| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sequences and Series |
| Type | Modelling with Recurrence Relations |
| Difficulty | Standard +0.3 This is a straightforward application of geometric sequences to a real-world context. Part (a) involves basic recurrence relation setup and solving $(1-r)^{12} = 0.00355$ using logarithms. Parts (b) and (c) require simple arithmetic with the same recurrence formula and qualitative reasoning about model assumptions. All techniques are standard for Further Maths students, with no novel problem-solving required beyond following the given structure. |
| Spec | 8.01h Modelling with recurrence: birth/death rates, INT function |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (i) |
| Answer | Marks |
|---|---|
| for twelve weeks. | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| (ii) | T = a12T |
| Answer | Marks |
|---|---|
| 1 – r = 120.00355= 0.62496… ⇒ r = 0.375 to 3s.f. | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.1b |
| 1.1 | a = r or 1 – r |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | (i) | After 16 weeks, the number of frogs is |
| Answer | Marks |
|---|---|
| 54.154... | B1 |
| Answer | Marks |
|---|---|
| [3] | 3.5c |
| Answer | Marks |
|---|---|
| 1.1 | Allow use of ‘T ’. |
| Answer | Marks |
|---|---|
| (ii) | E.g. The same weekly death-rate factor continues |
| Answer | Marks |
|---|---|
| exactly the same time. | B1 |
| [1] | 3.3 |
| (c) | E.g. |
| Answer | Marks | Guidance |
|---|---|---|
| smaller populations. | B1 | |
| [1] | 3.5a | No greater detail of analysis is required beyond “they would |
Question 7:
7 | (a) | (i) | E.g.
T = 100000 is the initial population as given
0
T = (1 – r)T because a death-rate of r means that 1
k + 1 k
– r of the population is left after each week.
0≤k ≤12 because the model given is only valid
for twelve weeks. | B1
B1
B1
[3] | 1.1
3.3
2.1
(ii) | T = a12T
12 0
1 – r = 120.00355= 0.62496… ⇒ r = 0.375 to 3s.f. | M1
A1
[2] | 3.1b
1.1 | a = r or 1 – r
AG
(b) | (i) | After 16 weeks, the number of frogs is
0.62496…16 × 100000 = 54.154 …
So 54.154 … × p ≥ 30
30
⇒ p≥ =0.5539...=0.554to 3 sf
54.154... | B1
M1
A1
[3] | 3.5c
3.1b
1.1 | Allow use of ‘T ’.
16
Or, starting again 0.62496…4 × 355
For ‘their population’ × p ≥ 30
(ii) | E.g. The same weekly death-rate factor continues
unchanged.
The females will all lay eggs.
Tadpoles instantly change to frogs and lay eggs at
exactly the same time. | B1
[1] | 3.3
(c) | E.g.
30 surviving females would produce 75000 eggs,
so the population is smaller than it was to start
with, so each ‘round’ will result in smaller and
smaller populations. | B1
[1] | 3.5a | No greater detail of analysis is required beyond “they would
appear to be dying out so the figure of 30 in the model is not a
good one”
PMT
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recorded or monitored7 In a conservation project, a batch of 100000 tadpoles which have just hatched from eggs is introduced into an environment which has no frog population. Previous research suggests that for every 1 million tadpoles hatched only 3550 will live to maturity at 12 weeks, when they become adult frogs.
It is assumed that the steady decline in the population of tadpoles, from all causes, can be explained by a weekly death-rate factor, $r$, which is constant across each week of this twelve-week period.
Let $\mathrm { T } _ { \mathrm { k } }$ denote the total number of tadpoles alive at the end of $k$ weeks after the start of this project.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Explain why a recurrence system for $\mathrm { T } _ { \mathrm { k } }$ is given by $T _ { 0 } = 100000$ and $\mathrm { T } _ { \mathrm { k } + 1 } = ( 1 - \mathrm { r } ) \mathrm { T } _ { \mathrm { k } }$ for $0 \leqslant k \leqslant 12$.
\item Show that $r = 0.375$, correct to 3 significant figures.
The proportion of females within each batch of tadpoles is $p$, where $0 < p < 1$. In a simple model of the frog population the following assumptions are made.
\begin{itemize}
\end{enumerate}\item The death rate factor for adult frogs is also $r$ and is the same for males and females.
\item The frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
\item \begin{enumerate}[label=(\roman*)]
\item Find the smallest value of $p$ for which the frog population will survive according to the model.
\item Write down one assumption that has been made in order to obtain this result.
\end{itemize}
Each surviving female will then lay a batch of eggs from which 2500 tadpoles are hatched.
\end{enumerate}\item By considering the total number of tadpoles hatched, give one criticism of the assumption that the frog population will survive provided there are at least thirty female frogs alive sixteen weeks after the start of this project.
\section*{END OF QUESTION PAPER}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2020 Q7 [10]}}