Year: 2023
Paper: 3
Question Number: 6
Course: UFM Pure
Section: Hyperbolic functions
No solution available for this problem.
The total entry was a marginal increase on that of 2022 (by just over 1%). Two questions were attempted by more than 90% of candidates, another two by 80%, and another two by about two thirds. The least popular questions were attempted by more than a sixth of candidates. All the questions were perfectly answered by at least three candidates (but mostly more than this), with one being perfectly answered by eighty candidates. Very nearly 90% of candidates attempted no more than 7 questions. One general comment regarding all the questions is that candidates need to make sure that they read the question carefully, paying particular attention to command words such as "hence" and "show that".
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item By considering the Maclaurin series for $\mathrm{e}^x$, show that for all real $x$,
\[\cosh^2 x \geqslant 1 + x^2.\]
Hence show that the function $\mathrm{f}$, defined for all real $x$ by $\mathrm{f}(x) = \tan^{-1} x - \tanh x$, is an increasing function.
Sketch the graph $y = \mathrm{f}(x)$.
\item Function $\mathrm{g}$ is defined for all real $x$ by $\mathrm{g}(x) = \tan^{-1} x - \frac{1}{2}\pi \tanh x$.
\begin{enumerate}
\item Show that $\mathrm{g}$ has at least two stationary points.
\item Show, by considering its derivative, that $(1+x^2)\sinh x - x\cosh x$ is non-negative for $x \geqslant 0$.
\item Show that $\dfrac{\cosh^2 x}{1+x^2}$ is an increasing function for $x \geqslant 0$.
\item Hence or otherwise show that $\mathrm{g}$ has exactly two stationary points.
\item Sketch the graph $y = \mathrm{g}(x)$.
\end{enumerate}
\end{questionparts}
The fourth most popular question being attempted by just over three quarters of the candidates, it was the second most successful with a mean of just over half marks. The best responses involved clear algebra and working, with the given results fully justified. Many candidates picked up marks by accurate differentiation, whilst the best candidates were able to sketch graphs showing all the main features and could carefully justify results. Many parts of the question asked candidates to show a given result, which meant that candidates needed to ensure they showed sufficient working before reaching the given result. In part (i) and part (ii)(b) candidates were required to use a specified method; candidates who did not use this method did not gain all the marks. There were some good answers to part (ii), but many candidates failed to show a stationary point of inflection at the origin, possibly as they assumed that they had shown the graph was strictly increasing rather than increasing. Failing to show the asymptote limits was another common mistake. Part (ii) (a) was found to be the hardest. Many candidates did not justify their results (such as the behaviour of g(x) or g'(x) as x tends to infinity). Some candidates drew graphs to help justify their result, but these generally did not explain why their graphs looked as they did. However, part (ii)(b) was generally done well, as was (c) by those that attempted it. Many candidates failed to gain the marks in part (d), mainly through failing to consider the symmetry of cosh²x/(1+x²). Candidates found the graph in part (e) easier to sketch than the one in part (i). The most common mistake here was to have the graph reflected, with g(x) positive when x is positive, incorrectly.