2020 LegaC MC Financial Modelling and Matrices

The recurrence relation given below is used to generate the terms of a sequence of numbers.

S₀ = 4, S_{n + 1} = –2S_n + 5

Which one of the following statements is not true for this sequence?

• Term S₃ = –17.

• Terms S₁ and S₅ are negative numbers.

• Terms S₂ and S₄ are positive numbers.

• S₁ + S₂ + S₃ + S₄ = 30.

• All terms of the sequence are negative numbers.

Brad purchased a truck for his business on a hire purchase agreement. He paid a deposit of $d and then $m per month for three years. By the end of the first year Michael had paid $52760 and by the end of the third year he had paid $148400.

The values of d and m are respectively:

• d = 6183, m = 4397

• d = 4397, m = 6183

• d = 4940, m = 3985

• d = 3985, m = 4940

• d = 4940, m = 47820

Consider the sequence defined by a_n = 2_n – 5. The first-order recurrence relation for this sequence is:

• a₀ = – 5, a_n+1 = 2a_n + 5

• a₀ = – 5, a_n+1 = a_n – 2

• a₀ = – 5, a_n+1 = a_n + 2

• a₀ = – 5, a_n+1 = 2a_n

• a₀ = – 5, a_n+1 = a_n + 5

Let C_n+1 = C_n − d, C₀ = $25800 be the recurrence relation that is used to calculate the depreciated value of a car after n years. Which one of the following expressions is not true?

• C₂ = C₃ + d

• C_n+1 < C_n

• C_n = C₀ – nd

• C₂ = C₁ − 2d

• C₄ − C₃ = −d

Jane deposits $15000 into an investment account. This account earns interest at a rate of 4.5% per annum compounded annually. The recurrence relations that could be used to calculate the balance of this investment account, Bn, after n years is

• B₀ = 15000, B_n+1 = 1.045B_n

• B₀ = 15000, B_n+1 = 1.45B_n

• B₀ = 15000, B_n+1 = 1.004B_n + 56.25

• B₀ = 15000, B_n+1 = 0.375B_n + 56.25

• B₀ = 15000, B_n+1 = 0.375B_n + 675

James inherited $130 000 in 2017. He invested this money straightaway in a perpetuity from which he receives quarterly payments. Over a period of two years James received payments totalling $10140.

The annual interest rate of this investment is

• 0.975%

• 1.95%

• 3.9%

• 7.8%

• 15.6%

James inherited $130 000 in 2017. He invested this money straightaway in a perpetuity from which he receives quarterly payments. Over a period of two years James received payments totalling $10140. After two years James decides to invest the balance of this account into an annuity, which pays interest at a rate of 3.75% per annum, compounded monthly. He will also deposit a monthly amount so that, at the end of the first year, the balance of this annuity, correct to the nearest cent, will be $136485.72. At the end of the first year James stops making monthly payments into this annuity. Which one of the following statements is not true for this situation?

• The opening balance of this annuity is $130 000.

• The interest earned over the first year from this annuity is $4985.72.

• The monthly amount deposited in this annuity is $125.

• The balance of this annuity at the end of the second year, correct to the nearest cent, is $141 692.83.

• The interest earned in the second year from this annuity is $4985.72.

Diana borrowed $370 000 from a financial institution to purchase a new property. She recorded all the transactions for this home loan, for the first four months of the loan, in the amortisation table. After four months, the financial institution decreased the interest rate of this home loan. Which one of the following statements is not true for this situation?

• The interest rate for this home loan, correct to one decimal place, is 3.5% per annum for the first four months.

• The principal reduction in the fifth month is $2839.19.

• The interest rate was reduced by 0.18% per annum in the fifth month.

• The interest rate for this home loan in the fifth month, correct to two decimal places, is 3.38% per annum.

• The amount still owed to the financial institution at the end of the fifth month is $356029.

Harrison took out a personal loan from a financial institution in order to go on a trip around the world for 6 months. He borrowed $56 000 at an interest rate of 4.9% per annum, compounding quarterly. He will make quarterly repayments of $3900. Harrison intends to fully repay this loan in four years. The last repayment of this loan, correct to the nearest cent is

• $3214.59

• $3227.72

• $3428.26

• $3470.26

• $3900.00

A building company has just purchased an excavator for $125 000. The excavator is going to be depreciated at a rate of 5.49% per annum compounded annually for the first 7 years, and at a rate of 15% per annum compounded annually thereafter. Which one of the following graphs best shows the value of the excavator as it depreciates over a period of 10 years from purchase?

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