| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Standard +0.8 This is a Further Maths question requiring systematic analysis of a rational function: finding intercepts, vertical/horizontal asymptotes, and asymptotic behavior for both positive and negative infinity. While each individual step uses standard techniques, the multi-part nature, the need to coordinate all information into a coherent sketch, and the asymptotic behavior analysis (part iii) requiring consideration of sign changes make this moderately challenging, above the typical A-level question but not requiring deep insight. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| \((-5,\ 0),\ (5,\ 0),\ \left(0,\ \frac{25}{24}\right)\) | B1, B1, B1 | \(-1\) for each additional point |
| Answer | Marks |
|---|---|
| \(x = 3,\ x = -4,\ x = -\frac{2}{3}\) and \(y = 0\) | B1, B1, B1, B1 |
| Answer | Marks |
|---|---|
| Some evidence of method needed e.g. substitute in 'large' values or argument involving signs | M1 |
| Large positive \(x\), \(y \to 0^+\) | B1 |
| Large negative \(x\), \(y \to 0^-\) | B1 |
| Answer | Marks |
|---|---|
| 4 branches correct | B1* |
| Asymptotic approaches clearly shown | B1dep* |
| Vertical asymptotes correct and labelled | B1 |
| Intercepts correct and labelled | B1 |
# Question 7:
## Part (i):
$(-5,\ 0),\ (5,\ 0),\ \left(0,\ \frac{25}{24}\right)$ | B1, B1, B1 | $-1$ for each additional point
**[3]**
## Part (ii):
$x = 3,\ x = -4,\ x = -\frac{2}{3}$ and $y = 0$ | B1, B1, B1, B1 |
**[4]**
## Part (iii):
Some evidence of method needed e.g. substitute in 'large' values or argument involving signs | M1 |
Large positive $x$, $y \to 0^+$ | B1 |
Large negative $x$, $y \to 0^-$ | B1 |
**[3]**
## Part (iv):
4 branches correct | B1* |
Asymptotic approaches clearly shown | B1dep* |
Vertical asymptotes correct and labelled | B1 |
Intercepts correct and labelled | B1 |
**[4]**
---7 A curve has equation $y = \frac { x ^ { 2 } - 25 } { ( x - 3 ) ( x + 4 ) ( 3 x + 2 ) }$.\\
(i) Write down the coordinates of the points where the curve crosses the axes.\\
(ii) Write down the equations of the asymptotes.\\
(iii) Determine how the curve approaches the horizontal asymptote for large positive values of $x$, and for large negative values of $x$.\\
(iv) Sketch the curve.
\hfill \mbox{\textit{OCR MEI FP1 2012 Q7 [14]}}