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Harvard Mathematics For Management Final Exam

Q1. The number of accidents at a dangerous intersection in Smalltown during each of the last six years is as follows: 0, 1, 1, 2, 3, 5. For this data set, which of the following is true?

Mean < Median < Mode
Mean < Mode < Median
Median < Mean < Mode
Mode < Mean < Median
Mode < Median < Mean

Q2. The Smalltown Limousine Company is going to buy an $80,000 limousine today. The executives estimate that the limousine will generate the following profits during each of the next 5 years.

finalexam question 2 table

The payback period for the limousine is

between 1 and 2 years
between 2 and 3 years
between 3 and 4 years
between 4 and 5 years
None of the above

Q3. Please download the file nba.xlsx. Rounded to the nearest $100,000, the median NBA player’s salary was

$2.5 million
$3.0 million
$2.0 million
$4.0 million
None of the above

Q4. Last year, a barber shop generated $100,000 in profit. Assume that the shop’s profits grow at 5% per year and that cash flows are discounted at 10% per year. If profits are received at the end of each year, what is the present value of all the shop’s future profits?

$1,500,000
$2,500,000
$2,100,000
$2,000,000
$3,000,000

Q5. Suppose that a total of $10 million will be allocated to the first three finishers in the Springfield Derby horse race in the ratio 7:2:1. The total amount received by the 2nd- and 3rd-place finishers in the race is

$7 million
$1 million
$3 million
$5 million
None of the above

Q6. Suppose that you draw two cards from a deck. After drawing the first card, you do not put the first card back in the deck. What is the probability (rounded to the nearest ten thousandth) that both cards are diamonds?

0.0543
0.0588
0.0625
0.0643
None of the above

Q7. The set of all values of x that satisfy 5(3 – x) ≥ 10 is expressed as

x ≥ 1
x ≤ −1
x ≤ 1
x ≤ 2
None of the above

Q8. When simplified, 18 + 4 × 9/22 + 5 equals _____________.

Q9. Suppose that 1% of all people have a particular disease. A test for the disease is 99% accurate. This means that a person who has the disease has a 99% chance of testing positive for the disease, while a person who doesn’t have the disease has a 99% chance of testing negative for the disease.

If a person tests positive for the disease, what is the chance (rounded to the nearest hundredth) that he or she actually has the disease?

0.99
0.40
0.50
0.45
None of the above

Q10. For x = 1/2 the second derivative of f(x) = 2x−2 is

3/4
100
192
25
None of the above

Q11. Let q = the annual demand in pounds for a drug and p = the price per pound of the drug. Suppose that q = 1,000,000p−6. Also suppose that you want to “invert the demand curve” and express price as a function of demand. Which of the following is an expression of the inverse demand curve?

p = 0.10q−1/6
p = 10q6
p = 10q−1/6
p = 0.10q−1/3/10
None of the above

Q12. A drug company believes that the annual demand for a drug will follow a normal random variable with a mean of 900 pounds and a standard deviation of 60 pounds. If the company produces 1000 pounds of the drug, what is the chance (rounded to the nearest hundredth) that it will run out of the drug? Assume that the only way to meet the demand for the drug is to use this year’s production number.

0.062
0.054
0.048
0.033
0.073

Q13. Please download the file nba.xlsx. Rounded to the nearest $100,000, the average NBA player’s salary was

$3.5 million
$3.7 million
$3.9 million
$4.5 million
None of the above

Q14. Suppose that the following cash flows are received:

finalexam question 14 table

If cash flows are discounted at 10% per period, the net present value of those cash flows (rounded to the nearest cent) is

−$2.39
−$2.63
−$5.12
−$5.63
None of the above

Q15. For x > 0, which of these statements best describe the graph of y = x0.25?

