| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure (Further Additional Pure) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Sketching surface sections |
| Difficulty | Standard +0.8 This is a Further Maths question requiring students to construct a function from geometric constraints involving a saddle point. While the setup is carefully scaffolded with clear parabolic sections and symmetry conditions, students must translate physical measurements into mathematical equations, determine two separate parabolas, and combine them into a single function z=f(x,y). The conceptual demand of working with saddle surfaces and combining cross-sections elevates this above routine Further Maths exercises, though the algebraic execution is straightforward once the approach is identified. |
| Spec | 8.05e Stationary points: where partial derivatives are zero |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (a) | The section x = 0, i.e. z = f(0, y) drawn |
| The section y = 0, i.e. z = f(x, 0) drawn | B1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 3.4 | -shaped parabola in y-z plane (axes labelled) |
| Answer | Marks |
|---|---|
| (b) | Suggestion z = ay2 – bx2 |
| Answer | Marks |
|---|---|
| z = –0.4 when y = 0, x = () 0.25 b = 6.4 | M1 * |
| Answer | Marks |
|---|---|
| [5] | 3.3 |
| Answer | Marks |
|---|---|
| 2.2a | M for quadratic (only) terms in both x and y |
Question 10:
10 | (a) | The section x = 0, i.e. z = f(0, y) drawn
The section y = 0, i.e. z = f(x, 0) drawn | B1
B1
[2] | 3.4
3.4 | -shaped parabola in y-z plane (axes labelled)
-shaped parabola in x-z plane (axes labelled)
(b) | Suggestion z = ay2 – bx2
(–0.25 x 0.25, –0.3 y 0.3)
z = 0.27 when x = 0, y = () 0.3
a = 3
z = –0.4 when y = 0, x = () 0.25 b = 6.4 | M1 *
A1
M1
*dep
A1
A1
[5] | 3.3
1.1
3.4
1.1
2.2a | M for quadratic (only) terms in both x and y
Details, including signs of coefficients
(Domain not required)
Use of “boundary” conditions to evaluate a, b
(at least one fully attempted)
PMT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
www.ocr.org.uk
For staff training purposes and as part of our quality assurance programme your call may be
recorded or monitored10\\
\includegraphics[max width=\textwidth, alt={}, center]{df94bc38-5187-4349-9005-f9b72691c70d-4_519_770_251_242}
A student wishes to model the saddle of a horse. They use a surface described by a function of the form $\mathrm { z } = \mathrm { f } ( \mathrm { x } , \mathrm { y } )$ with a saddle point at the origin $O$. The z -axis is vertically upwards. The $x$ - and $y$-axes lie in a horizontal plane, with the $x$-axis across the horse and the $y$-axis along the length of the horse (see diagram).
The arc $A O B$ is part of a parabola which lies in the $y z$-plane. The arc $C O D$ is part of a parabola which lies in the $x z$-plane. The saddle is symmetric in both the $x z$-plane and $y z$-plane.
The length of the saddle, the distance $A B$, is to be 0.6 m with both $A$ and $B$ at a height of 0.27 m above $O$. The width of the saddle, the distance $C D$, is to be 0.5 m with both $C$ and $D$ at a depth of 0.4 m below $O$.
\begin{enumerate}[label=(\alph*)]
\item On separate diagrams, sketch the sections $x = 0$ and $y = 0$.\\[0pt]
\item Determine a function f that describes the saddle. [You do not need to state the domain of function f .]
\section*{END OF QUESTION PAPER}
\section*{OCR \\
Oxford Cambridge and RSA}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure 2021 Q10 [7]}}