| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and tangent/normal |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard techniques: finding coordinates by substitution, computing a normal line using differentiation, and applying the volume of revolution formula. All steps are routine for P1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07m Tangents and normals: gradient and equations1.08h Integration by substitution4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(B = (0,1)\), \(C = (4,3)\) | B1, B1 | [2] If B0B0 then SCB1 for both \(y=1\) & \(x=4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{\delta y}{\delta x} = \frac{1}{2} \times 2(1+2x)^{-\frac{1}{2}}\) | M1A1 | \(-\frac{1}{2}\) required & at least one of \(\frac{1}{2} \times 2\) for M1 |
| Grad. of normal \(= -3\) | B1 | |
| \(y - 3 = -3(x-4)\) or \(y = -3x + 15\) oe | B1\(\sqrt{}\) | [4] Ft only from *their* C |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y^2 = 1 + 2x \Rightarrow x = \frac{1}{2(y^2 - 1)}\) SOI | B1 | \(\int x^2 \delta y\), square \(\frac{1}{2}(y^2-1)\) & attempt \(\int^n\) |
| \((\pi) \times \frac{1}{4} \times \int(y^4 - 2y^2 + 1)\delta y\) | M1 | |
| \((\pi) \times \frac{1}{4}\left[\frac{y^5}{5} - \frac{2y^3}{3} + y\right]\) | A1 | Apply limits \(0 \to\) *their* \(1\) (from *their* \(B\)) |
| \((\pi) \times \frac{1}{4}\left[\frac{1}{5} - \frac{2}{3} + 1\right]\) | DM1 | cao SCB1 for \(\int y^2 \delta x \to \frac{\pi}{4}\) (scores 1/5) |
| \(\frac{2}{15}\pi\) | A1 | [5] |
## Question 10:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $B = (0,1)$, $C = (4,3)$ | B1, B1 | [2] If B0B0 then SCB1 for both $y=1$ & $x=4$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{\delta y}{\delta x} = \frac{1}{2} \times 2(1+2x)^{-\frac{1}{2}}$ | M1A1 | $-\frac{1}{2}$ required & at least one of $\frac{1}{2} \times 2$ for M1 |
| Grad. of normal $= -3$ | B1 | |
| $y - 3 = -3(x-4)$ or $y = -3x + 15$ oe | B1$\sqrt{}$ | [4] Ft only from *their* C |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y^2 = 1 + 2x \Rightarrow x = \frac{1}{2(y^2 - 1)}$ SOI | B1 | $\int x^2 \delta y$, square $\frac{1}{2}(y^2-1)$ & attempt $\int^n$ |
| $(\pi) \times \frac{1}{4} \times \int(y^4 - 2y^2 + 1)\delta y$ | M1 | |
| $(\pi) \times \frac{1}{4}\left[\frac{y^5}{5} - \frac{2y^3}{3} + y\right]$ | A1 | Apply limits $0 \to$ *their* $1$ (from *their* $B$) |
| $(\pi) \times \frac{1}{4}\left[\frac{1}{5} - \frac{2}{3} + 1\right]$ | DM1 | cao SCB1 for $\int y^2 \delta x \to \frac{\pi}{4}$ (scores 1/5) |
| $\frac{2}{15}\pi$ | A1 | [5] |
---10\\
\includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-4_799_1390_255_376}
The diagram shows the curve $y = \sqrt { } ( 1 + 2 x )$ meeting the $x$-axis at $A$ and the $y$-axis at $B$. The $y$-coordinate of the point $C$ on the curve is 3 .\\
(i) Find the coordinates of $B$ and $C$.\\
(ii) Find the equation of the normal to the curve at $C$.\\
(iii) Find the volume obtained when the shaded region is rotated through $360 ^ { \circ }$ about the $\boldsymbol { y }$-axis.
\hfill \mbox{\textit{CAIE P1 2011 Q10 [11]}}