Mathematics · Oct / Nov 2021 · Paper 31 · Question 3
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As printed in the paper
The question as text
3. G O k t Figure 2 The total mass of gold, G tonnes, extracted from a mine is modelled by the equation G = 40 – 30e1–0.05t t k G 0 where t is the number of years after 1st January 1800. Figure 2 shows a sketch of G against t. Use the equation of the model to answer parts (a), (b) and (c).
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3(a)[3]
40
1 0.05
−
t
0
=
40
−
30e
⇒−
1 0.05
t
=
ln 30
(
) awrt 14.2
⇒
k
=
(ii)
March 1814
1 0.05 70
−
×
Attempts
G
=
40
−
30e
=
37.5
tonnes
40 (tonnes)
Mark (a)(i) and (ii) together
(a) (i)
G =
0
M1
Sets
and proceeds to a linear equation in t or k using a correct method by
taking logs of both sides, but condone slips in the rest of their rearrangement.
4
(
) awrt 14.2
A1
k =
or 14.25 but allow the exact value
20 1 ln 3
−
or other
4
ln
1
−
4
3
3
equivalent exact expressions such as
or
20 1 ln 3
−
or
20ln
e
. Isw
−
0.05
4
after a correct answer.
(a)(ii)
A1
March 1814 but allow April 1814 following ''correct'' k.
Condone “the third month of 1814” or “the fourth month of 1814” following
''correct'' k
Withhold this mark if they have two answers of which one is incorrect.
(b)
M1
Attempts to find G with t =70 (condone t =69) which may be implied by awrt 37.5
1 0.05 70
−
×
The numbers embedded in the equation is sufficient eg
G
=
40
−
30e
awrt 37.5tonnes (requires units)
A1
(c)
B1
40 (tonnes). Condone
G <
40
or
G „
40
. Do not allow
G
>
40
(
)
3(b)[2]
1 0.05 70
−
×
Attempts
G
=
40
−
30e
=
37.5
tonnes
3(c)[1]
40 (tonnes)
Mark (a)(i) and (ii) together
(a) (i)
G =
0
M1
Sets
and proceeds to a linear equation in t or k using a correct method by
taking logs of both sides, but condone slips in the rest of their rearrangement.
4
(
) awrt 14.2
A1
k =
or 14.25 but allow the exact value
20 1 ln 3
−
or other
4
ln
1
−
4
3
3
equivalent exact expressions such as
or
20 1 ln 3
−
or
20ln
e
. Isw
−
0.05
4
after a correct answer.
(a)(ii)
A1
March 1814 but allow April 1814 following ''correct'' k.
Condone “the third month of 1814” or “the fourth month of 1814” following
''correct'' k
Withhold this mark if they have two answers of which one is incorrect.
(b)
M1
Attempts to find G with t =70 (condone t =69) which may be implied by awrt 37.5
1 0.05 70
−
×
The numbers embedded in the equation is sufficient eg
G
=
40
−
30e
awrt 37.5tonnes (requires units)
A1
(c)
B1
40 (tonnes). Condone
G <
40
or
G „
40
. Do not allow
G
>
40
(
)
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