| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with exponential functions |
| Difficulty | Standard +0.3 This is a standard C3 volumes of revolution question with exponential functions. Part (a) requires simple substitution, (b) is algebraic expansion, (c) involves routine integration of exponential terms using the formula V = π∫y²dx, and (d) is a straightforward modulus graph sketch. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06d Natural logarithm: ln(x) function and properties4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Strip width \(h = 0.8\); midpoints at \(x = 0.4, 1.2, 2.0, 2.8, 3.6\) | M1 | |
| \(y\) values: \(\sqrt{27.064},\ \sqrt{28.728},\ \sqrt{35},\ \sqrt{48.952},\ \sqrt{73.656}\) | M1A1 | |
| Area \(\approx 0.8(5.202 + 5.360 + 5.916 + 6.997 + 8.582) \approx 25.6\) | A1 | 3 s.f. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The curve is convex (concave up), so mid-ordinate rule gives an underestimate | B1 | |
| Diagram showing rectangles below the curve | B1 |
## Question 5:
**(a)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Strip width $h = 0.8$; midpoints at $x = 0.4, 1.2, 2.0, 2.8, 3.6$ | M1 | |
| $y$ values: $\sqrt{27.064},\ \sqrt{28.728},\ \sqrt{35},\ \sqrt{48.952},\ \sqrt{73.656}$ | M1A1 | |
| Area $\approx 0.8(5.202 + 5.360 + 5.916 + 6.997 + 8.582) \approx 25.6$ | A1 | 3 s.f. |
**(b)**
| Answer/Working | Mark | Guidance |
|---|---|---|
| The curve is convex (concave up), so mid-ordinate rule gives an underestimate | B1 | |
| Diagram showing rectangles below the curve | B1 | |
---5 The diagram shows part of the graph of $y = \mathrm { e } ^ { 2 x } - 9$. The graph cuts the coordinate axes at ( $0 , a$ ) and ( $b , 0$ ).\\
\includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}
\begin{enumerate}[label=(\alph*)]
\item State the value of $a$, and show that $b = \ln 3$.
\item Show that $y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81$.
\item The shaded region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis. Find the volume of the solid formed, giving your answer in the form $\pi ( p \ln 3 + q )$, where $p$ and $q$ are integers.
\item Sketch the curve with equation $y = \left| \mathrm { e } ^ { 2 x } - 9 \right|$ for $x \geqslant 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 Q5}}