| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find dy/dx expression in terms of parameter |
| Difficulty | Challenging +1.2 This is a multi-part parametric equations question requiring calculator sketching, finding dy/dx using the chain rule, and solving a geometric condition. While it involves several techniques (parametric differentiation, finding loop widths from stationary points), each part follows standard FP2 methods without requiring novel insight. The proof in part (iii)(C) and numerical solving in part (iv) add moderate challenge, but this remains a typical Further Maths question testing routine application of parametric techniques. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch of cycloid (\(k=1\)) | B2 | |
| Features: cusps at \(y=0\), \(x=2n\pi\); maximum \(y=2\) at \(x=(2n+1)\pi\); smooth arches | B1 B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
Sketch for \(0| B1 |
| |
| No cusps (loops replaced by smooth curve not touching x-axis); curve stays above x-axis | B2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch for \(k=2\) showing loops | B2 | Loops below x-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dx}{d\theta} = 1-2\cos\theta\), \(\frac{dy}{d\theta} = \sin\theta\) | B1 | |
| \(\frac{dy}{dx} = \frac{\sin\theta}{1-2\cos\theta}\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Horizontal tangents when \(\sin\theta = 0\): but need endpoints of loop | M1 | |
| Loop exists between \(\theta\) values where \(x\) same: \(\frac{dx}{d\theta}=0 \Rightarrow \cos\theta = \frac{1}{2}, \theta = \pm\frac{\pi}{3}\) | M1 A1 | |
| Width \(= x(\frac{\pi}{3}) - x(-\frac{\pi}{3})\)... using symmetry | M1 | |
| \(= \frac{\pi}{3}-2\sin\frac{\pi}{3} - (-\frac{\pi}{3}+2\sin\frac{\pi}{3})\)... correction: self-intersection found | M1 | |
| Width \(= 2\sqrt{3} - \frac{2\pi}{3}\) | A1 | Completion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(k \approx 4.6\) (to 1 d.p.) | B2 | B1 for method shown on calculator |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch of cycloid ($k=1$) | B2 | |
| Features: cusps at $y=0$, $x=2n\pi$; maximum $y=2$ at $x=(2n+1)\pi$; smooth arches | B1 B1 | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch for $0<k<1$ | B1 | |
| No cusps (loops replaced by smooth curve not touching x-axis); curve stays above x-axis | B2 | |
## Part (iii)(A)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch for $k=2$ showing loops | B2 | Loops below x-axis |
## Part (iii)(B)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dx}{d\theta} = 1-2\cos\theta$, $\frac{dy}{d\theta} = \sin\theta$ | B1 | |
| $\frac{dy}{dx} = \frac{\sin\theta}{1-2\cos\theta}$ | B1 | |
## Part (iii)(C)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Horizontal tangents when $\sin\theta = 0$: but need endpoints of loop | M1 | |
| Loop exists between $\theta$ values where $x$ same: $\frac{dx}{d\theta}=0 \Rightarrow \cos\theta = \frac{1}{2}, \theta = \pm\frac{\pi}{3}$ | M1 A1 | |
| Width $= x(\frac{\pi}{3}) - x(-\frac{\pi}{3})$... using symmetry | M1 | |
| $= \frac{\pi}{3}-2\sin\frac{\pi}{3} - (-\frac{\pi}{3}+2\sin\frac{\pi}{3})$... correction: self-intersection found | M1 | |
| Width $= 2\sqrt{3} - \frac{2\pi}{3}$ | A1 | Completion |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $k \approx 4.6$ (to 1 d.p.) | B2 | B1 for method shown on calculator |5 A curve has parametric equations
$$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\roman*)]
\item For the case $k = 1$, use your graphical calculator to sketch the curve. Describe its main features.
\item Sketch the curve for a value of $k$ between 0 and 1 . Describe briefly how the main features differ from those for the case $k = 1$.
\item For the case $k = 2$ :\\
(A) sketch the curve;\\
(B) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$;\\
(C) show that the width of each loop, measured parallel to the $x$-axis, is
$$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
\item Use your calculator to find, correct to one decimal place, the value of $k$ for which successive loops just touch each other.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2006 Q5 [18]}}