Mathematics · Oct / Nov 2022 · Paper 31 · Question 5
YMA01
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The question as text
5. P O t Figure 2 The profit made by a company, £ P million, t years after the company started trading, is modelled by the equation t t 4 − 1 3 + 1 P = + ln ( ) 2 10 4 2 t + 1 The graph of P against t is shown in Figure 2. According to the model,
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5(a)[2]
P
ln
(
0.828...)
=
+
= −
2
10
4
3
P
= −
0.828...
(which is negative so a loss of £0.828 million,) so approximately £830
000 loss.
P(6) = −0.08799… and P(7) = 0.1975…
5(b)[2]
There is a sign change and hence as P is continuous on [6,7], so the root for t lies in
[6,7].
5(c)[2]
4
t
−
1
3
t
+
1
P
=
0
= −
ln
t
=
...
10
4
(
2
t
1
)
2
+
3
t
+
1
10
ln
+
1
−
(
)
2
−
1
4
2
t
+
1
1
30
t
+
1
t
=
=
+
ln
(
)
4
4
1
6
2
t
+
1
2
(
2
t
+
1
)
2
1
15 ln
=
t
+
*
4
8
t
+
1
1
15
13
2
5(d)[3]
(
)
t
=
+
ln
=
...
=
6.219978...
2
4
8
7
t =
awrt6.220
2
6.314
t =
6
"6.3..." 12
...
×
=
months, or repeated iteration to root 6.31487 gives 75.7785… or allow
5(e)[2]
"0.314..." 12
×
=
"3.768..."
So it will take 76 months. (Accept 75 or awrt 76 months)
Notes:
(a)
M1: Attempts to substitute t = 1 into the given formula. Allow if there is a slip but an attempt at substitution
is seen, or allow for sight of awrt 0.828...
−
A1*: Correct value for P(1) seen to at least 2.s.f. if substitution has been shown, or at least 3 s.f. if no
substitution was shown, followed by suitable conclusion that it is a loss of approximately £830 000, though
accept awrt £830 000. Must mention “loss” and include units (£ or pounds). Negative value given is A0.
(b)
M1: Attempts both P(6) and P(7) with at least one correct to 1 s.f. rounded or truncated. A tighter interval
could be used but must contain the root (6.31487)
A1: Both correct to1 s.f. rounded or truncated with suitable conclusion made. Must mention sign change and
continuity as well as conclusion about root in the interval.
(c)
M1: Attempts to isolate αt from the 4
t − after setting equal to zero.
1
10
A1*: Correct work to reach the given answer with no incorrect work seen and at least one intermediate step
with either
α=
t
...
reached or with the power law applied on the ln term (need not see the power explicitly
used, but must have been applied correctly).
(d)
M1: Attempts to use the formula with t1 =6. Accept with 6 embedded in formula followed by a value, or awrt
6.2 (or even 6.3) as implying the attempt.
A1: awrt 6.220 Accept 6.22.
A1: Correct and given to 3 d.p.
(e)
M1: Multiplies their final root from (d) by 12, or uses repeated iteration to narrow further then multiplies by
12. Allow the method if they multiply the fractional part only by 12 and get, e.g., 6 years 3 months.
A1: Accept 75 or 76 months, or anything that rounds to 76 months.
d cos
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