| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2024 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about y-axis, standard curve |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring: (a) substituting coordinates to find k (simple algebra), (b) setting up and evaluating a standard π∫x² dy integral with polynomial and root terms, (c) stating any reasonable model limitation, and (d) comparing calculated vs actual volume. All techniques are routine for Core Pure AS level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((0.4, 4) \Rightarrow 0.4 = k \times 4^2 + \sqrt{4} \Rightarrow k = \ldots\) | M1 | Substitutes \((0.4, 4)\) into the equation in attempt to find \(k\) |
| \(k = -0.1\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Cylinder volume \(= \pi \times 0.4^2 \times 0.5 = 0.08\pi = \frac{2}{25}\pi\) | B1 | Uses information given to establish correct cylinder volume |
| Volume by curve \(= \pi \int x^2 \, dy\); \(\pi \int (\sqrt{y} + ky^2)^2 \, dy = \pi \int (\sqrt{y} - 0.1y^2)^2 \, dy\) | M1 | Uses model and applies \(\pi \int x^2 \, dy\); \(dy\) not required, \(\pi\) may appear later |
| \(= \{\pi\} \int \left(y + 2ky^{\frac{5}{2}} + k^2 y^4\right) dy = \{\pi\} \int \left(y - 0.2y^{\frac{5}{2}} + 0.01y^4\right) dy\) | A1ft | Correct expansion; follow through on \(k\) value; indices need to be processed |
| \(\Rightarrow \{\pi\} \left[Ay^2 + By^{\frac{7}{2}} + Cy^5\right]\) at least one term with correct power | M1 | Attempts integration with at least one power raised by 1 |
| \(= \{\pi\} \left[\frac{y^2}{2} + \frac{4k}{7}y^{\frac{7}{2}} + \frac{k^2}{5}y^5\right] = \{\pi\}\left[\frac{y^2}{2} - \frac{2}{35}y^{\frac{7}{2}} + \frac{1}{500}y^5\right]\) | A1ft | Correct integration following through on \(x^2\) expression; need not be simplified |
| \(V = \pi\left(8 - \frac{256}{35} + \frac{256}{125}\right) - (0) + \frac{2}{25}\pi\); \(V = \frac{2392}{875}\pi + \frac{2}{25}\pi\) | M1 | Uses correct limits and finds sum of 2 volumes |
| \(V = \frac{2462\pi}{875} \text{ cm}^3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| E.g. The equation of the curve may not be a suitable model; the sides will not be perfectly smooth; there may be flaws/bubbles within the glass; the corner (ABC) may not be a perfect right angle | B1 | States acceptable limitation related to the curve/model; measurements may not be accurate; thickness-related comments score B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Makes appropriate comment consistent with their volume and \(9 \text{ cm}^3\): e.g. volume of \(8.84 \text{ cm}^3\) is only \(0.16 \text{ cm}^3\) less than \(9 \text{ cm}^3\) — good model (between 8.5 and 9.5); between 8 and 10 can be good or bad; less than 8 or more than 10 is a bad model | B1ft | Compares actual volume to part (b) answer and assesses model with reason; if using percentage error must use 9 as true volume |
## Question 8(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(0.4, 4) \Rightarrow 0.4 = k \times 4^2 + \sqrt{4} \Rightarrow k = \ldots$ | M1 | Substitutes $(0.4, 4)$ into the equation in attempt to find $k$ |
| $k = -0.1$ | A1 | |
---
## Question 8(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Cylinder volume $= \pi \times 0.4^2 \times 0.5 = 0.08\pi = \frac{2}{25}\pi$ | B1 | Uses information given to establish correct cylinder volume |
| Volume by curve $= \pi \int x^2 \, dy$; $\pi \int (\sqrt{y} + ky^2)^2 \, dy = \pi \int (\sqrt{y} - 0.1y^2)^2 \, dy$ | M1 | Uses model and applies $\pi \int x^2 \, dy$; $dy$ not required, $\pi$ may appear later |
| $= \{\pi\} \int \left(y + 2ky^{\frac{5}{2}} + k^2 y^4\right) dy = \{\pi\} \int \left(y - 0.2y^{\frac{5}{2}} + 0.01y^4\right) dy$ | A1ft | Correct expansion; follow through on $k$ value; indices need to be processed |
| $\Rightarrow \{\pi\} \left[Ay^2 + By^{\frac{7}{2}} + Cy^5\right]$ at least one term with correct power | M1 | Attempts integration with at least one power raised by 1 |
| $= \{\pi\} \left[\frac{y^2}{2} + \frac{4k}{7}y^{\frac{7}{2}} + \frac{k^2}{5}y^5\right] = \{\pi\}\left[\frac{y^2}{2} - \frac{2}{35}y^{\frac{7}{2}} + \frac{1}{500}y^5\right]$ | A1ft | Correct integration following through on $x^2$ expression; need not be simplified |
| $V = \pi\left(8 - \frac{256}{35} + \frac{256}{125}\right) - (0) + \frac{2}{25}\pi$; $V = \frac{2392}{875}\pi + \frac{2}{25}\pi$ | M1 | Uses correct limits and finds sum of 2 volumes |
| $V = \frac{2462\pi}{875} \text{ cm}^3$ | A1 | |
---
## Question 8(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| E.g. The equation of the curve may not be a suitable model; the sides will not be perfectly smooth; there may be flaws/bubbles within the glass; the corner (ABC) may not be a perfect right angle | B1 | States acceptable limitation related to the curve/model; measurements may not be accurate; thickness-related comments score B0 |
---
## Question 8(d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Makes appropriate comment consistent with their volume and $9 \text{ cm}^3$: e.g. volume of $8.84 \text{ cm}^3$ is only $0.16 \text{ cm}^3$ less than $9 \text{ cm}^3$ — good model (between 8.5 and 9.5); between 8 and 10 can be good or bad; less than 8 or more than 10 is a bad model | B1ft | Compares actual volume to part (b) answer and assesses model with reason; if using percentage error must use 9 as true volume |
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\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{18386c8a-6d2d-4c63-972a-bb9f78786b36-30_634_264_319_374}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{18386c8a-6d2d-4c63-972a-bb9f78786b36-30_762_609_260_1080}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 1 shows the central vertical cross-section, $O A B C D E O$, of the design for a solid glass ornament.
Figure 2 shows the finite region, $R$, which is bounded by the $y$-axis, the horizontal line $C B$, the vertical line $B A$, and the curve $A O$.
The ornament is formed by rotating the region $R$ through $360 ^ { \circ }$ about the $y$-axis.\\
The curve $A O$ is modelled by the equation
$$x = k y ^ { 2 } + \sqrt { y } \quad 0 \leqslant y \leqslant 4$$
where $k$ is a constant.\\
The point $A$ has coordinates ( $0.4,4$ ) and the point $B$ has coordinates ( $0.4,4.5$ )\\
The units are centimetres.
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $k$ according to this model.
\item Use algebraic integration to determine the exact volume of glass that would be required to make the ornament, according to the model.
\item State a limitation of the model.
When the ornament was manufactured, $9 \mathrm {~cm} ^ { 3 }$ of glass was required.
\item Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP AS 2024 Q8 [11]}}