Actuary Quiz

Section 1: Business related Mathematics

Q1. A plumber quoted $300, excluding GST (Goods and Services Tax), to complete a job. A GST of 10% is added to the price. The full price for the job will be

A. $3

B. $30

C. $303

D. $310

E. $330

Q2. Pia invests $800 000 in an ordinary perpetuity to provide an ongoing fortnightly pension for her retirement. The interest rate for this investment is 5.8% per annum. Assuming there are 26 fortnights per year, the amount she will receive at the end of each fortnight is closest to

A. $464

B. $892

C. $1422

D. $1785

E. $3867

Q3. A computer originally purchased for $6000 is depreciated each year using the reducing balance method. If the computer is valued at $2000 after four years, then the annual rate of depreciation is closest to

A. 17%

B. 24%

C. 25%

D. 28%

E. 33%

Q4. Sandra has purchased a $4200 plasma television under a hire-purchase agreement. She paid $600 deposit and will pay the balance in equal monthly instalments over one year. A flat interest rate of 6% per annum is charged. The amount of each monthly instalment is

A. $300

B. $303

C. $318

D. $350

E. $371

Q5. Sandra has purchased a $4200 plasma television under a hire-purchase agreement. She paid $600 deposit and will pay the balance in equal monthly instalments over one year. A flat interest rate of 6% per annum is charged. The annual effective interest rate that Sandra pays under this agreement is closest to

A. 10%

B. 11%

C. 12%

D. 13%

E. 14%

Q6. Sam and Charlie each invest $5000 for three years. Sam’s investment earns simple interest at the rate of 7.5% per annum. Charlie’s investment earns interest at the rate of 7.5% per annum compounding annually. At the conclusion of three years, correct to the nearest cent, Sam will have

A. $86.48 less than Charlie.

B. $86.48 more than Charlie.

C. $132.23 less than Charlie.

D. $132.23 more than Charlie.

E. the same as Charlie.

Q7. Ernie took out a reducing balance loan to buy a new family home. He correctly graphed the amount paid off the principal of his loan each year for the first five years. The shape of this graph (for the first five years of the loan) is best represented by

Q8. A loan of $300 000 is taken out to finance a new business venture. The loan is to be repaid fully over twenty years with quarterly payments of $6727.80. Interest is calculated quarterly on the reducing balance. The annual interest rate for this loan is closest to

A. 4.1%

B. 6.5%

C. 7.3%

D. 19.5%

E. 26.7%

Q9. An amount of $8000 is invested for a period of 4 years. The interest rate for this investment is 7.2% per annum compounding quarterly. The interest earned by the investment in the fourth year (in dollars) is given by

A.

B.

C.

D.

E.

Q10. A pensioner receives annual payments starting at £100 and increasing at 3% p.a. payable at the end of each year for 50 years. The present value of these payments at an interest rate of 5% p.a. is approximately

A. £1,010

B. £2,020

C. £3,030

D. £4,040

E. £5,050

Section 2: Sequences and Series

Q11. A sequence is generated by a first-order linear difference equation. The first four terms of this sequence are 1, 3, 7, 15. The next term in the sequence is

A. 17

B. 19

C. 22

D. 23

E. 31

Q12. For an examination, 8600 examination papers are to be printed at a rate of 25 papers per minute. After one hour, the number of examination papers that still need to be printed is

A. 1600

B. 2500

C. 6100

D. 7100

E. 8575

Q13. The values of the first seven terms of a geometric sequence are plotted on the graph below.

Values of a and r that could apply to this sequence are respectively

A. a = 90 r = –0.9

B. a = 100 r = –0.9

C. a = 100 r = –0.8

D. a = 100 r = 0.8

E. a = 90 r = 0.9

Q14. In 2008, there are 800 bats living in a park. After 2008, the number of bats living in the park is expected to increase by 15% per year. Let Bn represent the number of bats living in the park n years after 2008. A difference equation that can be used to determine the number of bats living in the park n years after 2008 is

A.

B.

C.

D.

E.

