Year: 2010
Paper: 2
Question Number: 7
Course: LFM Stats And Pure
Section: Polynomials
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. Of this number, more than 60 scored over 90% while, at the other end of the scale, almost 200 failed to score more than 40 marks. In hindsight, many of the pure maths questions were a little too accessible and lacked a sufficiently tough 'difficulty gradient', so that scores were slightly higher than anticipated. This was reflected in the grade boundaries for the "1" and the "2" (around ten marks higher than is generally planned) in particular. Next year's questions may be expected to be a little bit more demanding, but only in the sense that the final 5 or 6 marks on each question should have rather more bite to them: it should certainly not be the case that all questions are tougher to get into at the outset. Most candidates attempted the requisite number of questions (six), although many of the weaker brethren made seven or eight attempts, most of which were feeble at best and they generally only picked up a maximum of 5 or 6 marks per question. It is a truth universally acknowledged that practice maketh if not perfect then at least a whole lot better prepared, and choosing to waste time on a couple of extra questions is not a good strategy on the STEPs. The major down-side of the present modular examination system is that students are not naturally prepared to approach the subject holistically; ally this to the current practice of setting highly-structured, fully-guided questions requiring no imagination, insight, depth or planning from A-level candidates in a system that fails almost nobody and rewards even the most modestly able with high grades in a manner reminiscent of a dentist giving lollipops to kids who have done little more than been brave and seen the course through, it is even more important to ensure a full and thorough preparation for these papers. The 20% of the entry who seem to be either unprepared for the rigours of a STEP, or unwittingly possessed of only a smattering of basic advanced-level skills, seems to be remarkably steady year-on-year, even in a year when their more suitably prepared compatriots found the paper appreciably easier than usual. 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.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
\begin{questionparts}
\item
By considering the positions of its turning points, show that the curve
with equation
\[
y=x^3-3qx-q(1+q)\,,
\]
where $q>0$ and $q\ne1$, crosses the $x$-axis once only.
\item
Given that $x$ satisfies the cubic equation
\[
x^3-3qx-q(1+q)=0\,,
\]
and that
\[
x=u+q/u\,,
\]
obtain a quadratic equation
satisfied by $u^3$.
Hence find the real root of the cubic equation in the case $q>0$, $q\ne1$.
\item The quadratic equation
\[
t^2 -pt +q =0\,
\]
has roots $\alpha $ and $\beta$. Show that
\[
\alpha^3+\beta^3 = p^3 -3qp\,.
\]
It is given that one of these roots is the square of the other.
By considering the expression $(\alpha^2 -\beta)(\beta^2-\alpha)$,
find a relationship between $p$ and $q$.
Given further that $q>0$, $q\ne1$ and $p$ is real,
determine the value of $p$ in terms of $q$.
\end{questionparts}
This proved to be the second most popular question on the paper, both by choice and by success. I imagine that its helpful structure probably contributed significantly to both. Part of the problem is that there are ways to do this using methods not on single maths specifications, so it was necessary to be quite specific. Nonetheless, there were still areas where marks were commonly lost; in (i), candidates were required to show that both TPs lie below the x-axis and, while one of the y-coordinates was obviously negative (being the sum of three negative terms), the other one was only obviously so by completing the square. The problems found by candidates, even in the first case, just highlights the widespread difficulty found by students when dealing with inequalities.