Mathematics · Oct / Nov 2025 · Paper 31 · Question 5
YMA01
As printed in the paper
The question as text
5. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. The number of squirrels in a forest is being studied. The number of squirrels, N, in the forest, t years after the start of the study, is modelled by the equation 0 1 . t 4000 e N t 0 0 2 . t 19 e Use the equation of the model to answer parts (a), (b), (c) and (d).
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Show the answer and mark scheme
5(a)[1]
=
4000e
0
200
+
19
e
0
(
)
5(b)[2]
19
+
e
0.2
t
×
400e
0.1
t
−
4000e
0.1
t
×
0.2e
0.2
t
d
N
=
(
)
2
d
t
+
0.2
t
19
e
(
)
5(c)[2]
19
+
e
0.2
t
×
400e
0.1
t
−
4000e
0.1
t
×
0.2e
0.2
t
=
0
e
0.2
T
=
19
5(d)[3]
ln19
=
=
=
e
0.2
T
19
T
( 14.7...)
0.2
4000e
0.1"14.7"
N =
=
squirrels
458.8...
459
+
19
e
0.2"14.7"
(a)
B1:
200
(b)
M1:
Attempts to differentiate via the quotient rule (or the product rule). Look
for reaching
or
where A, B > 0
Condone missing brackets for the M mark.
A1:
Correct unsimplified derivative and isw. Any missing brackets must have
been recovered.
(c)
M1:
Sets their numerator of the form
(or unsimplified equivalent)
equal to zero and rearranges to
e
0.2
T
=
A
T =
A1:
and isw. Accept with t instead of T. Allow from answers to (b) with
e
0.2
19
an incorrect denominator but correct numerator or negative of the
numerator.
(d)
M1:
Takes lns of both sides and proceeds to find a value for T (accept awrt 14.7
if method is not shown). There must be evidence of using logarithms.
Alternatively score for stating or implying
dM1: Substitutes their positive value for T or
into the original equation to
find a value for N. It is dependent on the previous method mark.
T =
A1:
459 (accept 458) provided the Ms have been scored. The
e
0.2
have been genuinely found, not fluked from an incorrect numerator.
However, as long as the numerator was correct in (b), this mark may be
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