| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2018 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Challenging +1.8 This is a substantial multi-part question requiring 3D visualization of a paraboloid of revolution, understanding of conic sections in different planes, and application of the reflective property of parabolas. While it involves Further Maths content (3D surfaces, paraboloids), most parts are guided and methodical rather than requiring deep insight. The calculations in part (iv) are straightforward once the setup is understood, and part (v) tests conceptual understanding of the focus property. The main challenge is spatial reasoning and connecting multiple representations, placing it above average difficulty but not at the highest level. |
| Spec | 8.05c Sections and contours: sketch and relate to surface |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (i) | (a) |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | z = c or f(x, y) = c | B1 |
| Allow specific positive c values | Allow x2 + y2 = d |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | (a) | A parabola through O |
| condone continuations | If axes shown, must be x, z |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | z = kx2 or z = f(x, 0) | B1 |
| Answer | Marks |
|---|---|
| (iii) | a and b must be positive (since z 0) |
| Answer | Marks |
|---|---|
| by symmetry | B1 |
| Answer | Marks |
|---|---|
| E1 | 3.3 |
| Answer | Marks |
|---|---|
| 2.4 | Statement |
| Answer | Marks |
|---|---|
| (iv) | In (i) (b), c = 0.065 |
| Answer | Marks |
|---|---|
| x2 | B1 |
| Answer | Marks |
|---|---|
| A1 | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | Or diameter is 0.844 |
| Answer | Marks |
|---|---|
| Allow unsaid that k = a in (ii) (b) | Or d = 0.4222 = 0.178 FT |
| Answer | Marks |
|---|---|
| (v) | Draw at least two beams which after being |
| Answer | Marks |
|---|---|
| beams will miss the receiver o.e. | B1 |
| B1 | 3.2a |
| 3.5a | It is not sufficient to say only that the |
Question 7:
7 | (i) | (a) | A circle | B1 | 1.1 | If axes shown, must be x, y
[1]
(b) | z = c or f(x, y) = c | B1 | 3.3 | Not necessary to say that c (or d) > 0
Allow specific positive c values | Allow x2 + y2 = d
but d cannot be (OP)2
[1]
(ii) | (a) | A parabola through O | B1 | 1.1 | The parabola should stop at P but
condone continuations | If axes shown, must be x, z
[1]
(b) | z = kx2 or z = f(x, 0) | B1 | 1.1 | Allow specific positive k values
[1]
(iii) | a and b must be positive (since z 0)
a = b
by symmetry | B1
B1
E1 | 3.3
3.4
2.4 | Statement
Justification
[3]
(iv) | In (i) (b), c = 0.065
2 x = 2.652 x-coordinate of (e.g.) P is 0.422
0.065
Thus a = b = = 0.365 (to 3s.f.)
x2 | B1
B1
M1
A1 | 1.1
1.1
3.1a
1.1 | Or diameter is 0.844
Using this to find a, b
FT their x-coordinate of P
Allow unsaid that k = a in (ii) (b) | Or d = 0.4222 = 0.178 FT
No need to note b = a if
already noted in (iii)
[4]
(v) | Draw at least two beams which after being
reflected meet in B
Either the position of B moves accordingly or the
beams will miss the receiver o.e. | B1
B1 | 3.2a
3.5a | It is not sufficient to say only that the
parabola changes shape
[2]
PMT
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© OCR 20187 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical $z$-axis. The model is represented by part of the surface with equation $z = \mathrm { f } ( x , y )$ and $O$ is on the surface.
The point $P$ is on the rim of the dish and directly above the $x$-axis.\\
The object, $B$, modelled as a point on the $z$-axis is the receiving box which collects the TV signals reflected by the dish.\\
\includegraphics[max width=\textwidth, alt={}, center]{f2166e0a-cd4c-40af-b4b4-04ef4919d996-3_753_995_584_525}\\
(i) The horizontal plane $\Pi _ { 1 }$, containing the point $P$, intersects the surface of the model in a contour of the surface.
\begin{enumerate}[label=(\alph*)]
\item Sketch this contour in the Printed Answer Booklet.
\item State a suitable equation for this contour.\\
(ii) A second plane, $\Pi _ { 2 }$, containing both $P$ and the $z$-axis, intersects the surface of the model in a section of the surface.\\
(a) Sketch this section in the Printed Answer Booklet.\\
(b) State a suitable equation for this section.\\
(iii) A proposed equation for the surface is $z = a x ^ { 2 } + b y ^ { 2 }$. What can you say about the constants $a$ and $b$ within this equation? Justify your answers.\\
(iv) The real TV satellite dish has the following measurements (in metres): the height of $P$ above $O$ is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of
\begin{itemize}
\item $a$ and $b$,
\item any other constants stated within the answers to parts (i)(b) and (ii)(b).\\
(v) Incoming satellite signals arrive at the dish in linear "beams" travelling parallel to the $z$-axis. They are then 'bounced' off the dish to the receiving box at $B$.
\item On the diagram for part (ii)(a) in the Printed Answer Booklet draw some of these beams and mark $B$.
\item If the values of $a$ and $b$ were changed, what would happen?
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2018 Q7 [13]}}