| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Rational Function Asymptotes |
| Difficulty | Standard +0.3 This is a standard Further Pure 1 rational function question requiring routine techniques: finding intercepts by substitution, identifying vertical asymptote from denominator, and polynomial division for oblique asymptote. While it's FP1 content, the methods are algorithmic with no novel problem-solving required, making it slightly easier than average overall. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((-6,0), (-1,0)\) | B1 | States points of intersection with \(x\)-axis |
| \((0,-3)\) | B1 | States \(y\)-intercept |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| One asymptote is \(x = 2\) | B1 | |
| \(y = x + 9 + \frac{24}{x-2} \Rightarrow\) other asymptote is \(y = x+9\) | M1 A1 | By inspection or long division. A0 if error in division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch showing curve with asymptotes | B1 | Sketches axes and asymptotes, labelled or to scale |
| B1 | Upper branch correctly located and orientated | |
| B1 | Lower branch correctly located and orientated. Penalise at most 1 mark for poor forms at infinity |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(-6,0), (-1,0)$ | B1 | States points of intersection with $x$-axis |
| $(0,-3)$ | B1 | States $y$-intercept |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| One asymptote is $x = 2$ | B1 | |
| $y = x + 9 + \frac{24}{x-2} \Rightarrow$ other asymptote is $y = x+9$ | M1 A1 | By inspection or long division. A0 if error in division |
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch showing curve with asymptotes | B1 | Sketches axes and asymptotes, labelled or to scale |
| | B1 | Upper branch correctly located and orientated |
| | B1 | Lower branch correctly located and orientated. Penalise at most 1 mark for poor forms at infinity |
---4 The curve $C$ has equation
$$y = \frac { x ^ { 2 } + 7 x + 6 } { x - 2 }$$
(i) Find the coordinates of the points of intersection of $C$ with the axes.\\
(ii) Find the equation of each of the asymptotes of $C$.\\
(iii) Sketch C.
\hfill \mbox{\textit{CAIE FP1 2018 Q4 [8]}}