| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2022 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Show dy/dx equals given expression |
| Difficulty | Standard +0.3 Part (a) is a standard implicit differentiation exercise requiring product rule and chain rule application—routine for P4 students. Part (b) requires recognizing that furthest north/south occurs when dy/dx = 0, then solving the resulting cubic equation, which adds modest problem-solving but remains a typical exam question with clear signposting and standard techniques. |
| Spec | 1.07s Parametric and implicit differentiation |
\includegraphics{figure_4}
Figure 4 shows a sketch of the closed curve with equation
$$(x + y)^3 + 10y^2 = 108x$$
\begin{enumerate}[label=(\alph*)]
\item Show that
$$\frac{dy}{dx} = \frac{108 - 3(x + y)^2}{20y + 3(x + y)^2}$$ [5]
\end{enumerate}
The curve is used to model the shape of a cycle track with both $x$ and $y$ measured in km.
The points $P$ and $Q$ represent points that are furthest north and furthest south of the origin $O$, as shown in Figure 4.
Using the result given in part (a),
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find how far the point $Q$ is south of $O$. Give your answer to the nearest 100 m. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2022 Q11 [9]}}