Year: 2007
Paper: 2
Question Number: 5
Course: LFM Pure
Section: Trigonometry 2
Although the paper was by no means an easy one, it was generally found a more accessible paper than last year's, with most questions clearly offering candidates an attackable starting-point. The candidature represented the usual range of mathematical talents, with a pleasingly high number of truly outstanding students; many more who were able to demonstrate a thorough grasp of the material in at least three questions; and the few whose three-hour long experience was unlikely to have been a particularly pleasant one. However, even for these candidates, many were able to make some progress on at least two of the questions chosen. Really able candidates generally produced solid attempts at five or six questions, and quite a few produced outstanding efforts at up to eight questions. In general, it would be best if centres persuaded candidates not to spend valuable time needlessly in this way – it is a practice that is not to be encouraged, as it uses valuable examination time to little or no avail. Weaker brethren were often to be found scratching around at bits and pieces of several questions, with little of substance being produced on more than a couple. It is an important examination skill – now more so than ever, with most candidates now not having to employ such a skill on the modular papers which constitute the bulk of their examination experience – for candidates to spend a few minutes at some stage of the examination deciding upon their optimal selection of questions to attempt. As a rule, question 1 is intended to be accessible to all takers, with question 2 usually similarly constructed. In the event, at least one – and usually both – of these two questions were among candidates' chosen questions. These, along with questions 3 and 6, were by far the most popularly chosen questions to attempt. The majority of candidates only attempted questions in Section A (Pure Maths), and there were relatively few attempts at the Applied Maths questions in Sections B & C, with Mechanics proving the more popular of the two options. It struck me that, generally, the working produced on the scripts this year was rather better set-out, with a greater logical coherence to it, and this certainly helps the markers identify what each candidate thinks they are doing. Sadly, this general remark doesn't apply to the working produced on the Mechanics questions, such as they were. As last year, the presentation was usually appalling, with poorly labelled diagrams, often with forces missing from them altogether, and little or no attempt to state the principles that the candidates were attempting to apply.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1488.1
Banger Comparisons: 5
In this question,
$\f^2(x)$ denotes $\f(\f(x))$, $\f^3(x)$ denotes $\f( \f (\f(x)))\,$,
and so on.
\begin{questionparts}
\item
The function $\f$ is defined, for $x\ne \pm 1/ \sqrt3\,$,
by
$$ \f(x) = \ds \frac{x+\sqrt3} {1-\sqrt3\, x }\,.
$$
Find by direct calculation $\f^2(x) $ and $\f^3(x)$, and determine
$\f^{2007}(x)\,$.
\item Show that
$\f^n(x) = \tan(\theta + \frac 13 n\pi)$, where $x=\tan\theta$
and $n$ is any positive integer.
\item The function $\g(t)$ is defined, for $\vert t\vert\le1$ by
$\g(t) = \frac {\sqrt3}2 t + \frac 12 \sqrt {1-t^2}\,$.
Find an expression for $\g^n(t)$ for any positive integer $n$.
\end{questionparts}\begin{questionparts}
\item $\,$ \begin{align*}
&& f(x) &= \frac{x+\sqrt3}{1-\sqrt3x} \\
\Rightarrow && f(f(x)) &= \frac{f(x)+\sqrt3}{1-\sqrt3f(x)} \\
&&&= \frac{\frac{x+\sqrt3}{1-\sqrt3x}+\sqrt3}{1-\sqrt3 \frac{x+\sqrt3}{1-\sqrt3x}} \\
&&&= \frac{x+\sqrt{3}+\sqrt3(1-\sqrt3x)}{1-\sqrt3x-\sqrt3(x+\sqrt3)} \\
&&&= \frac{-2x+2\sqrt3}{-2-2\sqrt3x} \\
&&&= \frac{x-\sqrt3}{1+\sqrt3 x} \\
\\
&& f^3(x) &= f^2(f(x)) \\
&&&= \frac{f(x)-\sqrt3}{1+\sqrt3 f(x)} \\
&&&=\frac{\frac{x+\sqrt3}{1-\sqrt3x}-\sqrt3}{1+\sqrt3 \frac{x+\sqrt3}{1-\sqrt3x}} \\
&&&= \frac{(x+\sqrt3)-\sqrt3(1-\sqrt3 x)}{(1-\sqrt3x)+\sqrt3 (x+\sqrt3)} \\
&&&= \frac{-2x}{-2} = x \\
\\
&& f^{2007}(x) &= x
\end{align*}
\item If $x = \tan \theta$ then $f(x) = \frac{\tan \theta + \tan \frac{\pi}{3}}{1 - \tan \frac{\pi}{3} \tan \theta} = \tan (\theta + \frac{\pi}{3})$ and hence $f^n(x) = \tan (\theta + \frac{n \pi}{3})$
\item Note that if $t = \sin \theta$ then $g(t) = \cos \frac{\pi}{6} t\sin \theta + \frac12 \cos \theta = \sin(\theta + \frac{\pi}6)$ therefore $g^n(t) = \sin(\sin^{-1}(t) + \frac{n\pi}{6})$
\end{questionparts}
Although this was not a popular choice of question, those who attempted it generally did rather well on it. Finding f2 and f3 was a routine algebraic slog, and most attempts coped successfully with it. Spotting, and then exploiting, the periodicity of the function was then a relatively easy matter. Pretty much everyone used x = tan θ appropriately in (ii), with formal and informal induction approaches evenly mixed. Some shrewder candidates identified the two forms for the cases n = 1, 2, and 3 and then noted that the periodicity of the tan function accounted for everything thereafter. The final part of the question had intended to be a simple take on part (ii), but with t = sin θ this time, so that √(1 − t²) = cos θ, and attempts at this part of the question generally fell evenly into one of the two following camps: those who gave up, and those who proceeded as intended. In all, I think there were just three candidates who noticed the extra complication that can arise in this case, with just two or three more following a separate line of enquiry without realising the inherent dichotomy in the "powers" of the function g. A full inspection of the function exposes the fact that gn takes different forms depending upon which part of the domain of g is employed. This is because the √(1 − t²) bit should actually be |cos θ|, and this leads to different answers for g2 in the range ½ ≤ t ≤ 1 than in the rest of g's domain, so that candidates could get different answers from slightly different approaches. With so few candidates expected to attempt this last part of the question, and with the alternate route leading to a much easier answer (where the sequence gn turns out to be periodic with period 2), it was considered to be a suitable final part to the question. Candidates were not expected to take more than one route, nor to comment on the potential for different answers. In the event, none did the former, although a few gave a mention of the latter property.