Determine the equation of the following polynomial function:

[1] 34)( 2 xxxf [2] 34)( 2 xxxf

[3] 36)( 3 xxxf [4] 362)( 3 xxxf

_____ (m) Determine the equation of the following polynomial function:

[1] 64)( 2 xxxf [2] 64)( 2 xxxf

[3] 624)( 23 xxxxf [4] 362)( 23 xxxf

_____ (n) What kind of relationship might there be between the independent and dependent variables in this scatter plot?

[1] exponential

[2] quadratic [3] cubic [4] none of the above

_____ (o) Which of the following functions is an exponential function

[1] 3xy [2]

xy 5 [3] 2

1

xy [4] 5.0xy

_____ (p) How many x-intercepts does the exponential function f(x) = 2(5)x have? [1] 0 [2] 1 [3] 2 [4] 3 _____ (q) Match the following graph with its function.

[A] y = 3(0.5)x [B] y = 2(1.25)x

[C] y = 0.5(3)x [D] y = 2(0.75)x

_____ (r) A population of locusts has an initial population of 400 and increases at a rate of 8% per day. Which equation models the population after x days?

[A] xP )08.1(400 [B]

xP )8(400 [C] xP )92.0(400 [D]

xP )8.1(400

5 10 15 20 25 30 35 40 45 x

20

40

60

80

100

120

140

160

180

200y

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

____ (s) An arc subtends an angle of 4 radians at the centre of a circle that has radius 5cm. How long is the arc in centimeters?

[1] 0.8 [2] 1.25 [3] 9 [4] 20

_____ (t) What is the period of the function )2

(4cos2

xy ?

[1] 2

3

[2] 2 [3] 3 [4]

2

_____(u) Which function has period ?

[1] sin xy [2] sin xy [3] 2 sin xy [4] sin 2xy

_____ (v) What is the period of the function 25

)4(2cos4

xy

[1] 2 [2] 5 [3] 2 [4] 4

_____ (w) Given the following function 543

1sin6

xy , which of the following are true?

[1] amplitude = 3 [2] period = 6 [3] A phase shift 4

units left [4] all of the these

_____ (x) Convert 120 to radians.

[1] 6

[2]

3

[3]

2

3

[4]

4

3

_____ (y) Which one of these angles is in Quadrant III? [1] 1000 [2] 250 [3] 3100 [4] 2000

_____ (z) What is the range of the function 3cos2 5y x ?

[1] 28 y [2] 52 y [3] 82 y [4] 53 y

_____ (zz) What is the period of the function xy4

cos

?

[1] 4

[2] 8 [3] 4 [4] 4

1

3. For each of the following, place the correct answer in the space provided. (a) A sum of money is invested and earns 8% per year compounded quarterly. ________________ What is the interest rate per quarter, written as a percent? (b) What is 15 % of $300? ________________ (c) A credit card offers you a rebate of 5% on the first purchase. If you get the credit card and buy a $1500 computer system with the credit card. What is the value of the rebate? ________________ (d) Diane purchased a $250,000 house and she is making a 20% down payment. What is the value of the down payment? ________________

(e) Express 5 4 3 2 1 in factorial (!) notation. ________________

(f) How many different outfits can be made from 3 different blouses and 6 different skirts? ________________

(g) Evaluate !5

!10. ________________

(h) Evaluate 38 C . ________________

(i) What is the probability of rolling an even number on a standard 6 sided die? ________________ (j) One card is drawn from a standard 52 card deck. What is the probability of drawing a heart? ________________ (k) The odds in favor of Charles passing his driver’s test on the first try is 3:2. Determine the odds against Charles passing his driver’s test. ________________ (l) The probability rain today is 60%. Determine the probability of the complement to this event ________________

(m) Determine the leading coefficient of this polynomial function: 674 3 xxy ________________

(n) Determine the number of turning points of this function: 352 xxy ________________

(o) Determine the value of the y-intercept of this function: 638 3 xxy ________________

(p) The growth of a tree can be modeled by the function 55.09.1 th , h represents the

height in meters and t represents time in years. How tall will the tree be in 5 years? ________________ (q) The scatter plot below compares students’ absences from math class with the grade they obtained in the course. Draw a line of best fit and interpolate the math mark when a student misses 6 days of school.

________________ Number of absences (days) (r) Determine the dependent variable in the following function. The height (in meters) of a rider above the ground on a Ferris wheel ride over time (in seconds) since the ride began. ________________ (s) Is the following data exponential? (yes or no) ________________

x 0 1 2 3 4

y 3 18 108 648 3888 (t) Evaluate: log 1000= ________________

(u) How many x intercepts does the function xy log5 have? ________________

(v) Convert 1.8 radians to degrees. ________________

Math

Mark

(%

)

(w) Find the value of sin 0.52 to four decimal places. (Radian Mode) ________________

(x) For the function 1

2 sin 2 54

ty

, what is the period of the function? ________________

(y) What is the phase shift of the function 2 cos 3 1f x x ? ________________

(z) In a standard deck of cards, what is the complement of the spades? ________________ .

