| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Show dy/dx equals expression |
| Difficulty | Standard +0.3 This is a standard C4 parametric equations question with routine techniques: finding dy/dx using the chain rule, solving cos t = 0 for horizontal tangent, and applying the parametric area formula. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dx}{dt} = 1 + \cos t, \quad \frac{dy}{dt} = \cos t\) | M1 | |
| \(\frac{dy}{dx} = \frac{\cos t}{1 + \cos t}\) | M1 A1 | |
| (b) \(\frac{\cos t}{1 + \cos t} = 0, \quad \cos t = 0, \quad t = \frac{\pi}{2}\) | M1 A1 | |
| \(\therefore (\frac{\pi}{2} + 1, 1)\) | A1 | |
| (c) \(= \int_0^{\pi} \sin t \times (1 + \cos t) \, dt = \int_0^{\pi} (\sin t + \frac{1}{2}\sin 2t) \, dt\) | M1 A1 | |
| \(= [-\cos t - \frac{1}{4}\cos 2t]_0^{\pi}\) | M1 A1 | |
| \(= (1 - \frac{1}{4}) - (-1 - \frac{1}{4}) = 2\) | M1 A1 | (12) |
**(a)** $\frac{dx}{dt} = 1 + \cos t, \quad \frac{dy}{dt} = \cos t$ | M1 |
$\frac{dy}{dx} = \frac{\cos t}{1 + \cos t}$ | M1 A1 |
**(b)** $\frac{\cos t}{1 + \cos t} = 0, \quad \cos t = 0, \quad t = \frac{\pi}{2}$ | M1 A1 |
$\therefore (\frac{\pi}{2} + 1, 1)$ | A1 |
**(c)** $= \int_0^{\pi} \sin t \times (1 + \cos t) \, dt = \int_0^{\pi} (\sin t + \frac{1}{2}\sin 2t) \, dt$ | M1 A1 |
$= [-\cos t - \frac{1}{4}\cos 2t]_0^{\pi}$ | M1 A1 |
$= (1 - \frac{1}{4}) - (-1 - \frac{1}{4}) = 2$ | M1 A1 | (12) |
---\includegraphics{figure_2}
Figure 2 shows the curve with parametric equations
$$x = t + \sin t, \quad y = \sin t, \quad 0 \leq t \leq \pi.$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$ in terms of $t$. [3]
\item Find, in exact form, the coordinates of the point where the tangent to the curve is parallel to the $x$-axis. [3]
\item Show that the region bounded by the curve and the $x$-axis has area 2. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q6 [12]}}