The slope is greater than zero and is decreasing.
The slope is greater than zero and is increasing.
The slope is less than zero and is decreasing.
The slope is less than zero and is increasing.
None of the above

Q16. The equation of the line passing through the points (0, 3) and (2, 7) may be written as

2x – y = 3
x – 2y = 3
x + 2y = −3
y – 2x = 3
None of the above

Q17. A 3-year bond paying 4% annual coupons pays $1000 at maturity. Today the bond sells for $986.98. To the nearest one hundredth of one percent, the bond’s yield is

3.16%
4.87%
4.67%
4.47%
None of the above

Q18. Suppose that next year the U.S. will be in one of the following economic conditions: Boom, Moderate Growth, Recession, or Depression. The probability that each economic condition will occur, and that a jewelry store will earn profits within that broader economic condition are listed below:

finalexam question 18 table

The expected profit of the jewelry store during the next year is

$250,000
$220,000
$190,000
$170,000
None of the above

Q19. Your bank pays 5% annual interest, compounded quarterly. Rounded to the nearest one hundredth of a percent, the annual effective interest rate is

5.09%
5.00%
5.06%
5.12%
None of the above

Q20. Suppose that the following cash flows are received:

finalexam question 20 table

The Internal Rate of Return on the cash flows (rounded to the nearest percent) is

Q21. A roulette wheel contains the integers 1 through 36, 0, and 00. Suppose that you spin the wheel 6 times and that each time you bet on a single number. What is the probability (rounded to the nearest hundredth) that you win on at least one bet?

0.09
0.11
0.13
0.15
None of the above

Q22. The function f(x) = −x2 + 4x + 9 assumes its maximum value for which value of x?

2
1
3
0
None of the above

Q23. Suppose that you toss two dice. Let event A = the event that the first die shows a 4 and B = the event that the total on the two dice is 7.

Events A and B are independent events.

True
False

Q24. If a card is drawn from a deck, what is the chance that the card is an ace, a three, or a five?

2/13
11/52
1/2
1/4
None of the above

Q25. The fixed cost of developing a new drug is $3 billion. The unit cost of producing a single dose of the drug is $1, and patients are charged $11 per dose. In order to make a $1 billion profit on the drug, how many doses must be sold?

300,000,000
350,000,000
400,000,000
500,000,000
None of the above

Q26. Today you put $1000 in the bank. Your bank pays 5% interest, continuously compounded. In 3 years, how much money will you have in the bank (rounded to the nearest dollar)?

$1158
$1160
$1162
$1164
None of the above

Q27. The unit cost of producing a steak dinner at the Smalltown Inn is $6. If a restaurant charges p dollars for a steak dinner, customers will demand 200 – 5p steak dinners per week. To maximize the profit earned on steak dinners, what price should the inn charge?

Q28. If two dice are thrown, what is the probability that the first die shows a 4 or that the total on the two dice is 8?

11/36
1/3
1/2
5/18
None of the above

Q29. A 10-year bond paying 8% annual coupons pays $1000 at maturity. If the required rate of return on the bond is 7%, then today the bond will sell (rounded to the nearest cent) for

$1000.00
$1210.45
$987.48
$1070.24
None of the above

Q30. If f(x,y) = (2x)y, then f(4,3) is equal to ____________.

1296
512
256
64
None of the above

Q31.Smalltown Electronics buys MP3 players from its supplier for a price of $200 each. The store marks up the $200 by 50% to obtain the retail price. After Christmas, the retail price is marked down by 50%. After Christmas, the retail price will be ________ dollars.

Q32. An airline knows it will need to buy 100 million barrels of jet fuel 6 months from now. Of course, if the price of jet fuel increases, the airline will be in trouble. Suppose that put and call options on jet fuel are available for purchase.

To lower the airline’s risk associated with changes in jet fuel prices, the airline should purchase call options on jet fuel.

True
False

Q33. Evaluate

10
12
20
18
None of the above

Q34. What is the complete set of numbers for which f(x) = x3 + 2x2 – x is concave?

x ≥ −2/3
x ≥ −3/2
x ≥ 3/2
x ≥ 1
None of the above

Q35. Please download the file nba.xlsx to use in the following question. The file contains the salaries (in millions of dollars) of all NBA players during a recent season.

Which of the following choices best describes the salaries of NBA players?

symmetric
positively skewed
negatively skewed

Q36. A company made $1000 in profits this year. Its profits are increasing by 10% each year. How long will it take before its profits double?

between 5 and 6 years
between 6 and 7 years
between 7 and 8 years
between 8 and 9 years
None of the above

Q37. Suppose that the number of pounds of grapes sold by the Smalltown Co-op grocery store in a day is equally likely to be anywhere between 0 and 100 pounds (fractional values are possible). If you use a probability density function to describe the number of pounds of grapes sold daily by the store, the height of the function for any number of pounds between 0 and 100 is

0.01
0.02
0.005
0.10
None of the above

Q38. Consider a 9-month European call option with a strike price of $40 on a stock that sells for $35 today. If the annual risk-free rate (continuously compounded) is 8%, the stock pays no dividends, and the stock’s annual volatility is 40%, then the Black-Scholes price for this option (rounded to the nearest cent) is

$6.44
$3.77
$5.84
$8.12
None of the above

Q39. Please download the file nba.xlsx. If you had to express a typical NBA player’s salary with a single number (rounded to the nearest $100,000), you would use

$2.0 million
$2.5 million
$3.0 million
$3.5 million
None of the above

Q40. Suppose that Ln x = −3. Rounded to the nearest hundredth, what does x equal?

