Year: 2009
Paper: 1
Question Number: 10
Course: LFM Pure and Mechanics
Section: Motion on a slope
No solution available for this problem.
There were significantly more candidates attempting this paper again this year (over 900 in total), and the scores were pleasing: fewer than 5% of candidates failed to get at least 20 marks, and the median mark was 48. The pure questions were the most popular as usual; about two-thirds of candidates attempted each of the pure questions, with the exceptions of question 2 (attempted by about 90%) and question 5 (attempted by about one third). The mechanics questions were only marginally more popular than the probability and statistics questions this year; about one quarter of the candidates attempted each of the mechanics questions, while the statistics questions were attempted by about one fifth of the candidates. A significant number of candidates ignored the advice on the front cover and attempted more than six questions. In general, those candidates who submitted answers to eight or more questions did fairly poorly; very few people who tackled nine or more questions gained more than 60 marks overall (as only the best six questions are taken for the final mark). This suggests that a skill lacking in many students attempting STEP is the ability to pick questions effectively. This is not required for A-levels, so must become an important part of STEP preparation. Another "rubric"-type error was failing to follow the instructions in the question. In particular, when a question says "Hence", the candidate must make (significant) use of the preceding result(s) in their answer if they wish to gain any credit. In some questions (such as question 2), many candidates gained no marks for the final part (which was worth 10 marks) as they simply quoted an answer without using any of their earlier work. There were a number of common errors which appeared across the whole paper. These included a noticeable weakness in algebraic manipulations, sometimes indicating a serious lack of understanding of the mathematics involved. As examples, one candidate tried to use the misremembered identity cos β = sin √(1 − β²), while numerous candidates made deductions of the form "if a² + b² = c², then a + b = c" at some point in their work. Fraction manipulations are also notorious in the school classroom; the effects of this weakness were felt here, too. Another common problem was a lack of direction; writing a whole page of algebraic manipulations with no sense of purpose was unlikely to either reach the requested answer or gain the candidate any marks. It is a good idea when faced with a STEP question to ask oneself, "What is the point of this (part of the) question?" or "Why has this (part of the) question been asked?" Thinking about this can be a helpful guide. One aspect of this is evidenced by pages of formulæ and equations with no explanation. It is very good practice to explain why you are doing the calculation you are, and to write sentences in English to achieve this. It also forces one to focus on the purpose of the calculations, and may help avoid some dead ends. Finally, there is a tendency among some candidates when short of time to write what they would do at this point, rather than using the limited time to actually try doing it. Such comments gain no credit; marks are only awarded for making progress in a question. STEP questions do require a greater facility with mathematics and algebraic manipulation than the A-level examinations, as well as a depth of understanding which goes beyond that expected in a typical sixth-form classroom.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
A triangular wedge is fixed to a horizontal surface. The
base angles
of the wedge are $\alpha$ and $\frac\pi 2-\alpha$.
Two particles, of masses $M$ and $m$, lie on different faces of
the wedge, and are connected by a light inextensible string
which passes over a smooth pulley at the apex of the wedge, as
shown in the diagram.
The contacts between the particles and the wedge are smooth.
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\begin{questionparts}
\item
Show that if $\tan \alpha> \dfrac m M $
the particle of mass $M$ will slide down the face of the wedge.
\item
Given that $\tan \alpha = \dfrac{2m}M$, show that the magnitude of the
acceleration of the
particles is
\[
\frac{g\sin\alpha}{\tan\alpha +2}
\]
and that this is maximised at
$4m^3=M^3\,$.
\end{questionparts}
This was (marginally) the most popular of the mechanics questions, attempted by close to one third of the candidates. Those who attempted it generally gained reasonably good marks, and there were a few very pleasing answers. In the first part, however, there was a clear lack of understanding among the candidates of the basic principles of mechanics. While they showed a good understanding of resolving forces, the once commonplace mantra of "Apply Newton's Second Law to each particle, then combine as needed" seemed to be all but forgotten: too few candidates indicated the tension in the string on their diagram. Many therefore treated the two particles as a single system which could be regarded as one particle with various forces, but offered no justification for this assertion. In this case, such an approach does work, but had the pulley not been smooth, it would not have done. Furthermore, applying Newton's Second Law at each particle is a much more straightforward, less error-prone approach, and may have saved a number of candidates from unfortunate sign errors. Those who managed to write down a correct equation of motion were generally quite confident to show the required inequality. However, most fudged the argument, saying things like "the force on M is bigger than the force on m so Mg sin α > mg cos α," which does not show an appreciation of the effect of tension. The trigonometric manipulations required at the start of part (ii) were done fairly well, and those who got this far were very comfortable differentiating the expression using the quotient rule. However, having done so, very few were then able to perform the requisite manipulations on the derivative to reach the result that tan³ α = 2, and hence deduce the relationship between M and m. It was also disappointing that of those good candidates who had reached this point, only three realised that they needed to justify that this was a (local) maximum; it should be standard for students to ascertain the nature of any stationary point they have found if they require the maximum or minimum of a function.