Year: 2011
Paper: 2
Question Number: 11
Course: LFM Pure and Mechanics
Section: Vectors
No solution available for this problem.
There were just under 1000 entries for paper II this year, almost exactly the same number as last year. After the relatively easy time candidates experienced on last year's paper, this year's questions had been toughened up significantly, with particular attention made to ensure that candidates had to be prepared to invest more thought at the start of each question – last year saw far too many attempts from the weaker brethren at little more than the first part of up to ten questions, when the idea is that they should devote 25-40 minutes on four to six complete questions in order to present work of a substantial nature. It was also the intention to toughen up the final "quarter" of questions, so that a complete, or nearly-complete, conclusion to any question represented a significant (and, hopefully, satisfying) mathematical achievement. Although such matters are always best assessed with the benefit of hindsight, our efforts in these areas seem to have proved entirely successful, with the vast majority of candidates concentrating their efforts on four to six questions, as planned. Moreover, marks really did have to be earned: only around 20 candidates managed to gain or exceed a score of 100, and only a third of the entry managed to hit the half-way mark of 60. As in previous years, the pure maths questions provided the bulk of candidates' work, with relatively few efforts to be found at the applied ones. Questions 1 and 2 were attempted by almost all candidates; 3 and 4 by around three-quarters of them; 6, 7 and 9 by around half; the remaining questions were less popular, and some received almost no "hits". Overall, the highest scoring questions (averaging over half-marks) were 1, 2 and 9, along with 13 (very few attempts, but those who braved it scored very well). This at least is indicative that candidates are being careful in exercising some degree of thought when choosing (at least the first four) 'good' questions for themselves, although finding six successful questions then turned out to be a key discriminating factor of candidates' abilities from the examining team's perspective. Each of questions 4-8, 11 & 12 were rather poorly scored on, with average scores of only 5.5 to 6.6.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1487.5
Banger Comparisons: 3
Three non-collinear points $A$, $B$ and $C$ lie in
a horizontal ceiling. A particle $P$ of weight $W$
is suspended from this ceiling by means of three
light inextensible strings $AP$, $BP$ and $CP$,
as shown in the diagram. The point $O$ lies
vertically above $P$ in the ceiling.
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The angles $AOB$ and $AOC$ are $90^\circ+\theta$
and $90^\circ+\phi$, respectively, where $\theta$ and $\phi$
are acute angles such that $\tan\theta = \sqrt2$ and
$\tan\phi =\frac14\sqrt2$.
The strings $AP$, $BP$ and $CP$ make angles $30^\circ$, $90^\circ-\theta$
and $60^\circ$, respectively, with the vertical, and the tensions
in these strings have magnitudes $T$, $U$ and $V$ respectively.
\begin{questionparts}
\item
Show that the unit vector in the direction $PB$ can be written
in the form
\[
-\frac13\, {\bf i} - \frac{\sqrt2\,}3\, {\bf j} +
\frac{\sqrt2\, }{\sqrt3 \,} \,{\bf k}
\,,\]
where $\bf i\,$, $\, \bf j$ and $\bf k$ are the usual mutually perpendicular
unit vectors
with $\bf j$ parallel to $OA$ and $\bf k$ vertically upwards.
\item
Find expressions in vector form for the forces acting on $P$.
\item
Show that $U=\sqrt6 V$ and find $T$, $U$ and $V$ in terms of $W$.
\end{questionparts}
This was the least popular of all the questions on the paper, receiving under 40 "hits". The fact that it clearly involved both vectors and 3-dimensions was almost certainly responsible for the reluctance of candidates to give it a go. Those who managed to get past the initial stage of sorting out directions and components usually did very well, but most efforts foundered in the early stages. It was not helpful that some of these efforts confused angles to the vertical with those to the horizontal. Almost no-one verified that the given vector in (i) was indeed a unit vector.