| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Stationary Points of Rational Functions |
| Difficulty | Standard +0.8 This Further Pure 1 question requires finding asymptotes (vertical and oblique via polynomial division), differentiating a rational function using the quotient rule, solving a quadratic equation for stationary points, and producing an accurate sketch integrating all features. While systematic, it demands multiple techniques and careful algebraic manipulation across several parts, placing it moderately above average difficulty. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = \frac{1}{k}\) | B1 | States vertical asymptote |
| \(y = \frac{(kx-1)(k^{-1}x + k^{-2}) + k^{-2}}{kx-1}\) | M1 | Finds oblique asymptote |
| Oblique asymptote is \(y = k^{-1}x + k^{-2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y' = \frac{2x(kx-1) - kx^2}{(kx-1)^2} = 0 \Rightarrow kx^2 - 2x = 0\) | M1 | Differentiates and equates to 0 |
| \(x = 0,\ 2k^{-1}\) | A1 | Finds \(x\)-coordinates |
| \((0,0),\ \left(2k^{-1}, 4k^{-2}\right)\) | A1 | Finds \(y\)-coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Graph with correct axes and asymptotes | B1 | Axes and asymptotes correct |
| Upper branch correct | B1 | Upper branch correct |
| Lower branch correct | B1 | Lower branch correct; deduct at most 1 mark for poor forms at infinity |
## Question 6(i):
| $x = \frac{1}{k}$ | B1 | States vertical asymptote |
| $y = \frac{(kx-1)(k^{-1}x + k^{-2}) + k^{-2}}{kx-1}$ | M1 | Finds oblique asymptote |
| Oblique asymptote is $y = k^{-1}x + k^{-2}$ | A1 | |
## Question 6(ii):
| $y' = \frac{2x(kx-1) - kx^2}{(kx-1)^2} = 0 \Rightarrow kx^2 - 2x = 0$ | M1 | Differentiates and equates to 0 |
| $x = 0,\ 2k^{-1}$ | A1 | Finds $x$-coordinates |
| $(0,0),\ \left(2k^{-1}, 4k^{-2}\right)$ | A1 | Finds $y$-coordinates |
## Question 6(iii):
| Graph with correct axes and asymptotes | B1 | Axes and asymptotes correct |
| Upper branch correct | B1 | Upper branch correct |
| Lower branch correct | B1 | Lower branch correct; deduct at most 1 mark for poor forms at infinity |6 The curve $C$ has equation
$$y = \frac { x ^ { 2 } } { k x - 1 }$$
where $k$ is a positive constant.\\
(i) Obtain the equations of the asymptotes of $C$.\\
(ii) Find the coordinates of the stationary points of $C$.\\
(iii) Sketch $C$.
\hfill \mbox{\textit{CAIE FP1 2019 Q6 [9]}}