| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Stationary Points of Rational Functions |
| Difficulty | Standard +0.8 This FP1 question requires finding stationary points of a rational function by differentiating using the quotient rule, then analyzing when the discriminant of the resulting quadratic is negative. Part (ii) involves asymptote identification and curve sketching. While systematic, it demands careful algebraic manipulation across multiple steps and understanding of how discriminants relate to the existence of stationary points—more demanding than standard C1/C2 differentiation but typical for Further Maths. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y' = 0 \Rightarrow (x+1)(4x+k) - (2x^2 + kx) \times 1 = 0\) | 1 | M1 |
| \(\Rightarrow 4x^2 + (4+k)x + k - 2x^2 - kx = 0 \Rightarrow 2x^2 + 4x + k = 0\) | 1 | A1 |
| \(B^2 - 4AC < 0 \Rightarrow\) no stationary points \(\Rightarrow 16 - 8k < 0\) \(\Rightarrow k > 2\) for no stationary points | 3 | M1A1 A1 |
| Subtotal | 5 | |
| When \(k = 4\): Vertical asymptote: \(x = -1\) | 1 | B1 |
| Oblique asymptote: \(y = 2x + 2 - \dfrac{2}{x+1} \Rightarrow y = 2x + 2\) | 2 | M1A1 |
| Axes and asymptotes; Each branch | 3 | B1 B1B1 |
| Subtotal | 6 |
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y' = 0 \Rightarrow (x+1)(4x+k) - (2x^2 + kx) \times 1 = 0$ | 1 | M1 |
| $\Rightarrow 4x^2 + (4+k)x + k - 2x^2 - kx = 0 \Rightarrow 2x^2 + 4x + k = 0$ | 1 | A1 |
| $B^2 - 4AC < 0 \Rightarrow$ no stationary points $\Rightarrow 16 - 8k < 0$ $\Rightarrow k > 2$ for no stationary points | 3 | M1A1 A1 |
| **Subtotal** | **5** | |
| When $k = 4$: Vertical asymptote: $x = -1$ | 1 | B1 |
| Oblique asymptote: $y = 2x + 2 - \dfrac{2}{x+1} \Rightarrow y = 2x + 2$ | 2 | M1A1 |
| Axes and asymptotes; Each branch | 3 | B1 B1B1 |
| **Subtotal** | **6** | |
---8 The curve $C$ has equation $y = \frac { 2 x ^ { 2 } + k x } { x + 1 }$, where $k$ is a constant.\\
(i) Find the set of values of $k$ for which $C$ has no stationary points.\\
(ii) For the case $k = 4$, find the equations of the asymptotes of $C$ and sketch $C$, indicating the coordinates of the points where $C$ intersects the coordinate axes.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q8 [11]}}