# Harvard Mathematics For Management Pretest

Q1. Suppose that your investments earn an 8% rate of return. Today is January 1, 2015. On the first day of the years 2015, 2016, and 2017, you receive $10,000. How much money (rounded to the nearest dollar) will you have on January 1, 2018?

- $34,212
- $38,012
- $36,413
- $35,061
- None of the above

Q2. Let *q* = demand for seats on a 500-seat airplane and *p* = price charged per ticket. Suppose that *q* = 600 − 3*p* and assume that the unit cost of flying a passenger is $50. To maximize profit from the flight, the airline should charge how much per ticket?

- $100
- $125
- $150
- $175

Q3. Supply and demand for a product are both a linear function of price. Suppose that if a price of $8 is charged, 8 units will be demanded; that if a price of $5 is charged, 20 units will be demanded; that if a price of $3 is charged, 38 units will be supplied; and that if a price of $1 is charged, 26 units will be supplied. For what price will supply equal demand?

- $1
- $4
- $6
- $5
- None of the above

Q4. The numbers of goals scored by a soccer team during their last seven games were as follows: 0, 1, 2, 3, 4, 4, 5. For the given data set, which of the following is true?

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

Q5. Suppose that a stock is currently selling for $100. The change in the stock’s price during the next year follows a normal random variable with a mean of $10 and a standard deviation of $20. What is the probability (rounded to the nearest hundredth) that the stock will sell for at least $120 in a year’s time?

- 0.31
- 0.16
- 0.25
- 0.35
- 0.40

Q6. Consider the equation 2*x*2 − 7*x* + 3 = 0. If you divide the larger root of this equation by the smaller root, you obtain

- 4
- 6
- 8
- 2
- None of the above

Q7. Suppose that the following cash flows are received:

If cash flows are discounted at 20% per period, the net present value of these cash flows (rounded to the nearest cent) is

- $4.63
- $5.56
- $1.50
- $5.25
- None of the above

Q8. The function *f(x)* = *x*4 − 4*x*3 has how many inflection points?

- 0
- 1
- 2
- 3
- None of the above

Q9. The annual revenue-growth rates for a new tech startup during its first six years of operation were as follows:

Rounded to the nearest tenth of one percent, the startup’s 6-year Compound Annual Growth Rate was

- 3.3%
- 4.8%
- 11.7%
- 5.4%
- None of the above

Q10. Suppose two dice are tossed and that neither die shows a 4, 5, or 6. What is the probability that the total sum on the two dice is 6?

- 1/9
- 1/8
- 1/6
- 1/4
- None of the above

Q11. If f(*x,y*) = *x*2/3*y*−1/4, then *f*(27,16) equals

- 4
- 1.5
- 5
- 6.5
- None of the above

Q12. Rounded to the nearest percent, the Internal Rate of Return on the cash flows listed below is

- 28%
- 24%
- 26%
- 20%
- None of the above

Q13. When *x* = 3, the slope of the function *f(x)* = 4 + x + 2*x*2 is ___________.

Q14. Suppose that Log*b* 729 = 3 then, *b* must equal ___________.

Q15. Today is January 1, 2015. On the first day of the years 2016 through 2035, you will receive $10,000. 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

- $80,325
- $93.649
- $85,136
- $66,600
- None of the above

Q16. Suppose that a stock is currently selling for $100. The change in the stock’s price during the next year follows a normal random variable with a mean of $10 and a standard deviation of $20. What is the probability (rounded to the nearest hundredth) that the stock will sell for $85 or less in a year’s time?

- 0.13
- 0.14
- 0.17
- 0.20
- 0.11

Q17. The following table displays the probability of drinking a given number of bottles of diet soda in a day:

The average number of bottles of diet soda you drink in a day is

- 1.0
- 1.1
- 1.3
- 1.5
- None of the above

Q18. The equation of the straight line of slope 3 passing through the point (2, 5) may be written as

*y*= 3*x*+ 1*y*− 3*x*= −1*y*+ 3*x*= 1*y*− 3*x*= 2- None of the above

Q19. Suppose that 10% of all students at Cooley High play a winter sport, that 20% play a spring sport, and that 5% play both a winter and a spring sport. What is the probability that a randomly chosen student plays a winter or a spring sport?