Chapter I: Financial Mathematics: Investments 1. Tim has $5600 that he wants to invest for 12 years. Calculate the interest earned on each investment and clearly indicate which investment option will earn more interest. Round to nearest cent. Solve algebraically Option A: 3.5 % simple interest Option B: 3.1 % compound interest, compounded annually 2. An investment of $6 300 is paid an interest rate of 6.5 %, compounded monthly over 30 years.

a) Find the future value of the investment, to the nearest cent. Show all work, solve algebraically b) How much interest was earned on this investment? Round to the nearest cent.

c) What was the rate of return on the investment? Round to nearest tenth of a percent. Show work 3. An investment is paid an interest rate of 5.5 %, compounded semi-annually over 25 years. If the future value of the investment is $23,644.33. Calculate the principal, to the nearest cent. Solve algebraically.

4. Determine the regular monthly payment required to have an investment with a future value of $7000 at the end of 3 years, if the investment earns 5.3% interest compounded monthly. Round to the nearest cent.

N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END 5. Alex has an investment portfolio that includes: - A $5000 GIC, purchased 10 years ago, that earns 2.6% compounded annually - A $2000 Canada Savings Bond, purchased 5 years ago, that earns 3.1% compounded semi-annually - A savings account at 1.4%, compounded weekly, into which he has been making weekly deposits of $45 for 5 years. What is the current total value of Alex’ portfolio?

Chapter 2: Financial Mathematics: loans 1. Sandy’s bank has approved a loan of $4000 at 5.8 % compound interest, compounded quarterly, so that he can buy a lawn tractor. Sandy wants to repay the loan off with a single lump sum payment, at the end of 4 years. What will be the total repayment of the loan? 2. How many monthly payments are required to pay off a loan of $22,000 at an interest rate of 3.3%, compounded monthly, if each monthly payment is $350?

N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END

3. A student loan of $73,550 with an interest rate of 4.5%, compounded monthly, is to be paid off in 8 years with regular monthly payments. How much will each monthly payment be?

N = I% = PV = PMT = FV = P/Y = C/Y = PMT: END 4. A company replaces its trucks after the trucks have been used for 12 years. The company uses a depreciation rate of 25% per year. What will the truck be worth when the company replaces it? Initial value is $40,000. 5. Investments and loans can be in either simple or compound interest. When is it more beneficial to have simple interest? When is it more beneficial to have compound interest? Explain your answers. 6. Karen, a new teacher, is looking for a place to live for 10 months. She has two options:

Option A – She can rent a room with a kitchenette at a hotel for $900 per month, which includes cleaning services and utilities

Option B – She can take a 10-month lease of a furnished apartment for $750 per month. As well, she would have to pay $225 per month in utilities. Which is the better option? Do a cost-benefit analysis to explain your answer.

Chapter 3: Set Theory & Logic 1. Natasha drew a Venn diagram of plants in her Garden & categorized them with the following sets: G: Plants in her garden P: Perennials A: Annuals E: Edible plants a) List one pair of disjoint subsets.

_____________________________

b) Determine = ______ c) Is E’ equal to P? ________ d) Determine ( )n P A _______

2. Some table games use a board (B), dice (D), or cards (C), or a combination these. The Venn diagram shows the number of games that use these tools. Four games use none of them. Use the Venn diagram to answer the following questions.

a) = _____________

b) n ( (C D) B) =_____________

c) = ______________

3. A classroom teacher surveyed 30 students on their favorite music. - 16 liked pop music (P) - 13 liked country music (C) - 17 liked disco (D) - 3 liked country & pop - 7 liked country & disco - 6 liked pop & disco - 2 liked all three types of music How many students did not like any of these types of music? _________ 4. Consider the following Venn diagram of Outdoor and Indoor pets:

a) What is O I for this Venn Diagram? Circle

the correct set below:

A. {horse} B. {cat, dog} C. {bird, snake, turtle} D. {horse, cat, dog, bird, snake, turtle}

b) Determine n (O I ) _______ c) List the pets that are in O’ ______________________

5. 128 households were surveyed about cell phones. People were asked if they had Android or iPhone cell phones. 42 said they had neither. 30 had both. 35 had an Android only. How many have an iPhone only? 6. At a BBQ, hamburgers and hot dogs were served. Out of 50 people at the BBQ, 12 had neither, 25 had a Hamburger and 24 had a hot dog. How many people had one of each?

Chapter 4: Counting Methods 1. In 1985, Prince Edward Island license plates consisted of two letters followed by three digits, with the restriction that the first letter had to be P, Q or K. How many possible P.E.I. license plates were there in 1985? (repetition of digits and letters is allowed) 2. There are 6 horses in a race. How many different ways can they come in first, second and third place? 3. How many distinct permutations are there of the letters in the word MISSISSIPPI? 4. In how many distinct ways can a president, a secretary and a treasurer be chosen from a club with 24 members? 5. How many ways can a group of 3 people be chosen from a group of 10 people to go on a trip to Ontario? 6. How many 6-card hands are there having exactly 2 clubs and 4 hearts? 7. A committee is to be selected from the class with 10 women and 8 men. In how many ways can a committee of 5 people be formed if it must contain exactly 2 women and 3 men? 8. Simplify each of the following expressions.

