VCE General Maths Recursion and Financial Modelling 2019 Exam 2 Mini Test
VCAA General Maths Exam 2
This is the full VCE General Maths Exam with worked solutions. You can also try Mini-Tests, which are official VCAA exams split into short tests you can do anytime.
Number of marks: 12
Reading time: 3 minutes
Writing time: 18 minutes
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In all questions where a numerical answer is required, you should only round your answer when instructed to do so.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Recursion and financial modelling - 2019 - Exam 2
Recursion and financial modelling
Question 1 (4 marks)Phil is a builder who has purchased a large set of tools.
The value of Phil’s tools is depreciated using the reducing balance method.
The value of the tools, in dollars, after \(n\) years, \(V_n\), can be modelled by the recurrence relation shown below.
\(V_0 = 60\,000, \quad V_{n+1} = 0.9 V_n\)
a. Use recursion to show that the value of the tools after two years, \(V_2\), is $48 600. 1 mark
b. What is the annual percentage rate of depreciation used by Phil? 1 mark
c. Phil plans to replace these tools when their value first falls below $20 000.
After how many years will Phil replace these tools? 1 mark
d. Phil has another option for depreciation. He depreciates the value of the tools by a flat rate of 8% of the purchase price per annum.
Let \(V_n\) be the value of the tools after \(n\) years, in dollars.
Write down a recurrence relation, in terms of \(V_0\), \(V_{n+1}\) and \(V_n\), that could be used to model the value of the tools using this flat rate depreciation. 1 mark
Phil invests $200 000 in an annuity from which he receives a regular monthly payment.
The balance of the annuity, in dollars, after \(n\) months, \(A_n\), can be modelled by the recurrence relation
\(A_0 = 200\,000, \quad A_{n+1} = 1.0035 A_n - 3700\)
a. What monthly payment does Phil receive? 1 mark
b. Show that the annual percentage compound interest rate for this annuity is 4.2%. 1 mark
At some point in the future, the annuity will have a balance that is lower than the monthly payment amount.
c. What is the balance of the annuity when it first falls below the monthly payment amount?
Round your answer to the nearest cent. 1 mark
d. If the payment received each month by Phil had been a different amount, the investment would act as a simple perpetuity.
What monthly payment could Phil have received from this perpetuity? 1 mark
Phil would like to purchase a block of land.
He will borrow $350 000 to make this purchase.
Interest on this loan will be charged at the rate of 4.9% per annum, compounding fortnightly.
After three years of equal fortnightly repayments, the balance of Phil’s loan will be $262 332.33
a. What is the value of each fortnightly repayment Phil will make?
Round your answer to the nearest cent. 1 mark
b. What is the total interest Phil will have paid after three years?
Round your answer to the nearest cent. 1 mark
c. Over the next four years of his loan, Phil will make monthly repayments of $3517.28 and will be charged interest at the rate of 4.8% per annum, compounding monthly.
Let \(B_n\) be the balance of the loan \(n\) months after these changes apply.
Write down a recurrence relation, in terms of \(B_0\), \(B_{n+1}\) and \(B_n\), that could be used to model the balance of the loan over these four years. 2 marks
End of Question and Answer Book
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