Year: 2011
Paper: 2
Question Number: 5
Course: UFM Pure
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: 1484.0
Banger Comparisons: 1
The points $A$ and $B$ have position vectors $\bf a $ and $\bf b$
with respect to an origin $O$, and $O$, $A$~and~$B$ are non-collinear.
The point $C$, with position vector $\bf c$,
is the reflection of $B$ in the line through
$O$ and $A$. Show that $\bf c$ can be written in the
form
\[
\bf c = \lambda \bf a -\bf b
\]
where $\displaystyle \lambda = \frac{2\,{\bf a .b}}{{\bf a.a}}$.
The point $D$, with position vector $\bf d$, is the reflection of $C$ in
the line through $O$ and $B$.
Show that $\bf d$ can be written in the form
\[
\bf d = \mu\bf b - \lambda \bf a
\]
for some scalar $\mu$ to be determined.
Given that $A$, $B$ and $D$ are collinear, find the relationship
between $\lambda$ and $\mu$. In the case $\lambda = -\frac12$, determine
the cosine of $\angle AOB$ and describe the relative positions
of $A$, $B$ and $D$.
This vectors question was neither popular nor successful overall. For the most part this seemed to be due to the fact that candidates, although they are happy to work with scalar parameters – as involved in the vector equation of a line, for instance – they are far less happy to interpret them geometrically. Many other students clearly dislike non-numerical vector questions. Having said that, attempts generally fell into one of the two extreme camps of 'very good' or 'very poor'. More confident candidates managed the first result and realised that a "similarity" approach killed off the second part also, although efforts to tidy up answers were frequently littered with needless errors that came back to penalise the candidates when they attempted to use them later on. Many candidates noted that D was between A and B, but failed to realise it was actually the midpoint of AB. In the very final part, it was often the case that candidates overlooked the negative sign of cos, even when the remainder of their working was broadly correct.