| Date | May 2014 | Marks available | 5 | Reference code | 14M.2.sl.TZ2.4 |
| Level | SL only | Paper | 2 | Time zone | TZ2 |
| Command term | Calculate | Question number | 4 | Adapted from | N/A |
Question
Give your answers to parts (a) to (e) to the nearest dollar.
On Hugh’s 18th birthday his parents gave him options of how he might receive his monthly allowance for the next two years.
Option A \(\$60\) each month for two years
Option B \(\$10\) in the first month, \(\$15\) in the second month, \(\$20\) in the third month, increasing by \(\$5\) each month for two years
Option C \(\$15\) in the first month and increasing by \(10\%\) each month for two years
Option D Investing \(\$1500\) at a bank at the beginning of the first year, with an interest rate of \(6\%\) per annum, compounded monthly.
Hugh does not spend any of his allowance during the two year period.
If Hugh chooses Option A, calculate the total value of his allowance at the end of the two year period.
If Hugh chooses Option B, calculate
(i) the amount of money he will receive in the 17th month;
(ii) the total value of his allowance at the end of the two year period.
If Hugh chooses Option C, calculate
(i) the amount of money Hugh would receive in the 13th month;
(ii) the total value of his allowance at the end of the two year period.
If Hugh chooses Option D, calculate the total value of his allowance at the end of the two year period.
State which of the options, A, B, C or D, Hugh should choose to give him the greatest total value of his allowance at the end of the two year period.
Another bank guarantees Hugh an amount of \(\$1750\) after two years of investment if he invests $1500 at this bank. The interest is compounded annually.
Calculate the interest rate per annum offered by the bank.
Markscheme
The first time an answer is not given to the nearest dollar in parts (a) to (e), the final (A1) in that part is not awarded.
\(60 \times 24\) (M1)
Note: Award (M1) for correct product.
\( = 1440\) (A1)(G2)
[2 marks]
The first time an answer is not given to the nearest dollar in parts (a) to (e), the final (A1) in that part is not awarded.
(i) \(10 + (17 - 1)(5)\) (M1)(A1)
Note: Award (M1) for substituted arithmetic sequence formula, (A1) for correct substitution.
\( = 90\) (A1)(G2)
(ii) \(\frac{{24}}{2}\left( {2(10) + (24 - 1)(5)} \right)\) (M1)
OR
\(\frac{{24}}{2}\left( {10 + 125} \right)\) (M1)
Note: Award (M1) for correct substitution in arithmetic series formula.
\( = 1620\) (A1)(ft)(G1)
Note: Follow through from part (b)(i).
[5 marks]
The first time an answer is not given to the nearest dollar in parts (a) to (e), the final (A1) in that part is not awarded.
(i) \(15{(1.1)^{12}}\) (M1)(A1)
Note: Award (M1) for substituted geometric sequence formula, (A1) for correct substitutions.
\( = 47\) (A1)(G2)
Note: Award (M1)(A1)(A0) for \(47.08\).
Award (G1) for \(47.08\) if workings are not shown.
(ii) \(\frac{{15({{1.1}^{24}} - 1)}}{{1.1 - 1}}\) (M1)
Note: Award (M1) for correct substitution in geometric series formula.
\( = 1327\) (A1)(ft)(G1)
Note: Follow through from part (c)(i).
[5 marks]
The first time an answer is not given to the nearest dollar in parts (a) to (e), the final (A1) in that part is not awarded.
\(1500{\left( {1 + \frac{6}{{100(12)}}} \right)^{12(2)}}\) (M1)(A1)
Note: Award (M1) for substituted compound interest formula, (A1) for correct substitutions.
OR
\(N = 2\)
\(I\% = 6\)
\(PV = 1500\)
\(P/Y = 1\)
\(C/Y = 12\) (A1)(M1)
Note: Award (A1) for \(C/Y = 12\) seen, (M1) for other correct entries.
OR
\(N = 24\)
\(I\% = 6\)
\(PV = 1500\)
\(P/Y = 12\)
\(C/Y = 12\) (A1)(M1)
Note: Award (A1) for \(C/Y = 12\) seen, (M1) for other correct entries.
\( = 1691\) (A1)(G2)
[3 marks]
The first time an answer is not given to the nearest dollar in parts (a) to (e), the final (A1) in that part is not awarded.
Option D (A1)(ft)
Note: Follow through from their parts (a), (b), (c) and (d). Award (A1)(ft) only if values for the four options are seen and only if their answer is consistent with their parts (a), (b), (c) and (d).
[1 mark]
\(1750 = 1500{\left( {1 + \frac{r}{{100}}} \right)^2}\) (M1)(A1)
Note: Award (M1) for substituted compound interest formula equated to \(1750\), (A1) for correct substitutions into formula.
OR
\(N = 2\)
\(PV = 1500\)
\(FV = - 1750\)
\(P/Y = 1\)
\(C/Y = 1\) (A1)(M1)
Note: Award (A1) for \(FV = 1750\) seen, (M1) for other correct entries.
\( = 8.01\% {\text{ (8.01234}} \ldots \% ,{\text{ }}0.0801{\text{)}}\) (A1)(G2)
[3 marks]
