| Exam Board | Edexcel |
|---|---|
| Module | CP2 (Core Pure 2) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about y-axis, standard curve |
| Difficulty | Standard +0.8 This is a multi-step volumes of revolution problem requiring: (a) finding parameters in a rational function using two points, and (b) calculating volume by rotation about the y-axis involving both a rational curve and a circular arc. The integration requires expressing x² in terms of y for both curves, finding the circle equation, and combining two integrals with careful limits. While the techniques are standard Core Pure 2 content, the multi-part nature, parameter finding, and need to handle two different curves with correct setup makes this moderately challenging—above average but not requiring exceptional insight. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^24.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(1 = \frac{a}{0.5+b}\), \(0.5 = \frac{a}{2.5+b} \Rightarrow a = ..., b = ...\) | M1 | Uses the given coordinates correctly in the equation modelling the curve to obtain at least one correct equation and attempts to find the values of \(a\) and \(b\) |
| \(a = 2, b = 1.5\) | A1 | Correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(V_1 = \pi\int x^2\,\mathrm{d}y = \pi\int\left(\frac{"2"}{y+"1.5"}\right)^2\mathrm{d}y\) | B1ft | Uses the model to obtain \(\pi\int\left(\frac{\text{their }a}{y+\text{their }b}\right)^2\mathrm{d}y\). Note \(\pi\) can be recovered if it appears later. |
| \(\pi\int_{0.5}^{2.5}\left(\frac{"2"}{y+"1.5"}\right)^2\mathrm{d}y\) | M1 | Chooses limits appropriate to the model i.e. 0.5 and 2.5 |
| \(= \{4\pi\}\Big[-(y+1.5)^{-1}\Big]_{0.5}^{2.5} (=\pi)\) | M1 | Integrates to obtain an expression of the form \(k(y+"1.5")^{-1}\) |
| \(x^2 + (y-3)^2 = 0.5\) | B1 | Deduces the correct equation for the circle |
| \(V_2 = \pi\int x^2\,\mathrm{d}y = \pi\int\left(0.5-(y-3)^2\right)\mathrm{d}y\) or \(\pi\int\left(-y^2+6y-8.5\right)\mathrm{d}y\) | M1 | Uses their circle equation and \(\pi\int x^2\,\mathrm{d}y\) to attempt the top volume. Note \(\pi\) can be recovered if it appears later. |
| \(= \pi\int_{2.5}^{3+\frac{1}{\sqrt{2}}}\left(0.5-(y-3)^2\right)\mathrm{d}y\) or \(\pi\int_{2.5}^{3+\frac{1}{\sqrt{2}}}\left(-y^2+6y-8.5\right)\mathrm{d}y\) | M1 | Identifies limits appropriate to the model i.e. 2.5 and \(3 +\) their radius |
| \(= \{\pi\}\left[0.5y - \frac{1}{3}(y-3)^3\right]_{2.5}^{3+\frac{1}{\sqrt{2}}}\) or \(= \{\pi\}\left[-\frac{1}{3}y^3+3y^2-8.5y\right]_{2.5}^{3+\frac{1}{\sqrt{2}}}\) | A1 | Correct integration |
| \(V_1 + V_2 + \text{cylinder} = \pi + \pi\left(\frac{5}{24}+\frac{\sqrt{2}}{6}\right)+\frac{1}{2}\pi\) | dM1 | Uses the model to find the volume of the chess piece including the cylindrical base (dependent on all previous method marks) |
| \(= \pi\left(\frac{41}{24}+\frac{\sqrt{2}}{6}\right) \approx 6.11\,\text{cm}^3\) | A1 | Correct volume |
