| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 13 |
| Topic | Curve Sketching |
| Type | Rational functions with parameters: analysis depending on parameter sign/range |
| Difficulty | Standard +0.8 This is a multi-part question requiring asymptote analysis (vertical and oblique), differentiation using quotient rule, solving a quadratic for stationary points, and sketching curves with parameters for two cases. While systematic, it demands careful algebraic manipulation, understanding of rational function behavior, and synthesis of multiple techniques into coherent sketches—significantly above routine A-level but not requiring exceptional insight. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
**(i)** Vertical asymptote is $x = -\lambda$ B1
$y = x - \lambda + \frac{\lambda^2}{x+\lambda}$ M1
Oblique asymptote is $y = x - \lambda$ A1 **[3]**
**(ii)** For turning points $y' = 1 - \frac{\lambda^2}{(x+\lambda)^2} = 0$ M1A1
$x + \lambda = \pm\lambda \Rightarrow x = 0$ or $-2\lambda$
Turning points are $(0, 0)$ and $(-2\lambda, -4\lambda)$ A1A1 **[4]**
(If turning points are identified (on graph) with no working award B1 only)
(Differentiating a particular case, e.g. $\lambda = 1$, is M1A0A0)
**(iii)** First graph:
Axes and both asymptotes B1
Both branches B1B1
Second graph:
Axes and both asymptotes B1
Both branches B1B1 **[6]**
(If special cases, e.g. $\lambda = 1$, are drawn award max B2)
(Deduct at most 1 mark overall for bad forms at infinity)9 The curve $C$ has equation
$$y = \frac { x ^ { 2 } } { x + \lambda }$$
where $\lambda$ is a non-zero constant.\\
(i) Obtain the equation of each of the asymptotes of $C$.\\
(ii) Find the coordinates of the turning points of $C$.\\
(iii) In separate diagrams, sketch $C$ for the cases $\lambda > 0$ and $\lambda < 0$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q9 [13]}}