VCE General Maths Recursion and Financial Modelling 2020 Exam 2 Mini Test 1
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: 9
Reading time: 2 minutes
Writing time: 14 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 - 2020 - Exam 2 (Part 1)
Samuel has a reducing balance loan.
The first five lines of the amortisation table for Samuel’s loan are shown below.
| Payment number | Payment (\$) |
Interest (\$) |
Principal reduction (\$) |
Balance (\$) |
|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 0.00 | 320 000.00 |
| 1 | 1600.00 | 960.00 | 640.00 | 319 360.00 |
| 2 | 1600.00 | 958.08 | 641.92 | 318 718.08 |
| 3 | 1600.00 | 956.15 | 318 074.23 | |
| 4 | 1600.00 |
Interest is calculated monthly and Samuel makes monthly payments of \$1600.
Interest is charged on this loan at the rate of 3.6% per annum.
a. Using the values in the amortisation table
i. calculate the principal reduction associated with payment number 3 1 mark
ii. calculate the balance of the loan after payment number 4 is made.
Round your answer to the nearest cent. 1 mark
b. Let \(S_n\) be the balance of Samuel’s loan after \(n\) months.
Write down a recurrence relation, in terms of \(S_0\), \(S_{n+1}\) and \(S_n\), that could be used to model the month-to-month balance of the loan. 1 mark
Samuel opens a savings account.
Let \(B_n\) be the balance of this savings account, in dollars, \(n\) months after it was opened.
The month-to-month value of \(B_n\) can be determined using the recurrence relation shown below.
\( B_0 = 5000, \quad B_{n+1} = 1.003 B_n \)
a. Write down the value of \(B_4\), the balance of the savings account after four months.
Round your answer to the nearest cent. 1 mark
b. Calculate the monthly interest rate percentage for Samuel’s savings account. 1 mark
c. After one year, the balance of Samuel’s savings account, to the nearest dollar, is \$5183.
If Samuel had deposited an additional \$50 at the end of each month immediately after the interest was added, how much extra money would be in the savings account after one year?
Round your answer to the nearest dollar. 1 mark
Samuel now invests \$500 000 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 a recurrence relation of the form
\( A_0 = 500\,000, \quad A_{n+1} = kA_n - 2000 \)
a. Calculate the balance of this annuity after two months if \(k = 1.0024\). 1 mark
b. Calculate the annual compound interest rate percentage for this annuity if \(k = 1.0024\). 1 mark
c. For what value of \(k\) would this investment act as a simple perpetuity? 1 mark
End of Question and Answer Book
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