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Date	May 2018	Marks available	3	Reference code	18M.2.SL.TZ2.T_4
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	T_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.

[3]

Calculate the café’s total profit for the first 12 weeks.

[3]

Find the tea-shop’s profit during the 11th week.

[3]

Calculate the tea-shop’s total profit for the first 12 weeks.

[3]

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.

[4]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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).

(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 1^st (or 4^th) 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]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series

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22M.2.SL.TZ2.2a.i:
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22M.2.SL.TZ1.2a.i:
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17M.1.AHL.TZ2.H_3b:
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16N.1.SL.TZ0.S_9b:
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19M.1.AHL.TZ2.H_1:
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18M.2.SL.TZ2.T_4e:
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Find the value of m.
17M.1.AHL.TZ1.H_7b:
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18M.2.SL.TZ2.T_4b:
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19M.2.SL.TZ2.T_4d.iii:
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18M.1.SL.TZ1.T_7b:
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16N.2.AHL.TZ0.H_12d:
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19N.2.SL.TZ0.T_3e:
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19M.2.SL.TZ2.S_10d:
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21M.1.SL.TZ1.8a.i:
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Find the amount of fuel pumped into the tank in the $13th$ hour.
19N.2.SL.TZ0.T_3a:
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Find the total cost of buying 2 tickets in each of the first 16 rows.
17M.1.SL.TZ1.T_5a:
Find the distance from the base of this ladder to the top rung.
19N.2.SL.TZ0.T_3d:
Find the maximum number of complete levels that Maegan can build.
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21M.2.SL.TZ2.3a.ii:
the total number of seats in the concert hall.
21M.2.SL.TZ2.3a.i:
the number of seats in the last row.
17M.1.AHL.TZ2.H_3a:
the value of $d$ ;
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19M.2.SL.TZ1.S_7b.ii:
Find the value of $m$ .
19M.2.SL.TZ2.S_10c:
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Find the value of $n$ such that $u_{n} = 0$ .
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Write down an expression for ${A_n}$ in terms of $x$ on the day before Mary turned 18 years old showing clearly the value of $n$ .
17N.2.SL.TZ0.T_2c:
Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.
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Find an expression for the width of $U_{n}$ in centimetres.
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Find the value of the bicycle during the 5th year. Give your answer to two decimal places.
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A game is played in which the arrow attached to the centre of the disc is spun and the sector in which the arrow stops is noted. If the arrow stops in sector $1$ the player wins $10$ points, otherwise they lose $2$ points.

Let $X$ be the number of points won

Find $E (X)$ .
19M.2.SL.TZ2.S_10a:
Find the value of $k$ .
19M.2.SL.TZ1.S_7b.i:
Find the value of $r$ .
21M.1.SL.TZ1.8a.ii:
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EXN.3.AHL.TZ0.2b:
Given the width of a pixel is approximately $0.025 cm$ , find the number of squares in the final image.
21N.2.SL.TZ0.2a:
Calculate the percentage increase in applications from the first year to the second year.
21N.2.SL.TZ0.2b.ii:
Find an expression for $u_{n}$ .
21N.2.SL.TZ0.2b.iii:
Find the number of student applications the university expects to receive when $n = 11$ . Express your answer to the nearest integer.
21N.2.SL.TZ0.2c:
Write down an expression for $v_{n}$ .
21N.2.SL.TZ0.2d:
Calculate the total amount of acceptance fees paid to the university in the first $10$ years.
21N.2.SL.TZ0.2e:
Find $k$ .

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Topic 1—Number and algebra » SL 1.3—Geometric sequences and series

Topic 1—Number and algebra

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