- 30%
- 25%
- 35%
- 20%
- None of the above

Q20. Suppose that Joe Mauer has 150 hits in his 400 chances at bat. Assume that he will have 100 more chances at bat for the rest of the season. He needs ______ hits in his remaining chances at bat to ensure that he has hit .400. (Hitting .400 means he has gotten a hit during 40% of his at-bats.)

Q21. If f(*x*) = *x*2 + *x*3, then *f*(−3) equals ________________.

Q22. According to the Rule of Thumb, the data below lead you to believe that 95% of the time the monthly return on the S&P index will be between

- -6% and 8%
- -5% and 9%
- -4% and 6%
- -3% and 12%
- -8% and 9%

Q23. Your bank pays you 10% annual interest that is continuously compounded. Suppose you deposit $100 today. In 5 years, how much money will you have in the bank (rounded to the nearest cent)?

- $164.87
- $163.37
- $172.12
- $168.15
- None of the above

Q24. Suppose that for a sequence of cash flows, the first cash flow in the sequence is negative and that the cash flows have a unique Internal Rate of Return of 15% per year. Is the following statement true or false?

If you discount the cash flows using a rate of 15% per period, the NPV of the cash flows is positive.

- True
- False

Q25. When simplified, the expression 5 + 2 × 3 × (5 + 2) ÷ 3 + 1 equals ___________.

Q26. According to the Black-Scholes option pricing formula, what would be the correct price (rounded to the nearest cent) for a 6-month European put option, given the following parameters:

- $2.13
- $2.12
- $5.82
- $1.93
- None of the above

Q27. The following table displays the probability of drinking a given number of bottles of diet soda in a day:

The standard deviation of the number of bottles of diet soda drunk in a day (rounded to the nearest tenth) is

- 0.6 bottles
- 0.7 bottles
- 0.8 bottles
- 0.9 bottles
- None of the above

Q28. Consider a bond that matures in 10 years. It pays 9% annual coupons and $100 at maturity. If the required annual rate of return on the bond is 8%, then (rounded to the nearest cent) the bond will sell today for

- $93.77
- $106.71
- $100.00
- $131.18
- None of the above

Q29. Evaluate

- $93.77
- $106.71
- $100.00
- $131.18
- None of the above

Q30. For the data in the file sandp.xlsx, the mode of the monthly returns (rounded to the nearest one tenth of a percent) is

- -0.5%
- 0.0%
- 0.2%
- 0.4%
- None of the above

Q31. If two dice are tossed, what is the probability that a total of 8 is rolled?

- 3/36
- 4/12
- 5/36
- 6/36
- None of the above

Q32. For the data in the file sandp.xlsx, the median monthly return (rounded to the nearest one tenth of a percent) is

- 0.4%
- 0.5%
- 0.6%
- 1.0%
- 1.5%

Q33. Which of the following is the complete set of numbers for which the function *f(x)* = 3*x*3 − 6*x*2 is convex?

*x*≤ 0*x*≥ 0*x*≤ 1*x*≥ 1- None of the above

Q34. Suppose that your bank pays you a 10% annual interest rate, compounded quarterly, on your investments. If you deposit $100 today, in three years how much money will you have in the bank (rounded to the nearest cent)?

- $134.49
- $133.10
- $141.22
- $131.27
- None of the above

Q35. Let *q* = demand for a product and *p* = price in dollars charged for a product. Suppose that

*q* = 10,000*p*−4. If you wanted to “invert” this relationship to express *p* as a function of *q*, you would obtain

- p = 10q−0.25
- p = 100q−0.5
- p = 10q0.25
- p = q0.25/10
- None of the above

Q36. For what value of *x* does the function f(*x*) = 4*x*2 − 2*x* + 8 assume its minimum value?

- 0
- 1
- 2
- 1/4
- None of the above

Q37. When *x* = 2, the second derivative of *y* = 3*x*−1 equals

- .75
- .85
- .95
- 1
- None of the above

Q38. Suppose that a die is tossed three times. What is the probability that at least one 6 will be tossed?

