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FURTHER MATHEMATICS – MID YEAR HOLIDAY EXAMPART 3

Name:

Reading time 5 minutes Writing time 45 minutesRecursion & Financial Modelling Multiple-choice questions

Total: /20 marks1 The recurrence relation that generates the sequence 5, 9, 17, 33, 65, … is:

A V0 = 5, Vn+1 = 5Vn

B V0 = 5, Vn+1 = 14 – Vn D V0 = 5, Vn+1 = 2Vn – 1

C V0 = 5, Vn+1 = Vn + 4 E V0 = 5, Vn+1 = 3Vn – 6

2 The recurrence relation V0 = 10000, Vn+1 = 1.012Vn – 2000 models a reducing-balance loan

with payments of $2000 each quarter, where Vn is the value of the loan after n quarters. The

annual interest rate for this loan is:

A 1.2%

B 1.4% D 4.8%

C 3.12% E 14.4%

3 The recurrence relation V0 = 375 000, Vn+1 = 0.918 Vn is used to calculate the value of farm

machinery after n years, Vn , that is depreciating in value using a reducing-balance method. The

value of the machinery after eight years, V8, correct to the nearest dollar, is:

A 159 389

B 173 626 D 206 030

C 189 135 E 224 433

4 A car purchased for $25000 will be depreciated at the rate of 8.2% per annum, using a flat-rate

method. The car will be sold when it’s value first drops below $10000. The car will be sold after:

A 8 years

B 9 years D 11 years

C 10 years E 12 years

5 The number of years it will take an investment of $5000, earning compound interest at the rate

of 6.2% per annum to grow above $7000 is

A 5

B 6 D 8

C 7 E 9

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Use the following information to answer Questions 6 and 7.The value of a loan charged compound interest every year is modelled using the recurrence relation

V0 = 15 000, Vn+1 = 1.08Vn, where Vn is the value of the loan after n years.

6 A rule for the value of the loan after n years is

A Vn = 15 000 + 1.08n

B Vn = 15 000 + 1200n D Vn = (1.08)n 15 000

C Vn = (1.08)n 2000 E Vn = (15 000)n 1.08

7 The total amount of interest that is charged over the first five years of the loan is closest to:

A $5400

B $6000 D $8800

C $7040 E $22040

8 An investment of $28 000 has grown to $33578 after five years invested at r% per annum

compounding annually. The value of r is closest to:

A 2.3%

B 3.6% D 4.1%

C 3.7% E 4.3%

9 Which one of the following compound interest investments will earn the most interest after one

year?

A 3.4% per annum, compounding fortnightly

B 3.47% per annum, compounding quarterly

C 3.56% per annum, compounding weekly

D 3.64% per annum, compounding annually

E 3.66% per annum, compounding monthly

10 The number of trout in a fish farm pond after n months, Tn, can be modelled using the recurrence

relation T0 = 8000, Tn+1 = 0.96Tn + 400. Which of the following statements is not true?

A Each month, the number of trout in the pond is more than the number of trout in the pond in

the previous month.

B Each month, the number of trout added to the pond is 5% of the number of trout in the pond

in the previous month.

C Each month, 4% of the trout in the pond in the previous month are removed.

D Each month, 400 trout are added to the pond.

E Each month, the number of trout removed from the pond is more than 300.

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Use the following information to answer Questions 11 and 12.A reducing-balance loan of $6000 is charged compound interest at the rate of 7.68% per annum,

compounding monthly. The loan will be repaid with monthly payments of $400.

11 A recurrence relation that models the value of the loan after n months, Vn, is:

A V0 = 6000, Vn+1 = 0.9232Vn + 400

B V0= 6000,Vn+1 = 0.914Vn – 400

C V0= 6000,Vn+1 = 0.9936Vn + 400

D V0= 6000,Vn+1 = 1.086Vn – 400

E V0= 6000,Vn+1 = 1.0064Vn – 400

12 The number of payments required to fully repay the loan is closest to:

A 13

B 14 D 16

C 15 E 17

Use the following information to answer Questions 13 and 14.The amortisation table for a reducing-balance loan repaid with monthly payments is shown below.

Payment number Payment Interest

Principal reduction

Balance of loan

0 0 0.00 0.00 15 000.00

1 2520.00 48.00 2472.00 12 528.00

2 2520.00 40.09 2479.91 10 048.09

3 2520.00 32.15 2487.85 7560.24

4 2520.00 24.19 2495.81 5064.44

5 2520.00 16.21 2503.79 2560.64

6 p 8.19 2560.65 0.00

13 The annual interest rate for this loan is:

A 1.9%

B 3.2% D 4.8%

C 3.84% E 22.9%

14 The final payment for this loan, p, will be:

A $2511.21

B $2520.00 D $2560.65

C $2528.19 E $2568.84

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15 The value of an annuity investment of $160 000 after n years, Vn, with annual payments of

$30000 is modelled with the recurrence relation

V0 = 160 000, Vn+1 = 1.05 Vn 30 000.A graph of the value of the annuity is:

