| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | November |
| Marks | 10 |
| 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 completing the square (routine A-level skill), rearranging to make x the subject, and applying the standard formula V = π∫x²dy. All steps are procedural with no novel insight required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.08a Fundamental theorem of calculus: integration as reverse of differentiation4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((y =)(x+2)^2 - 1\) | B1 DB1 | 2nd B1 dependent on 2 in bracket |
| \(x + 2 = (\pm)(y+1)^{1/2}\) | M1 | |
| \(x = -2 + (y+1)^{1/2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^2 = 4 + (y+1) - /+ 4(y+1)^{\frac{1}{2}}\) | *M1A1 | SOI. Attempt to find \(x^2\). The last term can be \(-\) or \(+\) at this stage |
| \((\pi)\int x^2 \, dy = (\pi)\left[5y + \frac{y^2}{2} - \frac{4(y+1)^{\frac{3}{2}}}{\frac{3}{2}}\right]\) | A2,1,0 | |
| \((\pi)\left[15 + \frac{9}{2} - \frac{64}{3} - \left(-5 + \frac{1}{2}\right)\right]\) | DM1 | Apply \(y\) limits |
| \(\frac{8\pi}{3}\) or \(8.38\) | A1 |
## Question 11:
### Part 11(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(y =)(x+2)^2 - 1$ | B1 DB1 | 2nd B1 dependent on 2 in bracket |
| $x + 2 = (\pm)(y+1)^{1/2}$ | M1 | |
| $x = -2 + (y+1)^{1/2}$ | A1 | |
### Part 11(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 = 4 + (y+1) - /+ 4(y+1)^{\frac{1}{2}}$ | *M1A1 | SOI. Attempt to find $x^2$. The last term can be $-$ or $+$ at this stage |
| $(\pi)\int x^2 \, dy = (\pi)\left[5y + \frac{y^2}{2} - \frac{4(y+1)^{\frac{3}{2}}}{\frac{3}{2}}\right]$ | A2,1,0 | |
| $(\pi)\left[15 + \frac{9}{2} - \frac{64}{3} - \left(-5 + \frac{1}{2}\right)\right]$ | DM1 | Apply $y$ limits |
| $\frac{8\pi}{3}$ or $8.38$ | A1 | |11\\
\includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-18_650_611_260_762}
The diagram shows a shaded region bounded by the $y$-axis, the line $y = - 1$ and the part of the curve $y = x ^ { 2 } + 4 x + 3$ for which $x \geqslant - 2$.\\
(i) Express $y = x ^ { 2 } + 4 x + 3$ in the form $y = ( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants. Hence, for $x \geqslant - 2$, express $x$ in terms of $y$.\\
(ii) Hence, showing all necessary working, find the volume obtained when the shaded region 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.\\
\hfill \mbox{\textit{CAIE P1 2019 Q11 [10]}}