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Date	May 2021	Marks available	3	Reference code	21M.2.SL.TZ1.3
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	3	Adapted from	N/A

Question

Give your answers in parts (a), (d)(i), (e) and (f) to the nearest dollar.

Daisy invested $37 000$ Australian dollars ( $AUD$ ) in a fixed deposit account with an annual interest rate of $6.4 %$ compounded quarterly.

After $m$ months, the amount of money in the fixed deposit account has appreciated to more than $50 000 AUD$ .

Daisy is saving to purchase a new apartment. The price of the apartment is $200 000 AUD$ .

Daisy makes an initial payment of $25 %$ and takes out a loan to pay the rest.

The loan is for $10$ years, compounded monthly, with equal monthly payments of $1700 AUD$ made by Daisy at the end of each month.

For this loan, find

After $5$ years of paying off this loan, Daisy decides to pay the remainder in one final payment.

Calculate the value of Daisy’s investment after $2$ years.

[3]

Find the minimum value of $m$ , where $m \in ℕ$ .

[4]

Write down the amount of the loan.

[1]

the amount of interest paid by Daisy.

[2]

d.i.

the annual interest rate of the loan.

[3]

d.ii.

Find the amount of Daisy’s final payment.

[3]

Find how much money Daisy saved by making one final payment after $5$ years.

[3]

Markscheme

EITHER

$N = 2$
$P V = - 37 000$
$I % = 6.4$
$P / Y = 1$
$C / Y = 4$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct.

OR

$N = 8$
$P V = - 37 000$
$I % = 6.4$
$P / Y = 4$
$C / Y = 4$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct.

$F V = 37 000 \times {(1 + \frac{6.4}{100 \times 4})}^{4 \times 2}$ (M1)(A1)

Note: Award M1 for substitution into compound interest formula, (A1) for correct substitution.

$= 42 010 AUD$ A1

Note: Award (M1)(A1)A0 for unsupported $42009.87$ .

[3 marks]

EITHER

$P V = - 37 000$
$F V = 50 000$
$I % = 6.4$
$P / Y = 1$
$C / Y = 4$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct. The final mark can still be awarded for the correct number of months (multiple of $3$ ).

OR

$P V = - 37 000$
$F V = 50 000$
$I % = 6.4$
$P / Y = 4$
$C / Y = 4$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct.

$50 000 < 37 000 \times {(1 + \frac{6.4}{100 \times 4})}^{4 \times n}$ OR $50 000 < 37 000 \times {(1 + \frac{6.4}{100 \times 4})}^{n}$ (M1)(A1)

Note: Award M1 for the correct inequality, $50 000$ and substituted compound interest formula. Allow an equation. Award A1 for correct substitution.

THEN

$N = 4.74 (years) (4.74230 \dots)$ OR $N = 18.9692 \dots (quarters)$ (A1)

$m = 57$ months A1

Note: Award A1 for rounding their $m$ to the correct number of months. The final answer must be a multiple of $3$ . Follow through within this part.

[4 marks]

$150 000 AUD$ A1

[1 mark]

$120 \times 1700 - 150 000$ (M1)

$= 54 000 AUD$ A1

[2 marks]

d.i.

$N = 120$
$P V = - 150 000$
$P M T = 1700$
$F V = 0$
$P / Y = 12$
$C / Y = 12$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology or an attempt to use an annuity formula or $F V = 0$ seen. If a compound interest formula is equated to zero, award M1, otherwise award M0 for a substituted compound interest formula.
Award A1 for all entries correct in financial app or correct substitution in annuity formula, but award A0 for a substituted compound interest formula. Follow through marks in part (d)(ii) are contingent on working seen.

$r = 6.46 (%) (6.45779 \dots)$ A1

[3 marks]

d.ii.

$N = 60$
$I = 6.46 (6.45779 \dots)$
$P V = - 150 000$
$P M T = 1700$
$P / Y = 12$
$C / Y = 12$ (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology or an attempt to use an annuity formula. Award (M0) for a substituted compound interest formula. Award A1 for all entries correct. Follow through marks in part (e) are contingent on working seen.

$F V = 86973 AUD$ A1

[3 marks]

$204 000 - (60 \times 1700 + 86973)$ OR $204 000 - 188 973$ (M1)(A1)

Note: Award M1 for $60 \times 1700$ . Award M1 for subtracting their $(60 \times 1700 + 86973)$ from their $(204 000)$ . Award at most M1M0 for their $204 000 - (60 \times 1700)$ or M0M0 for their $204 000 - (86973)$ . Follow through from parts (d)(i) and (e). Follow through marks in part (f) are contingent on working seen.

$= 15 027 AUD$ A1

[3 marks]

Examiners report

[N/A]

d.i.

[N/A]

d.ii.

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.4—Financial apps – compound interest, annual depreciation

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22M.2.SL.TZ2.2d.i:
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SPM.2.SL.TZ0.1a.i:
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18M.2.SL.TZ1.T_3b:
Calculate 14 000 USD in INR.
SPM.2.AHL.TZ0.3a.ii:
Find the total amount paid for the car.
22M.2.SL.TZ2.2d.ii:
Describe what the value in part (d)(i) represents in this context.
22M.1.SL.TZ2.13b:
For option 2, find the minimum value of $x$ that Juliana would need to invest each year. Give your answer to the nearest dollar.
22M.1.SL.TZ2.13a:
For option 1, determine the minimum amount Juliana would need to invest. Give your answer to the nearest dollar.
21N.1.SL.TZ0.10b.i:
Find the amount Raul and Rosy will still owe the bank at the end of the first $10$ years.
17N.2.AHL.TZ0.H_12c:
Given that Phil’s aim is to own the house after 20 years, find the value for $P$ to the nearest dollar.
18M.1.SL.TZ2.T_4b:
Josh states: “Every integer is a natural number”.

