| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring analysis of rational functions and their transformations. Part (i) involves standard asymptote finding (vertical from denominator zeros, horizontal from degree comparison). Part (ii) requires understanding the y² transformation creates reflection symmetry and determining where the RHS is non-negative. While multi-step, the techniques are systematic applications of FM2 content without requiring novel geometric insight or complex algebraic manipulation. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.03d Circles: equation (x-a)^2+(y-b)^2=r^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x = \pm a\), \(y = 2\) | B1, B1, B1 | Must be \(=\); no working needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Two correct labelled asymptotes \(\parallel Ox\) and approaches | B1 | |
| Two correct labelled asymptotes \(\parallel Oy\) and approaches | B1 | |
| Crosses at \(\left(\frac{3}{2}a, 0\right)\) (and \((0,0)\) — may be implied) | B1 | |
| \(90°\) where it crosses \(Ox\); smoothly | B1 | |
| Symmetry in \(Ox\) | B1 |
## Question 6:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x = \pm a$, $y = 2$ | B1, B1, B1 | Must be $=$; no working needed |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Two correct labelled asymptotes $\parallel Ox$ and approaches | B1 | |
| Two correct labelled asymptotes $\parallel Oy$ and approaches | B1 | |
| Crosses at $\left(\frac{3}{2}a, 0\right)$ (and $(0,0)$ — may be implied) | B1 | |
| $90°$ where it crosses $Ox$; smoothly | B1 | |
| Symmetry in $Ox$ | B1 | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{268b605f-eb86-40df-946a-210da1355e83-3_716_1431_852_356}
The diagram shows the curve with equation $y = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } }$, where $a$ is a positive constant.\\
(i) Find the equations of the asymptotes of the curve.\\
(ii) Sketch the curve with equation
$$y ^ { 2 } = \frac { 2 x ^ { 2 } - 3 a x } { x ^ { 2 } - a ^ { 2 } } .$$
State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.
\hfill \mbox{\textit{OCR FP2 2007 Q6 [8]}}