| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Rational Function Asymptotes |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring multiple techniques: finding vertical and oblique asymptotes (via polynomial division), axis intercepts, differentiation using quotient rule, and synthesizing all information into a coherent sketch. While each individual step is standard, the combination and the oblique asymptote (rather than horizontal) elevates this above typical A-level questions to moderately challenging Further Maths territory. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x = 1\) | B1 | |
| \(y = \frac{x^2-3}{x-1} = \frac{(x-1)(x+1)-2}{x-1} = x+1\left[-\frac{2}{x-1}\right]\) | M1 | Or long division with quotient \(x+\ldots\) |
| \(\Rightarrow y = x+1\) | A1 | Must be stated |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((0,3)\), \((\sqrt{3},0)\) and \((-\sqrt{3},0)\) | B1 | All three; allow when \(x=0\), \(y=3\) etc but do NOT allow \(y=3\) etc |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{2x(x-1)-(x^2-3)}{(x-1)^2} = \frac{x^2-2x+3}{(x-1)^2}\) | M1, A1 | Differentiate; Gradient function |
| \(= \frac{(x-1)^2+2}{(x-1)^2} > 0\) for all \(x\), so no turning points | A1 | Conclusion; Alternative: \(\frac{dy}{dx} = 1 + \frac{2}{(x-1)^2} > 1\) so no turning points. Or "\(b^2-4ac\)"\(= -8 < 0\) so no roots |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct shape going through axes at correct points | B1 | Allow omission of \((0,3)\) if not in (ii) |
| Correct asymptotes included | B1 | Oblique asymptote can be \(y=x+c\) with \(c \neq 1\) |
| Approaches correct asymptotes correctly | B1 | |
| [3] |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x = 1$ | B1 | |
| $y = \frac{x^2-3}{x-1} = \frac{(x-1)(x+1)-2}{x-1} = x+1\left[-\frac{2}{x-1}\right]$ | M1 | Or long division with quotient $x+\ldots$ |
| $\Rightarrow y = x+1$ | A1 | Must be stated |
| **[3]** | | |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(0,3)$, $(\sqrt{3},0)$ and $(-\sqrt{3},0)$ | B1 | All three; allow when $x=0$, $y=3$ etc but do NOT allow $y=3$ etc |
| **[1]** | | |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = \frac{2x(x-1)-(x^2-3)}{(x-1)^2} = \frac{x^2-2x+3}{(x-1)^2}$ | M1, A1 | Differentiate; Gradient function |
| $= \frac{(x-1)^2+2}{(x-1)^2} > 0$ for all $x$, so no turning points | A1 | Conclusion; Alternative: $\frac{dy}{dx} = 1 + \frac{2}{(x-1)^2} > 1$ so no turning points. Or "$b^2-4ac$"$= -8 < 0$ so no roots |
| **[3]** | | |
### Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct shape going through axes at correct points | B1 | Allow omission of $(0,3)$ if not in (ii) |
| Correct asymptotes included | B1 | Oblique asymptote can be $y=x+c$ with $c \neq 1$ |
| Approaches correct asymptotes correctly | B1 | |
| **[3]** | | |
---2 The equation of a curve is $y = \frac { x ^ { 2 } - 3 } { x - 1 }$.\\
(i) Find the equations of the asymptotes of the curve.\\
(ii) Write down the coordinates of the points where the curve cuts the axes.\\
(iii) Show that the curve has no stationary points.\\
(iv) Sketch the curve and the asymptotes.
\hfill \mbox{\textit{OCR FP2 2013 Q2 [10]}}