| 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 | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Standard +0.8 This question requires understanding of 3D surfaces, trigonometric functions' ranges, and spatial reasoning about transformations. Part (a) demands critical analysis of a mathematical model against real-world constraints (identifying that peaks occur in a grid pattern rather than a line, and that sin x + sin y has range [-2,2] so peaks are equal height). Part (b) requires recognizing that sin x + sin y ≤ 2 with equality only at isolated points, making the contour effectively empty. While conceptually accessible, it requires synthesis across multiple ideas and non-routine problem-solving rather than standard technique application. |
| Spec | 8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives8.05c Sections and contours: sketch and relate to surface |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | a | It gives lines of peaks (etc.) extending in |
| Answer | Marks |
|---|---|
| direction, such as e − a x | B1 |
| Answer | Marks |
|---|---|
| [4] | 3.4 |
| Answer | Marks |
|---|---|
| 3.5c | Allow other intervals of length 2 |
| Answer | Marks |
|---|---|
| each | Allow ‘ratio’ or just |
| Answer | Marks |
|---|---|
| b | Max. (sin x + sin y) = 2 |
| Answer | Marks |
|---|---|
| of each island mountain) | B1 |
| Answer | Marks |
|---|---|
| [2] | 3.1a |
Question 5:
5 | a | It gives lines of peaks (etc.) extending in
both x- and y-directions
Restrict the domain to –π/2 ≤ y ≤ 3π/2
All the peaks are the same height
Introduce a (damping) factor in the x-
direction, such as e − a x | B1
B1
B1
B1
[4] | 3.4
3.3
3.5a
3.5c | Allow other intervals of length 2
Swapping x, y throughout is OK
OR a separate damping factor to
each | Allow ‘ratio’ or just
‘factor’ in conjunction
with ‘decreasing’
b | Max. (sin x + sin y) = 2
so contour z = 2 contains only (an infinite
number of) (isolated) points (the very peak
of each island mountain) | B1
B1
[2] | 3.1a
3.2a5 A research student is using 3-D graph-plotting software to model a chain of volcanic islands in the Pacific Ocean. These islands appear above sea-level at regular intervals, (approximately) distributed along a straight line. Each island takes the form of a single peak; also, along the line of islands, the heights of these peaks decrease in size in an (approximately) regular fashion (see Fig. 1.1).
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{06496165-0b83-4050-ae26-fa5a0614bd46-3_476_812_495_246}
\captionsetup{labelformat=empty}
\caption{Fig. 1.1}
\end{center}
\end{figure}
The student's model uses the surface with equation $\mathrm { z } = \sin \mathrm { x } + \sin \mathrm { y }$, a part of which is shown in Fig. 1.2 below. The surface of the sea is taken to be the plane $z = 0$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{06496165-0b83-4050-ae26-fa5a0614bd46-3_789_951_1270_242}
\captionsetup{labelformat=empty}
\caption{Fig. 1.2}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item - Describe two problems with this model.
\begin{itemize}
\item Suggest revisions to this model so that each of these problems is addressed.
\item Still using their original model, the student examines the contour $z = 2$ for their surface only to find that the software shows what appears to be an empty graph.
\end{itemize}
Explain what has happened.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2022 Q5 [6]}}