| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Tangent plane parallel to given direction |
| Difficulty | Challenging +1.3 This is a substantial Further Maths question requiring partial differentiation, solving simultaneous equations for stationary points, curve sketching, and finding points with specified normal vectors. While it involves multiple techniques and several steps, each individual component is relatively standard for FP3 level—the partial derivatives are straightforward polynomials, the stationary points require algebraic manipulation but no deep insight, and the normal vector condition is a direct application of the gradient. The length and multi-part nature elevate it above average, but it remains a methodical application of learned techniques rather than requiring novel problem-solving approaches. |
| Spec | 8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives8.05c Sections and contours: sketch and relate to surface8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero8.05g Tangent planes: equation at a given point on surface |
| Answer | Marks |
|---|---|
| (i) Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for \(z = xy^2 - 4x^2y - 2x^3 + 27x^2 - 36x + 20\) | [3] |
| (ii) Find coordinates of four stationary points, showing one is \((2,4,8)\) | [8] |
| (iii) Sketch sections at \(x=2\) and \(y=4\); deduce \((2,4,8)\) is neither max nor min | [6] |
| (iv) Show two points where normal is parallel to \(36\mathbf{i}+\mathbf{k}\); find coordinates | [7] |
## Question 2 (Option 2: Multi-variable calculus)
**(i)** Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ for $z = xy^2 - 4x^2y - 2x^3 + 27x^2 - 36x + 20$ | [3] |
**(ii)** Find coordinates of four stationary points, showing one is $(2,4,8)$ | [8] |
**(iii)** Sketch sections at $x=2$ and $y=4$; deduce $(2,4,8)$ is neither max nor min | [6] |
**(iv)** Show two points where normal is parallel to $36\mathbf{i}+\mathbf{k}$; find coordinates | [7] |
---2 A surface has equation $z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20$.\\
(i) Find $\frac { \partial z } { \partial x }$ and $\frac { \partial z } { \partial y }$.\\
(ii) Find the coordinates of the four stationary points on the surface, showing that one of them is $( 2,4,8 )$.\\
(iii) Sketch, on separate diagrams, the sections of the surface defined by $x = 2$ and by $y = 4$. Indicate the point $( 2,4,8 )$ on these sections, and deduce that it is neither a maximum nor a minimum.\\
(iv) Show that there are just two points on the surface where the normal line is parallel to the vector $36 \mathbf { i } + \mathbf { k }$, and find the coordinates of these points.
\hfill \mbox{\textit{OCR MEI FP3 2007 Q2 [24]}}