20.09
0.05
0.10
0.50
None of the above

Q41. Let p = the price in dollars charged for the dose of a drug. Suppose that the annual demand (in millions of doses) = 1000 – 3p. The annual supply (in millions of doses) = 600 + 2p. The annual supply will equal the annual demand if the price of a dose is ______ dollars.

Q42. Please download the file nba.xlsx. Rounded to the nearest $100,000, the mode of the NBA players’ salaries was

$500,000
$700,000
$900,000
$1,500,000
None of the above

Q43. For x = 2, the slope of the function f(x) = x4 – 3x2 is

10
15
25
30
None of the above

Q44. Smalltown Elevator produces elevator rails. To meet specifications, an elevator rail must be between 0.995 inches and 1.005 inches in diameter. Suppose that the diameter of an elevator rail follows a normal random variable with mean of 1 inch and standard deviation of 0.003 inches. Rounded to the nearest one tenth of one percent, what fraction of all elevator rails will meet specifications?

90.4%
95.2%
93.4%
97.3%
None of the above

Q45. The annual revenue-growth rates for a new tech startup during its first 4 years of operations were as follows:

finalexam question 45 table

Rounded to the nearest tenth of one percent, the startup’s 4-year Compound Annual Growth Rate was

-13.4%
0.0%
-25.0%
-10.5%
None of the above

Q46. During the last 4 months, the return on a stock and the return on the S&P market index was as follows.

finalexam question 46 table

Rounded to the nearest hundredth, the correlation between the monthly return on the stock and the S&P market index is

0.60
0.70
0.80
0.90
None of the above

Q47. Dylan Industries employs 50 workers. Blue collar workers are paid $600 per week, and white collar workers are paid $1000 per week. Dylan’s weekly payroll is $42,000. Dylan has _______ white collar employees.

Q48. The number of accidents at a dangerous intersection in Smalltown during each of the last six years is as follows: 0, 1, 1, 2, 3, 5. For this data set, the standard deviation of the number of accidents in a year (rounded to the nearest tenth) is

3.2
1.8
2.0
4.0
None of the above

Q49. Today you deposited $10,000 in a savings account that pays 6% annual interest, compounded semiannually. In 5 years, how much money will you have in your account (rounded to the nearest dollar)?

$13,121
$13,502
$13,382
$13,800
$13,439

Q50. If f(x) = x1/6 + x−1/3, then f(64) is equal to ________________.

Q51. Today is January 1, 2020. On January 1 of the years 2021 through 2030, you are to receive $50,000. If cash flows are discounted at 10% per year, the present value of these cash flows (as of today), rounded to the nearest dollar, is

$300,000
$337,951
$310,022
$307,228
None of the above

Q52. Suppose that next year the U.S. will be in one of the following economic conditions: Boom, Moderate Growth, Recession, or Depression. The probability that each economic condition will occur, and that a jewelry store will earn profits within that broader economic condition are listed below:

finalexam question 52 table

The standard deviation of the jewelry store’s profits next year (rounded to the nearest dollar) is

$102,442
$125,483
$149,873
$125,264
$263,818

Q53. On January 1 of the years 2015, 2016, and 2017, you received $15,000. If your investments earn 20% per year, how much money would you have on January 1, 2017?

$54,600
$65,520
$49,250
$63,000
None of the above

Q54. The inflection point for f(x) = x3 – 3x2 + x + 5 is

x = 0
x = 1
x = -1
x = 2
None of the above

Q55. Consider the equation 3x2 – 2x – 1 = 0. The sum of the roots of this equation is

1
2
-1/3
2/3
None of the above

Q56. Suppose that the average score on the GMAT exam is 500 and that the standard deviation of all scores is 100 points. You would expect approximately 95% of all GMAT scores to be between

400 and 600
300 and 700
200 and 800
350 and 650
250 and 750

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