Year: 2008
Paper: 2
Question Number: 7
Course: LFM Pure
Section: Differential equations
No solution available for this problem.
There were around 850 candidates for this paper – a slight increase on the 800 of the past two years – and the scripts received covered the full range of marks (and beyond!). The questions on this paper in recent years have been designed to be a little more accessible to all top A-level students, and this has been reflected in the numbers of candidates making good attempts at more than just a couple of questions, in the numbers making decent stabs at the six questions required by the rubric, and in the total scores achieved by candidates. Most candidates made attempts at five or more questions, and most genuinely able mathematicians would have found the experience a positive one in some measure at least. With this greater emphasis on accessibility, it is more important than ever that candidates produce really strong, essentially-complete efforts to at least four questions. Around half marks are required in order to be competing for a grade 2, and around 70 for a grade 1. The range of abilities on show was still quite wide. Just over 100 candidates failed to score a total mark of at least 30, with a further 100 failing to reach a total of 40. At the other end of the scale, more than 70 candidates scored a mark in excess of 100, and there were several who produced completely (or nearly so) successful attempts at more than six questions; if more than six questions had been permitted to contribute towards their paper totals, they would have comfortably exceeded the maximum mark of 120. While on the issue of the "best-six question-scores count" rubric, almost a third of candidates produced efforts at more than six questions, and this is generally a policy not to be encouraged. In most such cases, the seventh, eighth, or even ninth, question-efforts were very low scoring and little more than a waste of time for the candidates concerned. Having said that, it was clear that, in many of these cases, these partial attempts represented an abandonment of a question after a brief start, with the candidates presumably having decided that they were unlikely to make much successful further progress on it, and this is a much better employment of resources. As in recent years, most candidates' contributing question-scores came exclusively from attempts at the pure maths questions in Section A. Attempts at the mechanics and statistics questions were very much more of a rarity, although more (and better) attempts were seen at these than in other recent papers.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1472.0
Banger Comparisons: 2
\begin{questionparts}
\item By writing $y=u{(1+x^2)\vphantom{\dot A}}^{\frac12}$,
where $u$ is a function of $x$,
find the solution of the equation
\[
\frac 1 y \frac{\d y} {\d x} = xy + \frac x {1+x^2}
\]
for which $y=1$ when $x=0$.
\item Find the solution of the equation
\[
\frac 1 y \frac{\d y} {\d x} = x^2y + \frac {x^2 } {1+x^3}
\]
for which $y=1$ when $x=0$.
\item Give, without proof, a conjecture for
the solution of the equation
\[
\frac 1 y \frac{\d y} {\d x} = x^{n-1}y + \frac {x^{n-1} } {1+x^n}
\]
for which $y=1$ when $x=0$, where $n$ is an integer greater than 1.
\end{questionparts}
In many ways, part (i) of the question was very routine, requiring little more than technical competence to see the differential equation, using the given substitution, through to a correct solution. Part (ii) then required candidates to spot a slightly different substitution on the basis of having gained a feel for what had gone on previously. I had thought that many more candidates would try something involving the square root of 1 + x³ or the cube root of 1 + x², rather than cube root of 1 + x³, but many solutions that I saw went straight for the right thing. Once this had been successfully pushed through – with the working mimicking that of (i) very closely indeed – it was not difficult to spot the general answer required, unproven, in (iii). Overall, however, it seems that a lot of candidates failed to spot the right thing for part (ii) and their solutions stopped at this point. With almost 600 attempts, the mean score on this question was 10.