Write down whether Josh’s statement is correct. Justify your answer.
19N.1.SL.TZ0.T_8a:
Calculate the value of Siân’s investment after four years. Give your answer correct to two decimal places.
SPM.2.SL.TZ0.1a.iii:
Find the interest paid on the loan.
SPM.2.SL.TZ0.1b.i:
Find the amount to be borrowed for this option.
SPM.2.SL.TZ0.1a.ii:
Find the total amount paid for the car.
SPM.2.AHL.TZ0.3a.iii:
Find the interest paid on the loan.
SPM.2.AHL.TZ0.3a.i:
Find the repayment made each quarter.
EXM.1.SL.TZ0.1a:
Calculate the value of her savings after 10 years.
SPM.2.SL.TZ0.1b.ii:
Find the annual interest rate, $r$ .
21M.2.SL.TZ1.3b:
Find the minimum value of $m$ , where $m \in ℕ$ .
18M.2.SL.TZ1.T_3d:
Harinder chose option B. At the end of five years, Harinder converted this investment back to USD. The exchange rate, at that time, was 1 USD = 67.16 INR.

Calculate how much more money, in USD, Harinder earned by choosing option B instead of option A.
17N.2.AHL.TZ0.H_12d.ii:
Hence or otherwise, find the minimum value of $Q$ that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.
20N.1.SL.TZ0.T_8a:
Calculate the value of Imon’s investment after $5$ years.
SPM.2.AHL.TZ0.3b.i:
Find the amount to be borrowed for this option.
SPM.2.AHL.TZ0.3b.ii:
Find the annual interest rate, $r$ .
19N.1.SL.TZ0.T_8b:
After the four-year period, Siân withdraws $40\,000$ AUD from her savings account and uses this money to buy a car. It is known that the car will depreciate at a rate of $18$ % per year.

The value of the car will be $2500$ AUD after $t$ years.

Find the value of $t$ .
EXN.1.SL.TZ0.9b:
Find an approximation for the real interest rate for the money invested in the account.
SPM.2.SL.TZ0.1c:
State which option Bryan should choose. Justify your answer.
SPM.2.AHL.TZ0.3c:
State which option Bryan should choose. Justify your answer.
17N.2.AHL.TZ0.H_12a:
Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.
21M.2.SL.TZ1.3d.ii:
the annual interest rate of the loan.
17N.2.AHL.TZ0.H_12b:
Show that the total value of Phil’s savings after 20 years is $\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}$ .
18M.2.SL.TZ1.T_3c:
Calculate the amount of this investment, in INR, in this account after five years.
21M.1.SL.TZ2.10b:
Tommaso wants to invest his money in an account such that his investment will increase to $1.5$ times the initial amount in $5$ years. Assume the account pays a nominal annual interest of $r %$ compounded quarterly.

Determine the value of $r$ .
SPM.2.AHL.TZ0.3d:
Bryan chooses option B. The car dealership invests the money Bryan pays as soon as they receive it.

If they invest it in an account paying 0.4 % interest per month and inflation is 0.1 % per month, calculate the real amount of money the car dealership has received by the end of the 6 year period.
SPM.2.SL.TZ0.1d:
Bryan’s car depreciates at an annual rate of 25 % per year.

Find the value of Bryan’s car six years after it is purchased.
21M.2.SL.TZ1.3c:
Write down the amount of the loan.
21M.2.SL.TZ1.3d.i:
the amount of interest paid by Daisy.
21M.2.SL.TZ1.3f:
Find how much money Daisy saved by making one final payment after $5$ years.
21M.2.SL.TZ1.3a:
Calculate the value of Daisy’s investment after $2$ years.
21M.1.SL.TZ2.10a:
Calculate the amount Pietro will have in his account after $5$ years. Give your answer correct to $2$ decimal places.
21M.1.AHL.TZ2.5a:
Estimate the value of Roger’s laptop after $5$ years.
21M.1.AHL.TZ2.5c:
Comment on the validity of your answer to part (b).
18M.2.SL.TZ1.T_3a:
Calculate the value of r.
17N.2.AHL.TZ0.H_12d.i:
David wishes to withdraw $5000 at the end of each year for a period of $n$ years. Show that an expression for the minimum value of $Q$ is

$\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}$ .
EXM.1.SL.TZ0.1b:
The rate of inflation during this 10 year period is 1.5% per year.

Calculate the real value of her savings after 10 years.
EXN.1.SL.TZ0.9a:
Find the value of her investment after a period of $5$ years.
EXN.1.SL.TZ0.9c:
Hence find the real value of Sophia’s investment at the end of $5$ years.
20N.1.SL.TZ0.T_8b:
At the end of the $5$ years, Imon withdrew $x SGD$ from the fixed deposit account and reinvested this into a super-savings account with a nominal annual interest rate of $5.7 %$ , compounded half-yearly.

The value of the super-savings account increased to $20 000 SGD$ after $18$ months.

Find the value of $x$ .
21M.1.AHL.TZ2.5b:
Find the value of $k$ .
21N.1.SL.TZ0.10a:
Find the amount they will pay the bank each month.
21N.1.SL.TZ0.10b.ii:
Using your answers to parts (a) and (b)(i), calculate how much interest they will have paid in total during the first $10$ years.

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