A B C

D E

16 A perpetuity investment is established that will provide a monthly payment of $400. If the

investment is paid compounding interest at the rate of 6.1% per annum, compounding monthly,

the principal that must be invested is closest to:

A $6500

B $26 230 D $78 700

C $65 570 E $170 400

17 Brian invested $50 000 into an account that earns compound interest at the rate of 4.6% per

annum, compounding fortnightly. After one year, the value of the investment has grown to $58

997. The amount Brian is adding to his investment every fortnight is closest to:

A $250

B $270 D $650

C $540 E $890

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Use the following information to answer Questions 18, 19 and 20.The first 5 monthly payments of a reducing-balance loan are shown in the amortisation table below.

The third payment amount, p, is missing.

Payment number Payment Interest

Principal reduction

Balance of loan

0 0.00 0.00 0.00 150 000.00

1 980.00 717.50 262.50 149 737.50

2 980.00 716.24 263.76 149 473.74

3 p 714.98 485.02 148 988.73

4 1500.00 712.66 787.34 148 201.39

5 1500.00 708.90 791.10 147 410.29

18 The third payment amount, p, is:

A $980

B $1200 D $1500

C $1240 E $2250

19 If there are no further payment changes, the total number of remaining payments required to

fully repay this loan is closest to:

A 133

B 134 D 136

C 135 E 137

20 After six years, the annual interest rate for this loan was increased to 6.2% per annum. If the

loan is to be fully repaid after four more years, the new monthly payments are closest to:

A $1480

B $1500 D $1980

C $1645 E $2000

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Name:

Reading time 5 minutes Writing time 30 minutesRecursion & Financial Modelling Extended Response1 Phillip is a chef. He runs a cooking school from his home kitchen. A 10-week course will cost $1200 if

paid up-front.

Boon Soo enrolled in the course, but has negotiated to borrow the course fee from Phillip and pay it

back with $20 interest each week. Let Vn be the value of Boon Soo’s loan after n weeks.

a. Write down a recurrence relation that models the value of this simple-interest loan after n weeks.

1 mark

b. Use the recurrence relation to find the amount that Boon Soo will owe Phillip after 4 weeks.

1 mark

The rule Vn = 1200 + 20n can be used to calculate the value of Boon Soo’s loan after n weeks.

c. What is the value of Boon Soo’s loan after 6 weeks?

1 mark

d. Use the rule to calculate the number of weeks it takes for the value of Boon Soo’s loan to grow to

$1360.

1 mark

2. The ovens and other equipment in Phillip’s kitchen have been depreciated using a flat-rate depreciation

method. A recurrence relation that models the value of the kitchen equipment after n years, Vn, is

shown below.

V0 = 5800, Vn+1 = Vn – 350

a. What was the original purchase price of the kitchen equipment?

1 mark

b. What is the annual percentage rate of depreciation of the kitchen equipment? Write your answer

correct to two decimal places.

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1 mark

c. The kitchen equipment will be replaced when its value first drops below $2000.

i. After how many years will the kitchen equipment be replaced?

1 mark

ii. What is the value of the kitchen equipment when it is replaced?

1 mark

One of the ovens in Phillip’s kitchen will be depreciated using a unit-cost method, based on the number

of hours it is in use. After 100 hours of use, the value of the oven is $1170. After 260 hours of use, the

value of the oven is $1122.

d. i. Calculate the depreciation per unit (hour) for the oven.

1 mark

ii. Write down a rule for the value of the oven after n hours, Cn.

1 mark

iii. After how many hours of use will the value of the oven be half it’s purchase price?

1 mark

3. Phillip has borrowed some money from the bank to build a cooking school building in his backyard. The

amortisation table for this loan, showing the first 5 monthly payments on the reducing-balance loan, is

shown below. The letters A, B and C represent values from the table that are missing.

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Payment number

Payment amount

Interest paid

Principal reduction

Balance of loan

0 0 0 0 80 000.00

1 2000.00 352.00 A 78 352.00

2 2000.00 344.75 1655.25 76 696.75

3 2000.00 337.47 1662.53 75 034.22

4 B 330.15 1919.85 C

5 2250.00 321.70 1928.30 71 186.07


The values in the amortisation table have been rounded to the nearest cent, where necessary.

a. Use the values in the row for payment number 1 and 2 to show that the annual interest rate for this

loan is 5.28%.

1 mark

b. Write down the missing values of A, B and C.

1 mark

Phillip will continue to repay the loan with monthly payments of $2250.

c. Use a financial solver to determine the total number of payments Phillip must make, correct to the

nearest whole number, in order to fully repay this loan.

1 mark

d. After the fifth payment, the interest rate was increased to 5.32% per annum, compounding monthly.

If Phillip will increase his monthly payments so that he will fully repay the loan in the same time

period, what will his new payments be? Round your answer to the nearest cent.

1 mark

TOTAL: / 15 marks

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