- 80/108
- 1/2
- 100/216
- 91/216
- None of the above

Q39. Today is January 1, 2015. The annual discount rate is 10%. Beginning in 2016, you will receive $10,000 on the first day of every year. What is the present value of this infinite sequence of cash flows?

- $60,000
- $70,000
- $110,000
- $90,000
- None of the above

Q40. A zero-coupon bond pays $1000 in ten years and sells for $400 today. What is the bond’s yield (rounded to the nearest tenth of one percent)?

- 9.6%
- 7.5%
- 8.5%
- 10.2%
- None of the above

Q41. Suppose that a fixed cost of $100,000 is incurred in producing a product. The unit cost of producing each product is $50, and each unit of the product is sold for $90. To break even, ______ units of the product must be sold.

Q42. Suppose that in 2012, ABC Industries earned a profit of $1400. You know that in 2012 profits grew by 40% from their 2011 level. Also in 2011, profits grew by 25% from their 2010 level. The sum of the 2010 and the 2011 profits was

- $1600
- $1700
- $1800
- $1900
- None of the above

Q43. Suppose that the probability that it will rain on Saturday is 0.60. The probability that it will rain on Sunday is also 0.60. If it rains on one day, there is a 0.80 chance that it will rain the next day. What is the chance that it will rain at least one day during the weekend?

- 0.72
- 0.40
- 0.52
- 0.62
- None of the above

Q44. Britney is going to the candy store to buy 20 pieces of candy. She is going to buy chocolate candy and caramel candy. Each piece of chocolate candy costs 50 cents, and each piece of caramel candy costs 10 cents. You know that Britney spent $6.80 and bought 20 pieces of candy. She bought ______ pieces of chocolate.

Q45. Please download the file sandp.xlsx. The file contains monthly returns on the Standard & Poor’s Index. For example, in February 1881 the Standard & Poor’s Index dropped by 0.32%. Rounded to the nearest tenth of a percent, the mean monthly return on the S&P Index was

- 0.3%
- 0.4%
- 0.8%
- 1.0%
- 1.5%

Q46. Suppose that we create a histogram for the data in the file sandp.xlsx, with the boundaries for the bin ranges to the right.

The bin containing the most data points corresponds to monthly returns in the range

- -20% to -10%
- -10% to 0%
- 0% to 10%
- 10% to 20%
- 20% to 30%

Q47. You own 10,000 shares of Microsoft stock. A good way to hedge the risk involved in owning this stock would be to buy some call options on Microsoft stock.

- True
- False

Q48. If you want to graph a function in Excel, select the data and then choose Insert from the ribbon. Next, choose the ______________ type of graph to create the desired view. The answer you type below should be one word.

Q49. Let *q* = demand for a product and *p* = price of a product in dollars. Suppose that *q* = 10,000*p*−3. Then a 1% increase in price will decrease demand by approximately ______ percent.

Q50. Your bank pays you 10% annual interest that is continuously compounded. The annual effective interest rate (rounded to the nearest hundredth of a percent) is

- 10.00%
- 10.52%
- 10.25%
- 10.42%
- None of the above

Q51. The number of cookies that Jane ate during each of the last five days is as follows: 0, 1, 4, 7, 8. Rounded to the nearest tenth, the standard deviation of the number of cookies eaten daily is

- 4.0
- 3.8
- 3.6
- 3.5
- None of the above

Q52. Here are the heights and weights of four residents of Smalltown, Nevada:

The correlation between the height and the weight of these residents is

- .90
- .80
- .98
- .75
- .83

Q53. Suppose that a 3 of spades is drawn from a deck of cards. Let *A* = the event that the next card drawn is a heart and *B* = the event that the next card drawn is an ace. Is the following statement true or false?

Events *A* and *B* are independent.

- True
- False

Q54. The set of *x* satisfying −3*x* + 5 ≤ 17 is expressed as

*x*≤ −4*x*≤ 2- x ≥ −4
*x*≥ 4- None of the above

Q55. The executives of an auto plant are thinking of buying a machine that today costs $10 million. During each of the next 5 years, they expect the machine to generate the profits given in the following table.

The payback period for the machine is

- between 1 and 2 years
- between 2 and 3 years
- between 3 and 4 years
- exactly 4 years
- exactly 5 years

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