| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Tangent parallel to axis condition |
| Difficulty | Standard +0.3 This is a standard C4 parametric differentiation question requiring the chain rule (dy/dx = (dy/dt)/(dx/dt)) and solving a trigonometric equation. The techniques are routine for this module, though part (b) requires careful handling of multiple solutions in the given domain. Slightly easier than average due to straightforward differentiation and standard trig equation solving. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dx}{dt} = 4\cos\left(t + \frac{\pi}{6}\right)\) | B1 | Does not have to be simplified |
| \(\frac{dy}{dt} = -6\sin 2t\) | B1 | Does not have to be simplified |
| \(\frac{dy}{dx} = \frac{-6\sin 2t}{4\cos\left(t + \frac{\pi}{6}\right)}\) | B1√ oe | Follow through mark: their \(\frac{dy}{dt}\) divided by their \(\frac{dx}{dt}\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 0 \Rightarrow -6\sin 2t = 0\) | M1 oe | Candidate sets numerator equal to 0; numerator must be a trig function |
| Substitute found value of \(t\) to find \(x\) or \(y\) | M1 | First two M marks implied by ONE correct coordinate pair |
| At \(t=0\): \(x = 4\sin\left(\frac{\pi}{6}\right) = 2\), \(y = 3\cos 0 = 3 \rightarrow (2, 3)\) | A1 | At least TWO sets of coordinates |
| At \(t=\frac{\pi}{2}\): \(x = 4\sin\left(\frac{2\pi}{3}\right) = \frac{4\sqrt{3}}{2}\), \(y = 3\cos\pi = -3 \rightarrow (2\sqrt{3}, -3)\) | A1 | At least THREE sets of coordinates |
| At \(t=\pi\): \(x = 4\sin\left(\frac{7\pi}{6}\right) = -2\), \(y = 3\cos 2\pi = 3 \rightarrow (-2, 3)\) | A1 | ONLY FOUR correct sets; more than 4 sets award A0 |
| At \(t=\frac{3\pi}{2}\): \(x = 4\sin\left(\frac{5\pi}{3}\right) = \frac{4(-\sqrt{3})}{2}\), \(y = 3\cos 3\pi = -3 \rightarrow (-2\sqrt{3}, -3)\) | A1 | Coordinates must be exact; \((3.46...,-3)\) or \((-3.46...,-3)\) is A0 |
| [5] |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dx}{dt} = 4\cos\left(t + \frac{\pi}{6}\right)$ | B1 | Does not have to be simplified |
| $\frac{dy}{dt} = -6\sin 2t$ | B1 | Does not have to be simplified |
| $\frac{dy}{dx} = \frac{-6\sin 2t}{4\cos\left(t + \frac{\pi}{6}\right)}$ | B1√ oe | Follow through mark: their $\frac{dy}{dt}$ divided by their $\frac{dx}{dt}$ |
| | **[3]** | |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 0 \Rightarrow -6\sin 2t = 0$ | M1 oe | Candidate sets numerator equal to 0; numerator must be a trig function |
| Substitute found value of $t$ to find $x$ or $y$ | M1 | First two M marks implied by ONE correct coordinate pair |
| At $t=0$: $x = 4\sin\left(\frac{\pi}{6}\right) = 2$, $y = 3\cos 0 = 3 \rightarrow (2, 3)$ | A1 | At least TWO sets of coordinates |
| At $t=\frac{\pi}{2}$: $x = 4\sin\left(\frac{2\pi}{3}\right) = \frac{4\sqrt{3}}{2}$, $y = 3\cos\pi = -3 \rightarrow (2\sqrt{3}, -3)$ | A1 | At least THREE sets of coordinates |
| At $t=\pi$: $x = 4\sin\left(\frac{7\pi}{6}\right) = -2$, $y = 3\cos 2\pi = 3 \rightarrow (-2, 3)$ | A1 | ONLY FOUR correct sets; more than 4 sets award A0 |
| At $t=\frac{3\pi}{2}$: $x = 4\sin\left(\frac{5\pi}{3}\right) = \frac{4(-\sqrt{3})}{2}$, $y = 3\cos 3\pi = -3 \rightarrow (-2\sqrt{3}, -3)$ | A1 | Coordinates must be exact; $(3.46...,-3)$ or $(-3.46...,-3)$ is A0 |
| | **[5]** | |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8c963567-d751-4898-b7a7-7095d90514f0-07_687_1209_214_370}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of the curve $C$ with parametric equations
$$x = 4 \sin \left( t + \frac { \pi } { 6 } \right) , \quad y = 3 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $t$.
\item Find the coordinates of all the points on $C$ where $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2012 Q5 [8]}}