| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and related rates |
| Difficulty | Standard +0.8 This question combines volumes of revolution about the y-axis (requiring inversion of the function and careful setup of limits) with related rates involving chain rule. Part (i) requires algebraic manipulation to reach a specific form, and part (ii) involves differentiating the volume formula and applying chain rule with a given rate. While systematic, it requires multiple techniques and careful algebra beyond routine C3 questions. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08f Area between two curves: using integration1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt to express \(x\) or \(x^2\) in terms of \(y\) | M1 | |
| Obtain \(x^2 = \frac{1296}{(y+3)^4}\) | A1 | or (unsimplified) equiv |
| Obtain integral of form \(k(y+3)^{-3}\) | M1 | any constant \(k\) |
| Obtain \(-432\pi(y+3)^{-3}\) or \(-432(y+3)^{-3}\) | A1 | or (unsimplified) equiv |
| Attempt evaluation using limits 0 and \(p\) | M1 | for expression of form \(k(y+3)^{-n}\) obtained from integration attempt; subtraction correct way round |
| Confirm \(16\pi\!\left(1 - \frac{27}{(p+3)^3}\right)\) | A1 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or obtain \(\frac{dV}{dp} = 1296\pi(p+3)^{-4}\) | B1 | or equiv; perhaps involving \(y\) |
| Multiply \(\frac{dp}{dt}\) and attempt at \(\frac{dV}{dp}\) | *M1 | algebraic or numerical |
| Substitute \(p = 9\) and attempt evaluation | M1 | dep *M |
| Obtain \(\frac{1}{4}\pi\) or 0.785 | A1 | 4 |
# Question 8:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt to express $x$ or $x^2$ in terms of $y$ | M1 | |
| Obtain $x^2 = \frac{1296}{(y+3)^4}$ | A1 | or (unsimplified) equiv |
| Obtain integral of form $k(y+3)^{-3}$ | M1 | any constant $k$ |
| Obtain $-432\pi(y+3)^{-3}$ or $-432(y+3)^{-3}$ | A1 | or (unsimplified) equiv |
| Attempt evaluation using limits 0 and $p$ | M1 | for expression of form $k(y+3)^{-n}$ obtained from integration attempt; subtraction correct way round |
| Confirm $16\pi\!\left(1 - \frac{27}{(p+3)^3}\right)$ | A1 | **6** | AG; necessary detail required, including appearance of $\pi$ prior to final line |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or obtain $\frac{dV}{dp} = 1296\pi(p+3)^{-4}$ | B1 | or equiv; perhaps involving $y$ |
| Multiply $\frac{dp}{dt}$ and attempt at $\frac{dV}{dp}$ | *M1 | algebraic or numerical |
| Substitute $p = 9$ and attempt evaluation | M1 | dep *M |
| Obtain $\frac{1}{4}\pi$ or 0.785 | A1 | **4** | or greater accuracy |
---8\\
\includegraphics[max width=\textwidth, alt={}, center]{c940af95-e291-402a-856c-9090d13163d5-4_538_702_264_719}
The diagram shows the curve with equation
$$y = \frac { 6 } { \sqrt { x } } - 3$$
The point $P$ has coordinates $( 0 , p )$. The shaded region is bounded by the curve and the lines $x = 0$, $y = 0$ and $y = p$. The shaded region is rotated completely about the $y$-axis to form a solid of volume $V$.\\
(i) Show that $V = 16 \pi \left( 1 - \frac { 27 } { ( p + 3 ) ^ { 3 } } \right)$.\\
(ii) It is given that $P$ is moving along the $y$-axis in such a way that, at time $t$, the variables $p$ and $t$ are related by
$$\frac { \mathrm { d } p } { \mathrm {~d} t } = \frac { 1 } { 3 } p + 1 .$$
Find the value of $\frac { \mathrm { d } V } { \mathrm {~d} t }$ at the instant when $p = 9$.
\hfill \mbox{\textit{OCR C3 2009 Q8 [10]}}