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Date	May Specimen paper	Marks available	2	Reference code	SPM.2.AHL.TZ0.3
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	3	Adapted from	N/A

Question

In this question, give all answers to two decimal places.

Bryan decides to purchase a new car with a price of €14 000, but cannot afford the full amount. The car dealership offers two options to finance a loan.

Finance option A:

A 6 year loan at a nominal annual interest rate of 14 % compounded quarterly. No deposit required and repayments are made each quarter.

Finance option B:

A 6 year loan at a nominal annual interest rate of $r$ % compounded monthly. Terms of the loan require a 10 % deposit and monthly repayments of €250.

Find the repayment made each quarter.

[3]

a.i.

Find the total amount paid for the car.

[2]

a.ii.

Find the interest paid on the loan.

[2]

a.iii.

Find the amount to be borrowed for this option.

[2]

b.i.

Find the annual interest rate, $r$ .

[3]

b.ii.

State which option Bryan should choose. Justify your answer.

[2]

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.

[4]

Markscheme

N = 24
I % = 14
PV = −14000
FV = 0
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. Accept PV = 14000.

(€)871.82 A1

[3 marks]

a.i.

4 × 6 × 871.82 (M1)

(€) 20923.68 A1

[2 marks]

a.ii.

20923.68 − 14000 (M1)

(€) 6923.68 A1

[2 marks]

a.iii.

0.9 × 14000 (= 14000 − 0.10 × 14000) M1

(€) 12600.00 A1

[2 marks]

b.i.

N = 72

PV = 12600

PMT = −250

FV = 0

P/Y = 12

C/Y = 12 (M1)(A1)

Note: Award M1 for an attempt to use a financial app in their technology, award A1 for all entries correct. Accept PV = −12600 provided PMT = 250.

12.56(%) A1

[3 marks]

b.ii.

EITHER

Bryan should choose Option A A1

no deposit is required R1

Note: Award R1 for stating that no deposit is required. Award A1 for the correct choice from that fact. Do not award R0A1.

Bryan should choose Option B A1

cost of Option A (6923.69) > cost of Option B (72 × 250 − 12600 = 5400) R1

Note: Award R1 for a correct comparison of costs. Award A1 for the correct choice from that comparison. Do not award R0A1.

[2 marks]

real interest rate is 0.4 − 0.1 = 0.3% (M1)

value of other payments 250 + 250 × 1.003 + … + 250 × 1.003⁷¹

use of sum of geometric sequence formula or financial app on a GDC (M1)

= 20 058.43

value of deposit at the end of 6 years

1400 × (1.003)⁷² = 1736.98 (A1)

Total value is (€) 21 795.41 A1

Note: Both M marks can awarded for a correct use of the GDC’s financial app:

N = 72 (6 × 12)
I % = 3.6 (0.3 × 12)
PV = 0
PMT = −250
FV =
P/Y = 12
C/Y = 12

N = 72 (6 × 12)
I % = 0.3
PV = 0
PMT = −250
FV =
P/Y = 1
C/Y = 1

[4 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

Syllabus sections

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

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SPM.2.SL.TZ0.1a.i:
Find the repayment made each quarter.
18M.2.SL.TZ1.T_3b:
Calculate 14 000 USD in INR.
22M.2.SL.TZ2.2d.ii:
Describe what the value in part (d)(i) represents in this context.
22M.1.SL.TZ2.13b:
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22M.1.SL.TZ2.13a:
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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.TZ1.3e:
Find the amount of Daisy’s final payment.
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:
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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:
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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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Topic 1—Number and algebra

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