Year: 2008
Paper: 2
Question Number: 9
Course: LFM Pure and Mechanics
Section: Projectiles
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: 1484.0
Banger Comparisons: 1
In this question, use $g=10\,$m\,s$^{-2}$.
In cricket, a fast bowler
projects a ball at $40\,$m\,s$^{-1}$ from a point $h\,$m above the ground,
which is horizontal, and at an angle $\alpha$ above the
horizontal.
The trajectory is such that the ball will
strike the stumps at ground level a horizontal distance
of $20\,$m from the
point of projection.
\begin{questionparts}
\item
Determine, in terms of $h$, the two possible values of $\tan\alpha$.
Explain which of these two values is the more appropriate one, and
deduce
that the ball hits the stumps after approximately half a second.
\item State the range of values of $h$ for which the bowler
projects the ball below the horizontal.
\item In the case $h=2.5$, give an approximate value in degrees,
correct to two significant figures, for
$\alpha$. You need not justify the accuracy of your
approximation.
\end{questionparts}
[You may use the small-angle approximations $\cos\theta \approx 1$ and
$\sin\theta\approx \theta$.]
Of the applied maths questions, this was by far the most popular, with over 400 attempts. However, most of these were only partial efforts, with few candidates even getting around to completing part (i) successfully, and the mean score ended up at about 8. Most candidates were comfortable with the routine stuff to start with, quoting and using the trajectory equation and using the identity sec²α = 1 + tan²α to get a quadratic equation in tanα. For the remaining parts of the question, working was much less certain, even given the helpful information about small-angle approximations, and very few candidates were able to get a suitable approximation for tanα. Fewer still could turn an angle in radians into one in degrees.