| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2017 |
| Session | Specimen |
| Marks | 10 |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Challenging +1.2 This is a standard Further Maths multivariable calculus question requiring partial derivatives to find and classify a stationary point, then find a tangent plane equation. While beyond A-level Core content, these are routine techniques for Further Maths students with no novel problem-solving required—just systematic application of the Hessian determinant method and tangent plane formula. |
| Spec | 8.05e Stationary points: where partial derivatives are zero8.05f Nature of stationary points: classify using Hessian matrix8.05g Tangent planes: equation at a given point on surface |
A surface $S$ has equation $z = f(x, y)$, where $f(x, y) = 2x^2 - y^2 + 3xy + 17y$. It is given that $S$ has a single stationary point, $P$.
\begin{enumerate}[label=(\roman*)]
\item Determine the coordinates, and the nature, of $P$. [8]
\item Find the equation of the tangent plane to $S$ at the point $Q(1, 2, 38)$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2017 Q6 [10]}}