| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and area |
| Difficulty | Standard +0.3 This is a standard volumes of revolution question requiring finding an intersection point, applying the volume formula π∫y²dx, and calculating arc length. While multi-part with exact form requirements, these are routine A-level techniques with straightforward integration—slightly easier than average due to predictable structure and standard methods. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.08d Evaluate definite integrals: between limits4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 + (2x-1)^2 - 2[=0] \rightarrow 5x^2 - 4x - 1[=0]\) | \*M1 A1 | Or \(5y^2 + 2y - 7[=0]\) |
| \((5x+1)(x-1)[=0]\) or \((5y+7)(y-1)[=0]\) | DM1 | May see factors or formula or completing square |
| \(x = 1\), \(y = 1\) or \((1, 1)\) only | A1 | May be implied on the diagram |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\pi)\int(2-x^2)\,dx = (\pi)\left(2x - \frac{x^3}{3}\right)\) | \*M1 A1 | Attempt integration of \(y^2\), allow \(\int(2-y^2)\,dy\) |
| \((\pi)\left(2\sqrt{2} - \frac{(\sqrt{2})^3}{3}\right) - \left(2 - \frac{1}{3}\right)\) | DM1 | Apply limits \(1 \rightarrow \sqrt{2}\) |
| \(\frac{\pi}{3}(4\sqrt{2} - 5)\) | A1 | CAO, allow \(\frac{\pi}{3}(2\sqrt{8}-5)\), must be in given form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Arc length \(= \frac{1}{8}(2\pi\sqrt{2})\) or \(\frac{\pi\sqrt{2}}{4}\) | B1 | Must be exact |
| Perimeter \(= \sqrt{2} + \textit{their}\) arc length | B1 FT | Must be exact, do not allow inverse trig functions |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + (2x-1)^2 - 2[=0] \rightarrow 5x^2 - 4x - 1[=0]$ | \*M1 A1 | Or $5y^2 + 2y - 7[=0]$ |
| $(5x+1)(x-1)[=0]$ or $(5y+7)(y-1)[=0]$ | DM1 | May see factors or formula or completing square |
| $x = 1$, $y = 1$ or $(1, 1)$ only | A1 | May be implied on the diagram |
---
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\pi)\int(2-x^2)\,dx = (\pi)\left(2x - \frac{x^3}{3}\right)$ | \*M1 A1 | Attempt integration of $y^2$, allow $\int(2-y^2)\,dy$ |
| $(\pi)\left(2\sqrt{2} - \frac{(\sqrt{2})^3}{3}\right) - \left(2 - \frac{1}{3}\right)$ | DM1 | Apply limits $1 \rightarrow \sqrt{2}$ |
| $\frac{\pi}{3}(4\sqrt{2} - 5)$ | A1 | CAO, allow $\frac{\pi}{3}(2\sqrt{8}-5)$, must be in given form |
---
## Question 10(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Arc length $= \frac{1}{8}(2\pi\sqrt{2})$ or $\frac{\pi\sqrt{2}}{4}$ | B1 | Must be exact |
| Perimeter $= \sqrt{2} + \textit{their}$ arc length | B1 FT | Must be exact, do not allow inverse trig functions |
---\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of $A$.
\item Find the volume of revolution when the shaded region is rotated through $360 ^ { \circ }$ about the $x$-axis. Give your answer in the form $\frac { \pi } { a } ( b \sqrt { c } - d )$, where $a , b , c$ and $d$ are integers.
\item Find an exact expression for the perimeter of the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q10 [10]}}