| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Solve exponential equation using logarithms |
| Difficulty | Moderate -0.3 This is a multi-part C2 question covering standard exponential function techniques: finding intercepts (trivial substitution), verifying a solution, applying trapezium rule (routine numerical method), solving an exponential equation with logarithms (standard procedure), and describing transformations. While comprehensive, each individual part requires only direct application of learned techniques with no novel problem-solving or insight required. Slightly easier than average due to the straightforward nature of all parts. |
| 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 = b1.09f Trapezium rule: numerical integration |
6 The diagram shows a sketch of the curve with equation $y = 27 - 3 ^ { x }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f066f68a-e739-4da3-8ec1-e221461146b0-4_933_1074_376_484}
The curve $y = 27 - 3 ^ { x }$ intersects the $y$-axis at the point $A$ and the $x$-axis at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the $y$-coordinate of point $A$.
\item Verify that the $x$-coordinate of point $B$ is 3 .
\end{enumerate}\item The region, $R$, bounded by the curve $y = 27 - 3 ^ { x }$ and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of $R$.
\item \begin{enumerate}[label=(\roman*)]
\item Use logarithms to solve the equation $3 ^ { x } = 13$, giving your answer to four decimal places.
\item The line $y = k$ intersects the curve $y = 27 - 3 ^ { x }$ at the point where $3 ^ { x } = 13$. Find the value of $k$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Describe the single geometrical transformation by which the curve with equation $y = - 3 ^ { x }$ can be obtained from the curve $y = 27 - 3 ^ { x }$.
\item Sketch the curve $y = - 3 ^ { x }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C2 2006 Q6 [13]}}