Q15. The first term of a geometric sequence is 9. The third term of this sequence is 121. The second term of this sequence could be

A. –65

B. –33

C. 56

D. 65

E. 112

Q16. Kai commenced a 12-day program of daily exercise. The time, in minutes, that he spent exercising on each of the first four days of the program is shown in the table below.

If this pattern continues, the total time (in minutes) that Kai will have spent exercising after 12 days is

A. 59

B. 180

C. 354

D. 444

E. 468

Q17. The sequence 12, 15, 27, 42, 69, 111 . . . can best be described as

A. fibonacci-related

B. arithmetic with d > 1

C arithmetic with d < 1

D. geometric with r > 1

E geometric with r < 1

Q18. When placed in a pond, the length of a fish was 14.2 centimetres. During its first month in the pond, the fish increased in length by 3.6 centimetres. During its nth month in the pond, the fish increased in length by Gn centimetres, where Gn+1 = 0.75Gn

The maximum length this fish can grow to (in cm) is closest to

A. 14.4

B. 16.9

C. 19.0

D. 28.6

E. 71.2

Q19. When full, a swimming pool holds 50 000 litres of water. Due to evaporation and spillage the pool loses, on average, 2% of the water it contains each week. To help to make up this loss, 500 litres of water is added to the pool at the end of each week. Assume the pool is full at the start of Week 1. At the start of Week 5 the amount of water (in litres) that the pool contains will be closest to

A. 47 500

B. 47 600

C. 48 000

D. 48 060

E. 48 530

Q20. A pensioner receives a payment of £100 every month for the next 20 years. The pension provider funds for this with a lump sum of money which he places in a bank account receiving 6.5% p.a. interest. How much must he place in the account to ensure that there are sufficient funds to cover the payments?

A. 13400

B. 13500

C. 13600

D. 13700

E. 13800

Section 3: Matrices

Q21. If the d is equal to

A. -11

B. -10

C. 10

D. 7

E. 11

Q22. Apples cost $3.50 per kg, bananas cost $4.20 per kg and carrots cost $1.89 per kg. Ashley buys 3 kg of apples, 2 kg of bananas and 1 kg of carrots. A matrix product to calculate the total cost of these items is

A .

B.

C.

D.

E.

Q23. The cost prices of three different electrical items in a store are $230, $290 and $310 respectively. The selling price of each of these three electrical items is 1.3 times the cost price plus a commission of $20 for the salesman. A matrix that lists the selling price of each of these three electrical items is determined by evaluating

A.

B.

C.

D.

E.

Q24. Matrix A is a 1 × 3 matrix. Matrix B is a 3 × 1 matrix. Which one of the following matrix expressions involving A and B is defined?

A.

B.

C.

D.

E.

Q25. The determinant of is equal to 9. The value of x is

A. –7

B. –4.5

C. 1

D. 4.5

E. 7

Q26. The solution to the matrix equation is

A.

B.

C.

D.

E.

Infomation for Q27 and Q28

A large population of mutton birds igrates each year to a remote island to nest and breed. There are four nesting sites on the island, A, B, C and D.

Researchers suggest that the following transition matrix can be used to predict the number of mutton birds

nesting at each of the four sites in subsequent years. An equivalent transition diagram is also given.

Q27. Two thousand eight hundred mutton birds nest at site C in 2008. Of these 2800 mutton birds, the number that nest at site A in 2009 is predicted to be

A. 560

B. 980

C. 1680

D. 2800

E. 3360

Q28. This transition matrix predicts that, in the long term, the mutton birds will

A. nest only at site A.

B. nest only at site B.

C. nest only at sites A and C.

D. nest only at sites B and D.

E. continue to nest at all four sites.

Q29. Six thousand mutton birds nest at site B in 2008. Assume that an equal number of mutton birds nested at each of the four sites in 2007. The same transition matrix applies. The total number of mutton birds that nested on the island in 2007 was

A. 6 000

B. 8 000

C. 12 000

D. 16 000

E. 24 000

Q30. Consider the linear transformation given by the matrix the basis for this transformation under, which the matrix representation is diagonal, is given by:

A. {(5,3),(1,-1)}

B. {(3,5),(1,-1)}

C. {(3,5),(-1,1)}

D. {(5,3),(-1,1)}

E. {(3,5),(-1,-1)}

Section 4: Calculus

Q31. The first differential of the function is given by:

A.

