| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Rational curve sketching with asymptotes and inequalities |
| Difficulty | Standard +0.3 This is a standard Further Pure 1 question on rational functions requiring identification of asymptotes (straightforward from denominator zeros and degree comparison), basic limit behavior analysis, curve sketching, and solving a rational inequality. While it's a multi-part question worth several marks, each component follows routine procedures taught in FP1 with no novel problem-solving required. It's slightly above average difficulty due to being Further Maths content and requiring careful sign analysis for part (iv), but remains a textbook-style exercise. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = -\frac{7}{2}\), \(x = \frac{3}{2}\), \(y = 0\) | B1, B1, B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Large positive \(x\), \(y \to 0^+\) (e.g. \(x=100\)) | B1, B1 | |
| Large negative \(x\), \(y \to 0^-\) (e.g. \(x=-100\)) | M1 [3] | Evidence of method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Intercepts correct and labelled | B1 | |
| LH and central branches correct | B1 | |
| RH branch correct, with clear maximum | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x < -\frac{7}{2}\) | B1 | |
| or \(\frac{3}{2} < x \leq \frac{9}{5}\) | B2 [3] | Award B1 if only error relates to inclusive/exclusive inequalities |
# Question 7:
**Part (i):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = -\frac{7}{2}$, $x = \frac{3}{2}$, $y = 0$ | B1, B1, B1 **[3]** | |
**Part (ii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Large positive $x$, $y \to 0^+$ (e.g. $x=100$) | B1, B1 | |
| Large negative $x$, $y \to 0^-$ (e.g. $x=-100$) | M1 **[3]** | Evidence of method |
**Part (iii):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Intercepts correct and labelled | B1 | |
| LH and central branches correct | B1 | |
| RH branch correct, with clear maximum | B1 **[3]** | |
**Part (iv):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $x < -\frac{7}{2}$ | B1 | |
| or $\frac{3}{2} < x \leq \frac{9}{5}$ | B2 **[3]** | Award B1 if only error relates to inclusive/exclusive inequalities |
---7 A curve has equation $y = \frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) }$.\\
(i) Write down the equations of the two vertical asymptotes and the one horizontal asymptote.\\
(ii) Describe the behaviour of the curve for large positive and large negative values of $x$, justifying your answers.\\
(iii) Sketch the curve.\\
(iv) Solve the inequality $\frac { 5 x - 9 } { ( 2 x - 3 ) ( 2 x + 7 ) } \leqslant 0$.
\hfill \mbox{\textit{OCR MEI FP1 2010 Q7 [12]}}