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Date	May 2019	Marks available	1	Reference code	19M.2.SL.TZ2.T_4
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Write down	Question number	T_4	Adapted from	N/A

Question

On her first day in a hospital, Kiri receives $u_{1}$ ${u_1}$ milligrams (mg) of a therapeutic drug. The amount of the drug Kiri receives increases by the same amount, $d$ $d$ , each day. On the seventh day, she receives 21 mg of the drug, and on the eleventh day she receives 29 mg.

Kiri receives the drug for 30 days.

Ted is also in a hospital and on his first day he receives a 20 mg antibiotic injection. The amount of the antibiotic Ted receives decreases by 50 % each day. On the second day, Ted receives a 10 mg antibiotic injection, on the third day he receives 5 mg, and so on.

Write down an equation, in terms of $u_{1}$ ${u_1}$ and $d$ $d$ , for the amount of the drug that she receives on the seventh day.

[1]

a.i.

Write down an equation, in terms of $u_{1}$ ${u_1}$ and $d$ $d$ , for the amount of the drug that she receives on the eleventh day.

[1]

a.ii.

Write down the value of $d$ $d$ and the value of $u_{1}$ ${u_1}$ .

[2]

Calculate the total amount of the drug, in mg, that she receives.

[3]

Find the amount of antibiotic, in mg, that Ted receives on the fifth day.

[3]

d.i.

The daily amount of antibiotic Ted receives will first be less than 0.06 mg on the $k$ $k$ th day. Find the value of $k$ $k$ .

[3]

d.ii.

Hence find the total amount of antibiotic, in mg, that Ted receives during the first $k$ $k$ days.

[3]

d.iii.

Markscheme

(amount taken in the 7th day): $u_{1} + 6 d = 21$ ${u_1} + 6d = 21$ (A1)

Note: Accept $u_{1} + (7 - 1) d = 21$ ${u_1} + \left( {7 - 1} \right)d = 21$ . The equations do not need to be simplified. They should be given in terms of $u_{1}$ ${u_1}$ and $d$ $d$ for the marks to be awarded.

[1 mark]

a.i.

(amount taken in the 11th day): $u_{1} + 10 d = 29$ ${u_1} + 10d = 29$ (A1)

Note: Accept $u_{1} + (11 - 1) d = 29$ ${u_1} + \left( {11 - 1} \right)d = 29$ . The equations do not need to be simplified. They should be given in terms of $u_{1}$ ${u_1}$ and $d$ $d$ for the marks to be awarded.

[1 mark]

a.ii.

( $u_{1}$ ${u_1}$ =) 9 (A1)(ft)

( $d$ $d$ =) 2 (A1)(ft)

Note: Follow through from part (a), but only if values are positive and $u_{1}$ ${u_1}$ < 21.

[2 marks]

$(S_{30} =) \frac{30}{2} (2 \times 9 + (30 - 1) \times 2)$ $\left( {{S_{30}} = } \right)\frac{{30}}{2}\left( {2 \times 9 + \left( {30 - 1} \right) \times 2} \right)$ (M1)(A1)(ft)

Note: Award (M1) for substitution in the sum of an arithmetic sequence formula; (A1)(ft) for their correct substitution.

1140 (mg) (A1)(ft)(G3)

Note: Follow through from their $u_{1}$ ${u_1}$ and $d$ $d$ from part (b).

[3 marks]

20 × (0.5)⁴ (M1)(A1)

Note: Award (M1) for substitution into the geometric sequence formula, (A1) for correct substitution.

1.25 (mg) (A1)(G3)

[3 marks]

d.i.

$20 \times {(0.5)}^{k - 1} < 0.06$ $20 \times {\left( {0.5} \right)^{k - 1}} < 0.06$ (M1)(M1)

Note: Award (M1) for correct substitution into the geometric sequence formula; (M1) for comparing their expression to 0.06. Accept an equation instead of inequality.

( $k$ $k$ =) 10 (10th day) (A1)(ft)(G3)

Note: Follow through from part (d)(i), if 0 < $r$ $r$ < 1. Follow through answers must be rounded up for final mark.

[3 marks]

d.ii.

$\frac{20 (1 - {0.5}^{10})}{1 - 0.5}$ $\frac{{20\left( {1 - {{0.5}^{10}}} \right)}}{{1 - 0.5}}$ (M1)(A1)(ft)

Note: Award (M1) for substitution into sum of a geometric sequence formula, (A1)(ft) for correct substitution.
Follow through from their $u_{1}$ ${u_1}$ and $r$ $r$ in part (d)(i), if 0 < $r$ $r$ < 1. Follow through from their $k$ $k$ in part (d)(ii) but only if $k$ $k$ is a positive integer.