| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with implicit or parametric curves |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring: (a) finding circle-chord intersection by substituting x=0 and solving, yielding simple coordinates; (b) computing volume using standard π∫y² dx formula with clear limits. The circle equation is already in standard form, and the setup is direct with no tricky geometry or complex integration. Slightly easier than average due to the clean arithmetic and standard technique application. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((-2)^2 + y^2 = 8\) leading to \(y=2\) leading to \(A=(0,2)\) | B1 | |
| Substitute \(y = \) their \(2\) into circle leading to \((x-2)^2 + 4 = 8\) | M1 | Expect \(x=4\) |
| \(B = (4,2)\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to find \([\pi]\int(8-(x-2)^2)\,dx\) | *M1 | |
| \([\pi]\left[8x - \frac{(x-2)^3}{3}\right]\) or \([\pi]\left[8x - \left(\frac{x^3}{3} - 2x^2 + 4x\right)\right]\) | A1 | |
| \([\pi]\left(32 - \frac{16}{3}\right)\) or \([\pi]\left[32 - \left(\frac{64}{3} - 32 + 16\right)\right]\) | DM1 | Apply limits \(0 \to\) their \(4\) |
| Volume of cylinder \(= \pi \times 2^2 \times 4 = 16\pi\) | B1 FT | OR from \(\pi\int 2^2\,dx\) with their limits from (a). FT on their \(A\) and \(B\) |
| \([\text{Volume of revolution} = 26\frac{2}{3}\pi - 16\pi =]\ 10\frac{2}{3}\pi\) | A1 | Accept \(33.5\) |
| 5 |
## Question 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(-2)^2 + y^2 = 8$ leading to $y=2$ leading to $A=(0,2)$ | B1 | |
| Substitute $y = $ their $2$ into circle leading to $(x-2)^2 + 4 = 8$ | M1 | Expect $x=4$ |
| $B = (4,2)$ | A1 | |
| | **3** | |
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to find $[\pi]\int(8-(x-2)^2)\,dx$ | *M1 | |
| $[\pi]\left[8x - \frac{(x-2)^3}{3}\right]$ or $[\pi]\left[8x - \left(\frac{x^3}{3} - 2x^2 + 4x\right)\right]$ | A1 | |
| $[\pi]\left(32 - \frac{16}{3}\right)$ or $[\pi]\left[32 - \left(\frac{64}{3} - 32 + 16\right)\right]$ | DM1 | Apply limits $0 \to$ their $4$ |
| Volume of cylinder $= \pi \times 2^2 \times 4 = 16\pi$ | B1 FT | OR from $\pi\int 2^2\,dx$ with their limits from (a). FT on their $A$ and $B$ |
| $[\text{Volume of revolution} = 26\frac{2}{3}\pi - 16\pi =]\ 10\frac{2}{3}\pi$ | A1 | Accept $33.5$ |
| | **5** | |8\\
\includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-12_771_839_262_651}
The diagram shows the circle with equation $( x - 2 ) ^ { 2 } + y ^ { 2 } = 8$. The chord $A B$ of the circle intersects the positive $y$-axis at $A$ and is parallel to the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\item Find, by calculation, the coordinates of $A$ and $B$.
\item Find the volume of revolution when the shaded segment, bounded by the circle and the chord $A B$, is rotated through $360 ^ { \circ }$ about the $x$-axis.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q8 [8]}}