| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.8 This is a Further Maths multivariable calculus question requiring partial derivatives, solving simultaneous equations, and using the second derivative test (Hessian) to classify stationary points. While the techniques are standard for Further Maths, the multi-step nature, algebraic manipulation with parameter a, and the conceptual part (c) about the limiting case elevate it above routine exercises. |
| Spec | 8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero8.05f Nature of stationary points: classify using Hessian matrix |
6 The surface $C$ is given by the equation $z = x ^ { 2 } + y ^ { 3 } + a x y$ for all real $x$ and $y$, where $a$ is a non-zero real number.
\begin{enumerate}[label=(\alph*)]
\item Show that $C$ has two stationary points, one of which is at the origin, and give the coordinates of the second in terms of $a$.
\item Determine the nature of these stationary points of $C$.
\item Explain what can be said about the location and nature of the stationary point(s) of the surface given by the equation $z = x ^ { 2 } + y ^ { 3 }$ for all real $x$ and $y$.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2024 Q6 [13]}}