| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This question involves standard curve analysis techniques: finding asymptotes and intercepts (routine), then solving a quadratic condition to find range restrictions and using this to locate turning points. While part (ii) requires connecting the discriminant condition to the range of the function, this is a well-established A-level technique. The algebraic manipulation is straightforward, and the insight required (discriminant ≥ 0 for real solutions) is standard curriculum content, making this slightly easier than average overall. |
| Spec | 1.02p Interpret algebraic solutions: graphically1.07n Stationary points: find maxima, minima using derivatives |
The curve $C$ has equation $y = \frac{2x}{x^2 + 1}$.
\begin{enumerate}[label=(\roman*)]
\item Write down the equation of the asymptote of $C$ and the coordinates of any points where $C$ meets the coordinate axes. [2]
\item Show that the curve meets the line $y = k$ if and only if $-1 \leqslant k \leqslant 1$. Deduce the coordinates of the turning points of the curve. [5]
\end{enumerate}
[Note: You are NOT required to sketch $C$.]
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2013 Q3 [7]}}