## Question 7:
### Part (a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $1 = \frac{a}{0.5+b}$, $0.5 = \frac{a}{2.5+b} \Rightarrow a = ..., b = ...$ | M1 | Uses the given coordinates correctly in the equation modelling the curve to obtain at least one correct equation and attempts to find the values of $a$ and $b$ |
| $a = 2, b = 1.5$ | A1 | Correct values |
**(2 marks)**
---
### Part (b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $V_1 = \pi\int x^2\,\mathrm{d}y = \pi\int\left(\frac{"2"}{y+"1.5"}\right)^2\mathrm{d}y$ | B1ft | Uses the model to obtain $\pi\int\left(\frac{\text{their }a}{y+\text{their }b}\right)^2\mathrm{d}y$. Note $\pi$ can be recovered if it appears later. |
| $\pi\int_{0.5}^{2.5}\left(\frac{"2"}{y+"1.5"}\right)^2\mathrm{d}y$ | M1 | Chooses limits appropriate to the model i.e. 0.5 and 2.5 |
| $= \{4\pi\}\Big[-(y+1.5)^{-1}\Big]_{0.5}^{2.5} (=\pi)$ | M1 | Integrates to obtain an expression of the form $k(y+"1.5")^{-1}$ |
| $x^2 + (y-3)^2 = 0.5$ | B1 | Deduces the correct equation for the circle |
| $V_2 = \pi\int x^2\,\mathrm{d}y = \pi\int\left(0.5-(y-3)^2\right)\mathrm{d}y$ or $\pi\int\left(-y^2+6y-8.5\right)\mathrm{d}y$ | M1 | Uses their circle equation and $\pi\int x^2\,\mathrm{d}y$ to attempt the top volume. Note $\pi$ can be recovered if it appears later. |
| $= \pi\int_{2.5}^{3+\frac{1}{\sqrt{2}}}\left(0.5-(y-3)^2\right)\mathrm{d}y$ or $\pi\int_{2.5}^{3+\frac{1}{\sqrt{2}}}\left(-y^2+6y-8.5\right)\mathrm{d}y$ | M1 | Identifies limits appropriate to the model i.e. 2.5 and $3 +$ their radius |
| $= \{\pi\}\left[0.5y - \frac{1}{3}(y-3)^3\right]_{2.5}^{3+\frac{1}{\sqrt{2}}}$ or $= \{\pi\}\left[-\frac{1}{3}y^3+3y^2-8.5y\right]_{2.5}^{3+\frac{1}{\sqrt{2}}}$ | A1 | Correct integration |
| $V_1 + V_2 + \text{cylinder} = \pi + \pi\left(\frac{5}{24}+\frac{\sqrt{2}}{6}\right)+\frac{1}{2}\pi$ | dM1 | Uses the model to find the volume of the chess piece including the cylindrical base (dependent on all previous method marks) |
| $= \pi\left(\frac{41}{24}+\frac{\sqrt{2}}{6}\right) \approx 6.11\,\text{cm}^3$ | A1 | Correct volume |
**(9 marks)**
**Total: 11 marks**7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f6186130-0027-4670-a6ac-f8a722d2f5fc-24_691_896_255_587}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A student wants to make plastic chess pieces using a 3D printer. Figure 1 shows the central vertical cross-section of the student's design for one chess piece. The plastic chess piece is formed by rotating the region bounded by the $y$-axis, the $x$-axis, the line with equation $x = 1$, the curve $C _ { 1 }$ and the curve $C _ { 2 }$ through $360 ^ { \circ }$ about the $y$-axis.
The point $A$ has coordinates ( $1,0.5$ ) and the point $B$ has coordinates ( $0.5,2.5$ ) where the units are centimetres.
The curve $C _ { 1 }$ is modelled by the equation
$$x = \frac { a } { y + b } \quad 0.5 \leqslant y \leqslant 2.5$$
\begin{enumerate}[label=(\alph*)]
\item Determine the value of $a$ and the value of $b$ according to the model.
The curve $C _ { 2 }$ is modelled to be an arc of the circle with centre $( 0,3 )$.
\item Use calculus to determine the volume of plastic required to make the chess piece according to the model.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel CP2 2020 Q7 [11]}}