VCE General Maths Recursion and Financial Modelling 2022 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 - 2022 & 2020 - Exam 2
Pina owns workplace equipment, which she depreciates in value using flat rate depreciation.
The value of the equipment, in dollars, after \(n\) years, \(V_n\), can be determined using the rule
\(V_n = 200\,000 - 12\,500n\)
a. Determine \(V_1\), the value of the equipment after one year. 1 mark
b. After how many years will the equipment first have a value of zero? 1 mark
c. The value of the equipment, in dollars, after \(n\) years, \(V_n\), can also be modelled by a recurrence relation.
Write this recurrence relation in terms of \(V_0\), \(V_{n+1}\) and \(V_n\). 1 mark
d. Using Pina's depreciation model, the value of the equipment decreases by a fixed percentage of its original value each year.
Alternatively, the value of the equipment could have been depreciated by a fixed percentage of its current value each year.
What name is given to this type of depreciation? 1 mark
Pina invests $540 000 in an annuity paying 3% interest per annum, compounding monthly. Her annuity will provide a monthly payment of $5214.28 for 10 years.
Four lines of the amortisation table for Pina's annuity are shown below.
The information for payment number 3 is missing.
| Payment number | Payment ($) | Interest ($) | Principal reduction ($) | Balance ($) |
|---|---|---|---|---|
| 0 | 0.00 | 0.00 | 0.00 | 540 000.00 |
| 1 | 5214.28 | 1350.00 | 3864.28 | 536 135.72 |
| 2 | 5214.28 | 1340.34 | 3873.94 | 532 261.78 |
| 3 |
a. What is the value of payment number 3? 1 mark
b. Calculate the interest associated with payment number 3.
Round your answer to the nearest cent. 1 mark
c. Let \(P_n\) be the balance, in dollars, of Pina's annuity after \(n\) months.
Write a recurrence relation, in terms of \(P_0\), \(P_{n+1}\) and \(P_n\), that can model this balance from month to month. 1 mark
d. If Pina had invested the original $540 000 annuity as a simple perpetuity, what monthly payment would she have drawn? 1 mark
Later, Samuel took out a new reducing balance loan.
The interest rate for this loan was 4.1% per annum, compounding monthly.
The balance of the loan after four years of monthly repayments was \$329 587.25
The balance of the loan after seven years of monthly repayments was \$280 875.15
Samuel will continue to make the same monthly repayment.
To ensure the loan is fully repaid, to the nearest cent, the required final repayment will be lower.
In the first seven years, Samuel made 84 monthly repayments.
From this point on, how many more monthly repayments will Samuel make to fully repay the loan?
End of Question and Answer Book
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