VCE General Maths Recursion and Financial Modelling 2021 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: 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 - 2021 - Exam 2
Sienna invests $420 000 in a perpetuity from which she will receive a regular monthly payment of $1890.
The perpetuity earns interest at the rate of 5.4% per annum.
a. Determine the total amount, in dollars, that Sienna will receive after one year of monthly payments. 1 mark
b. Write down the value of the perpetuity after Sienna has received one year of monthly payments. 1 mark
c. Let \(S_n\) be the value of Sienna's perpetuity after \(n\) months.
Complete the recurrence relation, in terms of \(S_0\), \(S_{n+1}\) and \(S_n\), that would model the value of this perpetuity over time. Write your answers in the boxes provided. 1 mark
\(S_0 = \) , \(S_{n+1} = \) \( \times S_n - 1890 \)
Sienna owns a coffee shop.
A coffee machine, purchased for $12 000, is depreciated in value using the unit cost method.
The rate of depreciation is $0.05 per cup of coffee made.
The recurrence relation that models the year-to-year value, in dollars, of the coffee machine is
\(M_0 = 12\,000, \quad M_{n+1} = M_n - 1440\)
a. Calculate the number of cups of coffee that the machine produces per year. 1 mark
b. The recurrence relation above could also represent the value of the coffee machine depreciating at a flat rate.
What annual flat rate percentage of depreciation is represented? 1 mark
c. Complete the rule below that gives the value of the coffee machine, \(M_n\), in dollars, after \(n\) cups have been produced. Write your answers in the boxes provided. 1 mark
\(M_n = \) \( + \) \( \times n \)
For renovations to the coffee shop, Sienna took out a reducing balance loan of $570 000 with interest calculated fortnightly.
The balance of the loan, in dollars, after \(n\) fortnights, \(S_n\), can be modelled by the recurrence relation
\(S_0 = 570\,000, \quad S_{n+1} = 1.001S_n - 1193\)
a. Calculate the balance of this loan after the first fortnightly repayment is made. 1 mark
b. Show that the compound interest rate for this loan is 2.6% per annum. 1 mark
c. For the loan to be fully repaid, to the nearest cent, Sienna’s final repayment will be a larger amount.
Determine this final repayment amount.
Round your answer to the nearest cent. 1 mark
End of Question and Answer Book
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