| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | March |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about y-axis, region between two curves |
| Difficulty | Challenging +1.2 This is a multi-step volumes of revolution question requiring finding a line equation, then computing volume between curve and line rotated about y-axis. While it involves several techniques (line equation, integration with limits, difference of volumes), the setup is straightforward with clearly marked points and standard methods. The integration itself (polynomials) is routine. Slightly above average due to the multi-part nature and rotation about y-axis (less common than x-axis), but no novel insight required. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=04.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Gradient of \(AB = \frac{2-(-1)}{5-2}\) | M1 | Expect 1, must be from \(\Delta y / \Delta x\) |
| Equation of \(AB\) is \(y - 2 = 1(x-5)\) or \(y + 1 = 1(x-2)\) | A1 | OE. Expect \(y = x - 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| \([\pi]\int x^2\,dy = [\pi]\int(y^2+1)^2\,dy = [\pi]\int(y^4+2y^2+1)\,dy\) | M1 | For curve: attempt to square \(y^2+1\) and attempt integration. Subtracting curve equation from line equation before squaring is M0. Integration before squaring is M0 |
| \([\pi]\left(\frac{y^5}{5} + \frac{2y^3}{3} + y\right)\) | A2, 1, 0 | |
| \([\pi]\int(y+3)^2\,dy = [\pi]\int(y^2+6y+9)\,dy\) | M1 | For line: attempt to square *their* \(y+3\) and attempt integration |
| \([\pi]\left(\frac{y^3}{3} + 3y^2 + 9y\right)\) or \([\pi]\left(\frac{(y+3)^3}{3}\right)\) | A2, 1, 0 | Not available for incorrect line equations |
| \([\pi]\left\{\frac{8}{3}+12+18-\left(-\frac{1}{3}+3-9\right)\right\}\) or \([\pi]\left\{\frac{32}{5}+\frac{16}{3}+2-\left(-\frac{1}{5}-\frac{2}{3}-1\right)\right\}\) | DM1 | Apply limits \(-1 \to 2\) to either integral. Expect \(15\frac{3}{5}[\pi]\) and/or \(39[\pi]\). Evidence of substitution of both \(-1\) and \(2\) must be seen. Dependent on at least one of first 2 M1 marks |
| Volume \(= [\pi]\left(39 - 15\frac{3}{5}\right)\) | DM1 | Appropriate subtraction. Dependent on at least one of first 2 M1 marks |
| \(= 23\frac{2}{5}\pi\) or \(\frac{117}{5}\pi\) or awrt \(73.5[1327]\) | A1 |
## Question 11:
**Part 11(a):**
| Gradient of $AB = \frac{2-(-1)}{5-2}$ | M1 | Expect 1, must be from $\Delta y / \Delta x$ |
| Equation of $AB$ is $y - 2 = 1(x-5)$ or $y + 1 = 1(x-2)$ | A1 | OE. Expect $y = x - 3$ |
**Part 11(b):**
| $[\pi]\int x^2\,dy = [\pi]\int(y^2+1)^2\,dy = [\pi]\int(y^4+2y^2+1)\,dy$ | M1 | For curve: attempt to square $y^2+1$ and attempt integration. Subtracting curve equation from line equation before squaring is M0. Integration before squaring is M0 |
| $[\pi]\left(\frac{y^5}{5} + \frac{2y^3}{3} + y\right)$ | A2, 1, 0 | |
| $[\pi]\int(y+3)^2\,dy = [\pi]\int(y^2+6y+9)\,dy$ | M1 | For line: attempt to square *their* $y+3$ and attempt integration |
| $[\pi]\left(\frac{y^3}{3} + 3y^2 + 9y\right)$ or $[\pi]\left(\frac{(y+3)^3}{3}\right)$ | A2, 1, 0 | Not available for incorrect line equations |
| $[\pi]\left\{\frac{8}{3}+12+18-\left(-\frac{1}{3}+3-9\right)\right\}$ or $[\pi]\left\{\frac{32}{5}+\frac{16}{3}+2-\left(-\frac{1}{5}-\frac{2}{3}-1\right)\right\}$ | DM1 | Apply limits $-1 \to 2$ to either integral. Expect $15\frac{3}{5}[\pi]$ and/or $39[\pi]$. Evidence of substitution of both $-1$ and $2$ must be seen. Dependent on at least one of first 2 M1 marks |
| Volume $= [\pi]\left(39 - 15\frac{3}{5}\right)$ | DM1 | Appropriate subtraction. Dependent on at least one of first 2 M1 marks |
| $= 23\frac{2}{5}\pi$ or $\frac{117}{5}\pi$ or awrt $73.5[1327]$ | A1 | |11\\
\includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-16_599_780_274_671}
The diagram shows the curve with equation $x = y ^ { 2 } + 1$. The points $A ( 5,2 )$ and $B ( 2 , - 1 )$ lie on the curve.
\begin{enumerate}[label=(\alph*)]
\item Find an equation of the line $A B$.
\item Find the volume of revolution when the region between the curve and the line $A B$ is rotated through $360 ^ { \circ }$ about the $\boldsymbol { y }$-axis.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2023 Q11 [11]}}