| Date | November 2017 | Marks available | 3 | Reference code | 17N.2.sl.TZ0.2 |
| Level | SL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 2 | Adapted from | N/A |
Question
Rosa joins a club to prepare to run a marathon. During the first training session Rosa runs a distance of 3000 metres. Each training session she increases the distance she runs by 400 metres.
A marathon is 42.195 kilometres.
In the \(k\)th training session Rosa will run further than a marathon for the first time.
Carlos joins the club to lose weight. He runs 7500 metres during the first month. The distance he runs increases by 20% each month.
Write down the distance Rosa runs in the third training session;
Write down the distance Rosa runs in the \(n\)th training session.
Find the value of \(k\).
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
Find the distance Carlos runs in the fifth month of training.
Calculate the total distance Carlos runs in the first year.
Markscheme
3800 m (A1)
[1 mark]
\(3000 + (n - 1)400{\text{ m}}\)\(\,\,\,\)OR\(\,\,\,\)\(2600 + 400n{\text{ m}}\) (M1)(A1)
Note: Award (M1) for substitution into arithmetic sequence formula, (A1) for correct substitution.
[2 marks]
\(3000 + (k - 1)400 > 42195\) (M1)
Notes: Award (M1) for their correct inequality. Accept \(3 + (k - 1)0.4 > 42.195\).
Accept \( = \) OR \( \geqslant \). Award (M0) for \(3000 + (k - 1)400 > 42.195\).
\((k = ){\text{ }}99\) (A1)(ft)(G2)
Note: Follow through from part (a)(ii), but only if \(k\) is a positive integer.
[2 marks]
\(\frac{{50}}{2}\left( {2 \times 3000 + (50 - 1)(400)} \right)\) (M1)(A1)(ft)
Note: Award (M1) for substitution into sum of an arithmetic series formula, (A1)(ft) for correct substitution.
\(640\,000{\text{ m}}\) (A1)
Note: Award (A1) for their \(640\,000\) seen.
\( = 640{\text{ km}}\) (A1)(ft)(G3)
Note: Award (A1)(ft) for correctly converting their answer in metres to km; this can be awarded independently from previous marks.
OR
\(\frac{{50}}{2}\left( {2 \times 3 + (50 - 1)(0.4)} \right)\) (M1)(A1)(ft)(A1)
Note: Award (M1) for substitution into sum of an arithmetic series formula, (A1)(ft) for correct substitution, (A1) for correctly converting 3000 m and 400 m into km.
\( = 640{\text{ km}}\) (A1)(G3)
[4 marks]
\(7500 \times {1.2^{5 - 1}}\) (M1)(A1)
Note: Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.
\( = 15\,600{\text{ m }}(15\,552{\text{ m}})\) (A1)(G3)
OR
\(7.5 \times {1.2^{5 - 1}}\) (M1)(A1)
Note: Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.
\( = 15.6{\text{ km}}\) (A1)(G3)
[3 marks]
\(\frac{{7500({{1.2}^{12}} - 1)}}{{1.2 - 1}}\) (M1)(A1)
Notes: Award (M1) for substitution into sum of a geometric series formula, (A1) for correct substitutions. Follow through from their ratio (\(r\)) in part (d). If \(r < 1\) (distance does not increase) or the final answer is unrealistic (eg \(r = 20\)), do not award the final (A1).
\( = 297\,000{\text{ m }}(296\,853 \ldots {\text{ m}},{\text{ }}297{\text{ km}})\) (A1)(G2)
[3 marks]
