GRADE 12 FET
PRELIMINARY EXAMINATION 2016

BRIDGE HOUSE

MATHEMATICS DEPARTMENT

Advanced Programme Mathematics:Finance and Modelling

Time: 1 hour100 marks

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

  1. This question paper consists of 6 pages and 8 questions. Please check that your paper is complete.
  2. Please make sure you get a separate formula sheet.
  3. Read the questions carefully.
  4. Answer all the questions.
  5. Number youranswers exactly as the questions are numbered.
  6. You may use an approved non-programmable and non-graphical calculator, unless a specific question prohibits the use of a calculator.
  7. Round youranswer to two decimal digits where necessary.
  8. All the necessary working details must be clearly shown.
  9. It is in your own interest to write legibly and to present your work neatly.

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GRADE 12: APM: Finance and Modelling
PRELIMINARY EXAMINATIONSPage 1 of 6

QUESTION 1

John opens a savings account and deposits R5 000 into the account immediately. Five months later he invests a further Rx into the account. The interest rate for the first four months is 18% per annum compounded monthly, 21% per annum compounded monthly for the next two months and 24% p.a. compounded monthly thereafter. The investment has a future value of R100 000 at the end of the 8 month. Calculate the value of [9]

QUESTION 2

An amount P was invested at per annum compounded monthly for two years. The accumulatedamount, R1 196,41 was reinvested at per annum compounded quarterly. After a further 3 years,the investment was worth R1 609,04. Determine and evaluate P.

[8]

QUESTION 3

A contactor buys a truck for R850000. The value of the truck depreciates by 20% p.a. on a reducing balance. This truck will need to be replaced at the end of 4 years. The value of a new truck is expected to appreciate by 15% p.a.

  1. Calculate the resale value of the truck in 4 years’ time.(4)
  2. What should be the value of a sinking fund that needs to be set up to pay for the new truck, if the old truck is used as a trade-in? (5)
  3. Monthly payments are made into a sinking fund account which earns interest at 12% p.a. compounded monthly. Payments commence in 3 months’ time, and finish in 4 years’ time. Calculate the monthly payment. (5)

[14]

QUESTION 4

John obtains a loan from the bank for his new home. He is charged 6,2% interest per annum, compounded monthly. He intends repaying the loan (at the end of every month) with equal monthly instalments of R9456 over a period of 20 years.

  1. Calculate the value of his loan.(6)
  2. Calculate how much of his first instalment is used to pay off loan capital. (not interest). (5)
  3. Calculate the equivalent annual interest rate, compounded daily, as a percentage, correct to four decimal places. (365 days per year). (8)

[19]

QUESTION 5

The population of an ant colony is modelled by using the Logistic model given below:

Where is in days and and .

  1. What is the limit of the number of ants in the colony?(2)
  2. Determine the value of correct to 2 decimal places.(3)

[5]

QUESTION 6

The formulae governing the number of rabbits and foxes in a Predator-Prey model is given as:

  1. Explain the presence of the term .(4)
  2. Calculate the equilibrium point for the above model given the following parameters:
    (15)

[19]

QUESTION 7

The favourite food of the blue whale is a called krill. The blue whale consumes huge quantities of this tiny animal as its main source of food. The maximum sustainable population for krill is 500 tons per acre of ocean. If there are no whales and the ocean is not overcrowded, the krill will increase at 25% p.a.

Under ideal conditions the whales have an average life span of 50 years.

If K is used for the krill population and W for the blue whale population, then the following equations model their populations:

  1. Determine the values of and .(4)
  2. Show why .(2)

In a certain ocean, the initial population of blue whales has been greatlyreduced due to whaling operations and is thus currently only at 5 000 whaleswhilst the krill has reached 750 tons per acre. The values of the parameters band f are given as and . The graph showing thechange of the krill and blue whale populations in this environment is givenbelow.

(Assume that all whaling operations cease immediately.)

  1. Explain why the rate of decrease in the krill population changes
    dramatically at A.(2)
  2. Estimate from the graph the krill population when the blue whale population is increasing most rapidly. (2)
  3. Estimate the stable populations for blue whales and krill.(2)
  4. If the krill’s intrinsic growth rate reduces to 10% p.a., determine the new equilibrium point. (6)

[18]

QUESTION 8

Von Koch’s Snowflake is a well-known fractal, based on a recursive theme.

This is obtained by the original line segment in Step 1 being trisected. On the middle of the three sections an equilateral triangle is constructed so that all the line segments are equal in length.

This is obtained by each of the four line segments in Step 2 again being trisected. On each of the middle sections of each of the four line segments in Step 2 another equilateral triangle has been constructed.

  1. Write down a first order recursive formula that represents the length of the
    fractal shape.(4)
  2. Determine which step will be the first where the length exceeds 1000 units. Also give this length, correct to three decimal places. (4)

[8]

[100]

TOTAL FOR THIS PAPER: 100 MARKS

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