VCE General Maths Recursion and Financial Modelling 2023 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: 10
Reading time: 2.5 minutes
Writing time: 15 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 - 2023 & 2021 - Exam 2
Arthur borrowed $30 000 to buy a new motorcycle.
Interest on this loan is charged at the rate of 6.4% per annum, compounding quarterly.
Arthur will repay the loan in full with quarterly repayments over six years.
a. How many repayments, in total, will Arthur make? 1 mark
The balance of the loan, in dollars, after \(n\) quarters, \(A_n\), can be modelled by the recurrence relation
\(A_0 = 30000, \quad A_{n+1} = 1.016A_n - 1515.18\)
b. Showing recursive calculations, determine the balance of the loan after two quarters. Round your answer to the nearest cent. 1 mark
c. The final repayment required will differ slightly from all the earlier repayments of $1515.18
Determine the value of the final repayment.
Round your answer to the nearest cent. 1 mark
Arthur invests $600 000 in an annuity that provides him with a monthly payment of $3973.00
Interest is calculated monthly.
Three lines of the amortisation table for this annuity are shown below.
| Payment number | Payment ($) | Interest ($) | Principal reduction ($) | Balance ($) |
|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 0.00 | 600 000.00 |
| 1 | 3973.00 | 2520.00 | 1453.00 | 598 547.00 |
| 2 | 3973.00 | 2513.90 | 1459.10 | 597 087.90 |
a. The interest rate for the annuity is 0.42% per month.
Determine the interest rate per annum. 1 mark
b. Using the values in the table, complete the next line of the amortisation table.
Write your answers in the spaces provided in the table below.
Round all values to the nearest cent. 1 mark
| Payment number | Payment ($) | Interest ($) | Principal reduction ($) | Balance ($) |
|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 0.00 | 600 000.00 |
| 1 | 3973.00 | 2520.00 | 1453.00 | 598 547.00 |
| 2 | 3973.00 | 2513.90 | 1459.10 | 597 087.90 |
| 3 |
c. Let \(V_n\) be the balance of Arthur’s annuity, in dollars, after \(n\) months.
Write a recurrence relation in terms of \(V_0\), \(V_{n+1}\) and \(V_n\) that can model the value of the annuity from month to month. 1 mark
d. The amortisation tables on page 11 show that the balance of the annuity reduces each month.
If the balance of an annuity remained constant from month to month, what name would be given to this type of annuity? 1 mark
Sienna invests $152 431 into an annuity from which she will receive a regular monthly payment of $900 for 25 years. The interest rate for this annuity is 5.1% per annum, compounding monthly.
a. Let \(V_n\) be the balance of the annuity after \(n\) monthly payments. A recurrence relation written in terms of \(V_0\), \(V_{n+1}\) and \(V_n\) can model the value of this annuity from month to month.
Showing recursive calculations, determine the value of the annuity after two months.
Round your answer to the nearest cent. 2 marks
b. After two years, the interest rate for this annuity will fall to 4.6%.
To ensure that she will still receive the same number of $900 monthly payments, Sienna will add an extra one-off amount into the annuity at this time.
Determine the value of this extra amount that Sienna will add.
Round your answer to the nearest cent. 1 mark
End of Question and Answer Book
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