| Date | May 2018 | Marks available | 3 | Reference code | 18M.2.sl.TZ2.4 |
| Level | SL only | Paper | 2 | Time zone | TZ2 |
| Command term | Calculate | Question number | 4 | Adapted from | N/A |
Question
A new café opened and during the first week their profit was $60.
The café’s profit increases by $10 every week.
A new tea-shop opened at the same time as the café. During the first week their profit was also $60.
The tea-shop’s profit increases by 10 % every week.
Find the café’s profit during the 11th week.
Calculate the café’s total profit for the first 12 weeks.
Find the tea-shop’s profit during the 11th week.
Calculate the tea-shop’s total profit for the first 12 weeks.
In the mth week the tea-shop’s total profit exceeds the café’s total profit, for the first time since they both opened.
Find the value of m.
Markscheme
60 + 10 × 10 (M1)(A1)
Note: Award (M1) for substitution into the arithmetic sequence formula, (A1) for correct substitution.
= ($) 160 (A1)(G3)
[3 marks]
\(\frac{{12}}{2}\left( {2 \times 60 + 11 \times 10} \right)\) (M1)(A1)(ft)
Note: Award (M1) for substituting the arithmetic series formula, (A1)(ft) for correct substitution. Follow through from their first term and common difference in part (a).
= ($) 1380 (A1)(ft)(G2)
[3 marks]
60 × 1.110 (M1)(A1)
Note: Award (M1) for substituting the geometric progression nth term formula, (A1) for correct substitution.
= ($) 156 (155.624…) (A1)(G3)
Note: Accept the answer if it rounds correctly to 3 sf, as per the accuracy instructions.
[3 marks]
\(\frac{{60\left( {{{1.1}^{12}} - 1} \right)}}{{1.1 - 1}}\) (M1)(A1)(ft)
Note: Award (M1) for substituting the geometric series formula, (A1)(ft) for correct substitution. Follow through from part (c) for their first term and common ratio.
= ($)1280 (1283.05…) (A1)(ft)(G2)
[3 marks]
\(\frac{{60\left( {{{1.1}^n} - 1} \right)}}{{1.1 - 1}} > \frac{n}{2}\left( {2 \times 60 + \left( {n - 1} \right) \times 10} \right)\) (M1)(M1)
Note: Award (M1) for correctly substituted geometric and arithmetic series formula with n (accept other variable for “n”), (M1) for comparing their expressions consistent with their part (b) and part (d).
OR
(M1)(M1)
Note: Award (M1) for two curves with approximately correct shape drawn in the first quadrant, (M1) for one point of intersection with approximate correct position.
Accept alternative correct sketches, such as
Award (M1) for a curve with approximate correct shape drawn in the 1st (or 4th) quadrant and all above (or below) the x-axis, (M1) for one point of intersection with the x-axis with approximate correct position.
17 (A2)(ft)(G3)
Note: Follow through from parts (b) and (d).
An answer of 16 is incorrect. Award at most (M1)(M1)(A0)(A0) with working seen. Award (G0) if final answer is 16 without working seen.
[4 marks]
