| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric curves and Cartesian conversion |
| Type | Properties of specific curves |
| Difficulty | Standard +0.3 This is a standard C4 parametric differentiation question. Part (a) requires routine application of dy/dx = (dy/dt)/(dx/dt). Part (b) involves finding where y=0, determining the tangent equation, and calculating a triangle area—all straightforward multi-step work with no novel insights required. Slightly easier than average due to the mechanical nature of the steps. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| \(\frac{dx}{dt} = \frac{1}{2}at^{-\frac{1}{2}}, \quad \frac{dy}{dt} = a(1-2t)\) | M1 |
| \(\frac{dy}{dx} = \frac{a(1-2t)}{\frac{1}{2}at^{-\frac{1}{2}}} = 2\sqrt{t}(1-2t)\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y=0 \Rightarrow t=0\) (at \(O\)) or \(1\) (at \(A\)) | B1 | |
| \(t=1, x=a, y=0,\) grad \(=-2\) | M1 | |
| \(\therefore y-0 = -2(x-a)\) | A1 | |
| at \(B, x=0 \therefore y=2a\) | M1 | |
| area \(= \frac{1}{2} \times a \times 2a = a^2\) | M1 A1 | (9 marks) |
## (a)
$\frac{dx}{dt} = \frac{1}{2}at^{-\frac{1}{2}}, \quad \frac{dy}{dt} = a(1-2t)$ | M1 |
$\frac{dy}{dx} = \frac{a(1-2t)}{\frac{1}{2}at^{-\frac{1}{2}}} = 2\sqrt{t}(1-2t)$ | M1 A1 |
## (b)
$y=0 \Rightarrow t=0$ (at $O$) or $1$ (at $A$) | B1 |
$t=1, x=a, y=0,$ grad $=-2$ | M1 |
$\therefore y-0 = -2(x-a)$ | A1 |
at $B, x=0 \therefore y=2a$ | M1 |
area $= \frac{1}{2} \times a \times 2a = a^2$ | M1 A1 | **(9 marks)**
---\includegraphics{figure_1}
Figure 1 shows the curve with parametric equations
$$x = a\sqrt{t}, \quad y = at(1-t), \quad t \geq 0,$$
where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac{dy}{dx}$ in terms of $t$. [3]
\end{enumerate}
The curve meets the $x$-axis at the origin, $O$, and at the point $A$. The tangent to the curve at $A$ meets the $y$-axis at the point $B$ as shown.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the area of triangle $OAB$ is $a^2$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q4 [9]}}