| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | June |
| Marks | 6 |
| Topic | Implicit equations and differentiation |
| Type | Show dy/dx equals given expression |
| Difficulty | Standard +0.8 This is a Further Maths implicit differentiation question requiring multiple techniques: differentiating x³+y³=xy implicitly (product rule on right side), finding the maximum point by setting dy/dx=0 and solving simultaneously with the curve equation, then using the parabola condition. The multi-step reasoning and algebraic manipulation elevate it above standard implicit differentiation exercises, but it follows a clear path once the setup is understood. |
| Spec | 1.07s Parametric and implicit differentiation |
Fig. 10 shows the graph of $x^3 + y^3 = xy$.
\includegraphics{figure_10}
\begin{enumerate}[label=(\roman*)]
\item Find an expression for $\frac{dy}{dx}$ in terms of $x$ and $y$.
[4]
\item P is the maximum point on the curve. The parabola $y = kx^2$ intersects the curve at P. Find the value of the constant $k$.
[2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q7 [6]}}