Mathematics · May / June 2024 · Paper 21 · Question 10
YMA01
As printed in the paper
The question as text
10. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The number of dormice and the number of voles on an island are being monitored. Initially there are 2000 dormice on the island. A model predicts that the number of dormice will increase by 3% each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
10(a)[2]
2000 1.03
×
=
awrt 2390
3470
4
7
3
(
)
3690
=
,3470
=
=
=
0.94...
10(b)[3]
ab
ab
b
3690
3 3470
3690
Hence
0.9797...
a =
dM1
b =
=
3690
4
0.9797..
t
N =
4000 0.98
×
T
1.03
4000
T
T
Sets
2000 1.03
×
=
4000 0.98
×
=
10(c)[3]
0.98
2000
1.03
log
log 2
...
T
=
T
=
0.9
8
T
=
13.9
(a)
5
6
M1:
Attempts
2000 1.03
×
or
2000 1.03
×
. May use successive iterations (finding each
year) but end up at one these in effect.
A1:
Awrt 2390
(b)
M1:
Makes progress in using the model and attempting to find one of the variables a or b3
(or b). Allow if r is used instead of b for the M’s.
3470
3
6
3
Allow from the indexing error
3690
=
,3470
=
=
ab
ab
b
3690
dM1: Full method to find values for a and b.
3
6
Condone this being scored from
3690
=
,3470
=
ab
ab
4000 0.98
t
A1:
N =
×
Accept awrt 4000 to 2sf,, awrt 0.98. Must be seen as an equation not
just values of a and b but allow if the equation is seen (c) (including the N = ).
Condone n or T used the variables.
(c)
M1:
Makes progress towards solving the problem by forming a single term with single
power T. Follow through on their equations provided that they are of the correct form.
T
T
Alternatively takes logs of both sides 2000 1.03
×
=
4000 0.98
×
and uses addition
law and power law of logs, e.g ln 2000
ln1.03
ln 4000
ln0.98
+
T
=
+
T
T
1.03
4000 1.03
T
−
1
T
Award from an attempt on
2000 1.03
×
=
4000 0.98
×
=
0.98
similar index error.
dM1: Full and complete attempt to find T. Dependent upon previous method but may be
implied for the correct value for their
a
=
b
T
A1:
cao
T =
13.9
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