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Date November 2017 Marks available 3 Reference code 17N.2.SL.TZ0.T_2
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Calculate Question number T_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;

[1]
a.i.

Write down the distance Rosa runs in the n th training session.

[2]
a.ii.

Find the value of k .

[2]
b.

Calculate the total distance, in kilometres, Rosa runs in the first 50 training sessions.

[4]
c.

Find the distance Carlos runs in the fifth month of training.

[3]
d.

Calculate the total distance Carlos runs in the first year.

[3]
e.

Markscheme

3800 m     (A1)

[1 mark]

a.i.

3000 + ( n 1 ) 400  m OR 2600 + 400 n  m     (M1)(A1)

 

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

 

[2 marks]

a.ii.

3000 + ( k 1 ) 400 > 42195     (M1)

 

Notes:     Award (M1) for their correct inequality. Accept 3 + ( k 1 ) 0.4 > 42.195 .

Accept = OR . Award (M0) for 3000 + ( k 1 ) 400 > 42.195 .

 

( k = )   99     (A1)(ft)(G2)

 

Note:     Follow through from part (a)(ii), but only if k is a positive integer.

 

[2 marks]

b.

50 2 ( 2 × 3000 + ( 50 1 ) ( 400 ) )     (M1)(A1)(ft)

 

Note:     Award (M1) for substitution into sum of an arithmetic series formula, (A1)(ft) for correct substitution.

 

640 000  m     (A1)

 

Note:     Award (A1) for their 640 000 seen.

 

= 640  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

50 2 ( 2 × 3 + ( 50 1 ) ( 0.4 ) )     (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  km     (A1)(G3)

[4 marks]

c.

7500 × 1.2 5 1     (M1)(A1)

 

Note:     Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.

 

= 15 600  m  ( 15 552  m )     (A1)(G3)

OR

7.5 × 1.2 5 1     (M1)(A1)

 

Note:     Award (M1) for substitution into geometric series formula, (A1) for correct substitutions.

 

= 15.6  km     (A1)(G3)

[3 marks]

d.

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  m  ( 296 853  m ,   297  km )     (A1)(G2)

[3 marks]

e.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.
[N/A]
d.
[N/A]
e.

Syllabus sections

Topic 1—Number and algebra » SL 1.2—Arithmetic sequences and series
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Topic 1—Number and algebra » SL 1.3—Geometric sequences and series
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