| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Finding x from given y value |
| Difficulty | Moderate -0.8 This is a straightforward exponential growth question requiring standard substitution and logarithm manipulation. Part (a) is simple interpretation, part (b) involves routine algebraic steps to find k using two given points, and part (c) is direct substitution. All techniques are standard AS-level material with no novel problem-solving required, making it easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(A\) represents the initial population of the island. | B1 | AO3 |
| (b) \(100 = Ae^{2k}\) | B1 | AO1 (both values) |
| \(160 = Ae^{12k}\) | B1 | AO1 |
| Dividing to eliminate \(A\) | M1 | AO1 |
| \(1.6 = e^{10k}\) | A1 | AO1 |
| \(k = \frac{1}{10} \ln 1.6 = 0.047\) | A1 | AO1 (convincing) |
| (c) \(A = 91(0283)\) | B1 | AO1 (o.e.) |
| When \(t = 20\), \(N = 91(0283) \times e^{0.94}\) | M1 | AO1 (f.t. candidate's derived value for \(A\)) |
| \(N = 233\) | A1 | AO3 (c.a.o.) |
**(a)** $A$ represents the initial population of the island. | B1 | AO3
**(b)** $100 = Ae^{2k}$ | B1 | AO1 (both values)
$160 = Ae^{12k}$ | B1 | AO1
Dividing to eliminate $A$ | M1 | AO1
$1.6 = e^{10k}$ | A1 | AO1
$k = \frac{1}{10} \ln 1.6 = 0.047$ | A1 | AO1 (convincing)
**(c)** $A = 91(0283)$ | B1 | AO1 (o.e.)
When $t = 20$, $N = 91(0283) \times e^{0.94}$ | M1 | AO1 (f.t. candidate's derived value for $A$)
$N = 233$ | A1 | AO3 (c.a.o.)
**Total: [8]**
---The size $N$ of the population of a small island at time $t$ years may be modelled by $N = Ae^{kt}$, where $A$ and $k$ are constants. It is known that $N = 100$ when $t = 2$ and that $N = 160$ when $t = 12$.
\begin{enumerate}[label=(\alph*)]
\item Interpret the constant $A$ in the context of the question. [1]
\item Show that $k = 0.047$, correct to three decimal places. [4]
\item Find the size of the population when $t = 20$. [3]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 Q15 [8]}}