| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and stationary points |
| Difficulty | Standard +0.3 This is a straightforward two-part question combining basic differentiation (using chain rule on a square root) and a standard volume of revolution integral. Part (i) requires routine application of the chain rule and setting dy/dx = 0, while part (ii) is a direct application of the volume formula with a simple polynomial integrand after squaring. Both techniques are core P1 skills with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08d Volumes of revolution: about x and y axes |
**(i)**
- B1: $\frac{dy}{dx} = \frac{1}{2}(8x - x^2)^{-1/2} \times (8 - 2x)$. B1 for everything but $\times (8 - 2x)$; B1 for $\times (8 - 2x)$, even if B0
- B1: $= 0$ when $x = 4$. Sets to $0$ + attempt at solution
- M1: $\Rightarrow (4, 4)$. co – A0 if fortuitous because of B0 earlier
- A1: co
**(ii)**
- B1: $y = 0$ when $x = 0$ or $8$. Anywhere
- B2,1: Vol $= \pi \int (8x - x^2) \, dx$. $-1$ for each error (not including $\pi$)
- B1: $= \pi \left[ 4x^2 - \frac{x^3}{3} \right]$
- A1: $\Rightarrow \frac{256\pi}{3}$. co
---8 The equation of a curve is $y = \sqrt { } \left( 8 x - x ^ { 2 } \right)$. Find\\
(i) an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$, and the coordinates of the stationary point on the curve,\\
(ii) the volume obtained when the region bounded by the curve and the $x$-axis is rotated through $360 ^ { \circ }$ about the $x$-axis.
\hfill \mbox{\textit{CAIE P1 2011 Q8 [8]}}