| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Topic | Sequences and series, recurrence and convergence |
| Type | Applied recurrence modeling |
| Difficulty | Challenging +1.2 This question involves iterating recurrence relations (straightforward calculation), finding equilibrium points by solving P = f(P) (standard technique), and analyzing stability using derivatives (A-level Further Maths content but routine application). The models are given explicitly, requiring no modeling insight, and the analysis follows textbook methods with no novel problem-solving required. |
| Spec | 1.04e Sequences: nth term and recurrence relations |
(i) Model 1: Attempt iteration
$P_3 = 687$
M1 At least twice
A1 Allow decimal values, or 686
Model 2: Attempt iteration
$P_3 = 927$
M1 At least twice
A1 [4] Allow decimal values, or 926
(ii) Model 1
B1 Identify Model 1, with minimal explanation e.g. decreasing rate of increase
converges to 693
B1 [2] Identify that it converges to 693 oe (could justify that $P_t \approx P_{t-1}$)
(iii) appears to settle down to periodic (values 926, 561, 980 and 429)
B1 [1] State as periodic oe – need to see 4 values, or refer to period of 4 years9 A new lake is stocked with fish. Let $P _ { t }$ be the population of fish in the lake after $t$ years. Two models using recurrence relations are proposed for $P _ { t }$, with $P _ { 0 } = 550$.
$$\begin{aligned}
& \text { Model } 1 : P _ { t } = 2 P _ { t - 1 } \mathrm { e } ^ { - 0.001 P _ { t - 1 } } \\
& \text { Model } 2 : P _ { t } = \frac { 1 } { 2 } P _ { t - 1 } \left( 7 - \frac { 1 } { 160 } P _ { t - 1 } \right)
\end{aligned}$$
(i) Evaluate the population predicted by each model when $t = 3$.\\
(ii) Identify, with evidence, which one of the models predicts a stable population in the long term.\\
(iii) Describe the long term behaviour of the population for the other model.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2014 Q9 [7]}}