(a) 10 4P (b) 20 17C (c) !97

!100

9.How many pathways are there from A to B in the following diagram if you can only travel right or down?

A

B

10. A pizza restaurant offers 5 veggies and 5 meats for its pizzas. In how many ways can a 3-topping pizza have at least 2 veggies?

Chapter 5: Probability 1. A single card is drawn from a shuffled deck of 52 cards. What is the probability that the card is red OR a king? Express answer as a fraction in lowest terms. 2. A green and red die are rolled at the same time. What is the probability of rolling a 4 on the green die and a 5 on the red die? Express the answer as a fraction in lowest terms. 3. Two cards are drawn without replacement from a standard deck of 52 cards. What is the probability of drawing 2 face cards? Express the answer as a decimal rounded off to four decimal places

4. A hockey team’s coach says that their odds of winning the next game is 5:3, the odds of losing is 1:7, and the odds of a tie is 1:3. Is this information possible? 5. Six boys and seven girls apply to go to a conference. Only four will be selected. Determine the probability of each: (a) Only girls are selected. (b) There will be two girls and two boys. 6. The Venn Diagram shows the number of Canadian athletes winning medals at the Olympics from 1996 to 2010. S = {athletes who have won two or more medals at the Summer Olympics} W = {athletes who have won two or more medals at the Winter Olympics} O = {athletes who have won at least one Olympic Medal} a) Are the two events (S and W) mutually exclusive? Explain. O

S W 20 1 47 b) A Canadian athlete who won a medal at the Summer Olympics from 1996 to 2010 is selected at random. Determine the odds in favour of this athlete having won two or more medals. 239

Chapter 6: Polynomial Functions 1. Darcy is planning to build a stable for 15 horses. He has found the area of other reputable stables.

a) Using the graphing calculator, use linear regression to determine the equation of the line of best fit for the data. Round off all values to two decimal places. b) Predict the area of the stable required to house 15 horses using the rounded equation above. Show all work algebraically. Round your answer to the nearest tenth of a unit. c) Using the graphing calculator, predict the number of horses that could be housed in stable of 2845 ft2. Round to the nearest whole number and include units. 2. Bob likes to solve puzzles on the internet. He recorded the times he took to solve the puzzles with different difficulty levels.

a) Using the graphing calculator, use quadratic regression to determine the equation of the curve of best fit for the data. Round off all values to two decimal places b) Predict how long it would take Jen to solve a 3.5 star puzzle using the rounded equation above. Solve algebraically. Round to the tenths place and include units.

Chapter 7: Exponential and Logarithmic Functions 1. Graph the following function and complete the table of values for the function y = 3x.

2. The height of a sunflower was recorded every seven days as it grew.

(a) Using the graphing calculator, use exponential regression to determine the equation that models the growth of the sunflower over time. Round all values to three decimal places. (b) Estimate the height of the sunflower, to the nearest tenth of a centimeter, on day 10 using the rounded equation above. Show work algebraically and include units. (c) Using the graphing calculator, determine on what day would you expect the sunflower to reach a height of 247 cm? Round to nearest whole number and include units.

3. The population of a Prince Edward Island is currently 137, 000 and is increasing exponentially at a rate of 1.2%. Assuming the growth rate continues, what will be the population of people in PEI in 10 years. Round to the nearest whole number and include proper units.

4. The population of Elmsdale, PEI is 340 and is decreasing at a rate of 2% per year. What will the population be in 10 years? Round to nearest whole number and include proper units.

x y

–2

–1

0

1

2

3

5. (a) What is the equation that models this exponential function? xaby

x 0 1 2 3 4

y 7 28 112 448 1792

(b) Use your function to predict the value of y when x is 6. Show all work. 6. Match the equations to the graphs.

A. xy )5.0(6 B. 4242 3 xxxy C. xy )2(4

D. xy log2 E. 542 xxy F. 24 xy

7. Use your knowledge of simple and compound interest to answer the following questions. a) On the graph: Identify the curve that represents simple interest and the curve that represents compound interest. b) What type of function is the simple interest graph? Why? c) What type of function is the compound interest graph? Why?

Chapter 8: TRIGONMETRIC FUNCTIONS 1. State the period of the graph Period _____________

2. Determine an equation of the form dcxbay )(cos having the following characteristics:

amplitude = 3, period =4

, phase shift right =

3

, and vertical displacement up 5.

3. Determine the following for the given equation: 1

3 sin 22

y x

Amplitude ______________

Period ______________

Phase Shift ______________ Describe in words

Vertical Displacement ______________ Describe in words

4. Given the equation of the image, describe all of the transformations to the original sine graph. Use words left or right, up or down, etc. to describe transformations

10)(3

1sin5 xy

5. Write one equation to represent the function below. Use dcxay )cos( OR dcxay )sin(

6. On the grid provided, sketch one period (wave) of the function: y = 3 cos 2x + 4

Please remember – this review is an excellent way to prepare for the exam, but don’t forget to study your old tests and assignments. Good luck, and please see me any time if you have questions!