| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Stationary points of surface (multivariable) |
| Difficulty | Challenging +1.2 This is a multivariable calculus problem requiring partial differentiation and solving simultaneous equations to find stationary points. While it involves Further Maths content (making it inherently harder than standard A-level), the technique is straightforward: set ∂z/∂x = 0 and ∂z/∂y = 0, then solve the resulting system. The algebra is moderately involved but routine for Further Maths students, and the question explicitly tells you there are exactly two stationary points with one already given, reducing the problem-solving demand. |
| Spec | 8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero |
2 The surface $S$ has equation $z = x ^ { 3 } + y ^ { 3 } - 2 x ^ { 2 } - 5 y ^ { 2 } + 3 x y$.\\
It is given that $S$ has two stationary points; one at the origin, $O$, and the other at the point $A$.\\
Determine the coordinates of $A$.
\hfill \mbox{\textit{OCR Further Additional Pure AS 2022 Q2 [6]}}