| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | February |
| Marks | 9 |
| Topic | Volumes of Revolution |
| Type | Applied context: real-world solid |
| Difficulty | Standard +0.8 This is a solid of revolution problem requiring students to (a) find constants from boundary conditions and (b) integrate π∫y²dx to find volume. While the setup requires careful interpretation of the geometry and the integration involves expanding (ax²+b)² then integrating term-by-term, these are standard A-level techniques. The 7-mark allocation for part (b) indicates extended working but not exceptional difficulty. Slightly above average due to the geometric modeling aspect and algebraic manipulation required. |
| Spec | 1.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_9}
A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section $ABCDEFGHA$ of the bottle with measurements taken by the student.
The horizontal cross-section between $CF$ and $DE$ is a circle of diameter 8 cm and the horizontal cross-section between $BG$ and $AH$ is a circle of diameter 2 cm.
The student thinks that the curve $GF$ could be modelled as a curve with equation
$$y = ax^2 + b \qquad 1 \leq x \leq 4$$
where $a$ and $b$ are constants and $O$ is the fixed origin, as shown in Figure 2.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$ and the value of $b$ according to the model.
[2]
\item Use the model to find the volume of water that the bottle can contain.
[7]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q9 [9]}}