B.

C.

D.

E.

Q32. Solve the following differential equation , y(0)=0 and hence decide which of the following is closest to the value of y(1).

A. 16.3

B. 17.3

C. 18.3

D. 19.3

E. 20.3

Q33. By solving the corresponding characteristic equation show that the general solution to the linear differential equation is:

A.

B.

C.

D.

E.

Q34. The general solution to the differential equation is given by

A.

B.

C.

D.

E.

Q35. The differential equation in above is used to model the assets, £S million, of a bank t years after it was set up. Given that the initial assets of the bank were £200 million, and using your answer to the above, the assets of the bank 10 years after it was set up, to the nearest £ million are:

A. £524

B. £534

C. £544

D. £554

E. £564

Q36. By using integration show that the volume of the shape made by rotating the function , between x=-1 and x=1, 360 degrees about the x axis is given by:

A.

B.

C.

D.

E.

Q37. The curve C has polar equation . The area of the region bounded by C is given by.

A. a/2

B. a/4

C. a

D. 2a

E. 4a

Q38. The integral of the function is given by:

A.

B.

C.

D.

E.

Q39. Using partial fractions or otherwise show that the following integral: is closest to:

A. 0.04

B. 0.05

C. 0.06

D. 0.07

E. 0.08

Q40. Using integration by substitution, state which of the following gives the closest answer to the integral of

A. 1/10

B. 1/15

C. 1/20

D. -1/15

E. -1/10

Section 5: Probability and Statistics

Q41. A fair die is rolled until a 1 appears. The probability that an odd number of rolls are needed is:

A. 1/6

B. 5/6

C. 6/11

D. 5/11

E. 1/2

Q42. If the distribution function F is given by then the density function of Y=15+2X is:

A.

B.

C.

D.

E.

Q43. The moment generating function for the normal distribution with mean and standard deviation is given by:

A.

B.

C.

D.

E.

Q44. Experience shows that the number of accidents on a particular road is distributed with a poisson distribution with a mean of 2 accidents per week. Using the central limit theorem what is the approximate probability that there will less than 100 accidents in a year.

A. 0.25

B. 0.33

C. 0.5

D. 0.67

E. 0.75

Q45. Assume the amount of rainfall recorded in a day is uniformly distributed on the interval (0,b). If a sample of 10 years’ records shows that the following amounts of rainfall were recorded on that date, compute using the method of moments and estimate of b. 0,0,0.7,1,0.1,0,0.2,0.5,0,0.6 measurements are in inches.

A. 0.42

B. 0.47

C. 0.52

D. 0.57

E. 0.62

Q46. If a die is thrown 50 times what is the expected number of sixes scored?

A. 8.33

B. 7.33

C. 6.33

D. 5.33

E. 4.33

Q47. The discrete random variable X takes the values 0 and 1 with probabilities of ½ each. The Probability Generating Function, Moment Generating Function and Standard deviation (in that order) are given by:

A.

B.

C.

D.

E.

Q48. The random variables X and Y are distributed with means and variances . The correlation coefficient of X and Y is . If Z=X+Y then the mean and variance of Z are:

A.

B.

C.

D.

E.

Q49. The mean, mean deviation, standard deviation and mid-spread of the observations 16, 5, 7, 13, 2, 9, 3, 20, 13, 6, 5 is:

A. 18, 5.73, 4.49, 8

B. 18, 4.73, 5.49, 8

C. 8, 4.73, 5.49, 18

D. 8, 5.49, 4.73, 18

E. 18, 4.73, 5.49, 18

Q50. Ignoring leap years what is the value of N such that the following probability is less than1/2: "The probability that none of a group of N or more randomly chosen people has a birthday on the same day as anyone else in the group"

A. 21

B. 22

C. 23

D. 24

E. 25