| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Simple exponential equation solving |
| Difficulty | Moderate -0.8 This is a straightforward C2 question testing basic exponential function properties: sketching y=a^x (routine), solving a^x=b using logarithms (standard technique), and recognizing that reflection in the y-axis transforms f(x) to f(-x). All parts are textbook exercises requiring only direct application of standard methods with no problem-solving or insight needed. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Graph shown | B1 | Correct graph, must clearly go below the intersection pt and an indication of correct behaviour of curve for large positive and large negative values of \(x\). Ignore any scaling on axes. |
| B1 | Only one y-intercept, marked/stated as 1 or as coords (0, 1) with graph having no other intercepts on either axes. | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(9^x = 15 \Rightarrow x \log 9 = \log 15\) | M1 | OE eg \(x = \log_9 15\) |
| \((x =) 1.23(2486...) = 1.23\) to 3sf | A1 | Condone > 3sf. Must see evidence of logs used so NMS scores 0/2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\{f(x) \} = 9^{-x}\) | B1 | OE |
| Total | 5 |
**4(a)**
Graph shown | B1 | Correct graph, must clearly go below the intersection pt and an indication of correct behaviour of curve for large positive and large negative values of $x$. Ignore any scaling on axes.
| B1 | Only one y-intercept, marked/stated as 1 or as coords (0, 1) with graph having no other intercepts on either axes. | 2
**4(b)**
$9^x = 15 \Rightarrow x \log 9 = \log 15$ | M1 | OE eg $x = \log_9 15$
$(x =) 1.23(2486...) = 1.23$ to 3sf | A1 | Condone > 3sf. Must see evidence of logs used so NMS scores 0/2 | 2
**4(c)**
$\{f(x) \} = 9^{-x}$ | B1 | OE | 1
Total | | 5
---4
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = 9 ^ { x }$, indicating the value of the intercept on the $y$-axis.\\
(2 marks)
\item Use logarithms to solve the equation $9 ^ { x } = 15$, giving your value of $x$ to three significant figures.
\item The curve $y = 9 ^ { x }$ is reflected in the $y$-axis to give the curve with equation $y = \mathrm { f } ( x )$. Write down an expression for $\mathrm { f } ( x )$.\\
(l mark)
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2013 Q4 [5]}}