2007 Paper 2 Q2
Year: 2007
Paper: 2
Question Number: 2
Course: LFM Stats And Pure
Section: Curve Sketching
Problem
- Express \(b\) and \(c\) in terms of \(p\) and \(q\).
- Sketch the curve. Mark on your sketch the point of inflection and shade the region \(R\). Describe the symmetry of the curve.
- Show that \(m-n=(q-p)^3\).
- Show that the area of \(R\) is \(\frac12 (q-p)^4\).
Solution
- \(\,\) \begin{align*} && y &= 2x^3-bx^2+cx \\ \Rightarrow && y' &= 6x^2-2bx+c \end{align*} We must have \(p, q\) are the roots of this equation, ie \(\frac13b = p+q, \frac16c = pq\)
- The point of inflection will be at \(\frac{b}6\)
The graph will have rotational symmetry of \(180^{\circ}\) about the point of inflection.
- \begin{align*} && m-n &= 2(p^3-q^3)-b(p^2-q^2)+c(p-q) \\ &&&= (p-q)(2(p^2+qp+q^2)-b(p+q)+c) \\ &&&= (p-q)(2(p^2+qp+q^2)-3(p+q)^2+6pq) \\ &&&= (p-q)(-p^2-q^2+2pq) \\ &&&= (q-p)^3 \end{align*}
- The area of \(R\) is \begin{align*} A &= \frac12 bh \\ &= \frac12 (q-p)(m-n) = \frac12(q-p)^4 \end{align*} as required.
Examiner's report
— 2007 STEP 2, Question 2Although the paper was by no means an easy one, it was generally found a more accessible paper than last year's, with most questions clearly offering candidates an attackable starting-point. The candidature represented the usual range of mathematical talents, with a pleasingly high number of truly outstanding students; many more who were able to demonstrate a thorough grasp of the material in at least three questions; and the few whose three-hour long experience was unlikely to have been a particularly pleasant one. However, even for these candidates, many were able to make some progress on at least two of the questions chosen. Really able candidates generally produced solid attempts at five or six questions, and quite a few produced outstanding efforts at up to eight questions. In general, it would be best if centres persuaded candidates not to spend valuable time needlessly in this way – it is a practice that is not to be encouraged, as it uses valuable examination time to little or no avail. Weaker brethren were often to be found scratching around at bits and pieces of several questions, with little of substance being produced on more than a couple. It is an important examination skill – now more so than ever, with most candidates now not having to employ such a skill on the modular papers which constitute the bulk of their examination experience – for candidates to spend a few minutes at some stage of the examination deciding upon their optimal selection of questions to attempt. As a rule, question 1 is intended to be accessible to all takers, with question 2 usually similarly constructed. In the event, at least one – and usually both – of these two questions were among candidates' chosen questions. These, along with questions 3 and 6, were by far the most popularly chosen questions to attempt. The majority of candidates only attempted questions in Section A (Pure Maths), and there were relatively few attempts at the Applied Maths questions in Sections B & C, with Mechanics proving the more popular of the two options. It struck me that, generally, the working produced on the scripts this year was rather better set-out, with a greater logical coherence to it, and this certainly helps the markers identify what each candidate thinks they are doing. Sadly, this general remark doesn't apply to the working produced on the Mechanics questions, such as they were. As last year, the presentation was usually appalling, with poorly labelled diagrams, often with forces missing from them altogether, and little or no attempt to state the principles that the candidates were attempting to apply.
Rating Information
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Show LaTeX source
Problem source
A curve has equation $y=2x^3-bx^2+cx$. It has a maximum point at $(p,m)$ and a minimum point at $(q,n)$ where $p>0$ and $n>0$.
Let $R$ be the region enclosed by the curve, the line $x=p$ and the line $y=n$.
\begin{questionparts}
\item Express $b$ and $c$ in terms of $p$ and $q$.
\item Sketch the curve. Mark on your sketch the point of inflection and shade the region $R$. Describe the symmetry of the curve.
\item Show that $m-n=(q-p)^3$.
\item Show that the area of $R$ is $\frac12 (q-p)^4$.
\end{questionparts}Solution source
\begin{questionparts}
\item $\,$ \begin{align*}
&& y &= 2x^3-bx^2+cx \\
\Rightarrow && y' &= 6x^2-2bx+c
\end{align*}
We must have $p, q$ are the roots of this equation, ie $\frac13b = p+q, \frac16c = pq$
\item The point of inflection will be at $\frac{b}6$
The graph will have rotational symmetry of $180^{\circ}$ about the point of inflection.
\begin{center}
\begin{tikzpicture}
\def\functionf(#1){2*(#1)^3-5.3*(#1)^2+3.6*(#1)};
\def\xl{-.5};
\def\xu{2.5};
\def\yl{-2};
\def\yu{3};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{10/\xrange}
\pgfmathsetmacro{\yscale}{10/\yrange}
% Define the styles for the axes and grid
\tikzset{
axis/.style={very thick, ->},
grid/.style={thin, gray!30},
x=\xscale cm,
y=\yscale cm
}
% Define the bounding region with clip
\begin{scope}
% You can modify these values to change your plotting region
\clip (\xl,\yl) rectangle (\xu,\yu);
% Draw a grid (optional)
% \draw[grid] (-5,-3) grid (5,3);
\draw[thick, blue, smooth, domain=\xl:\xu, samples=201]
plot ({\x}, {\functionf(\x)});
\filldraw[opacity=0.25, blue, domain=0.458742:1.30792, samples=201]
(0.458742, {\functionf(1.30792)}) -- plot ({\x}, {\functionf(\x)});
\filldraw (0.458742, {\functionf(0.458742)}) circle (1.5pt) node[above] {$(p,m)$};
\filldraw (1.30792, {\functionf(1.30792)}) circle (1.5pt) node[below] {$(q,n)$};
\filldraw ({5.3/6}, {\functionf(5.3/6)}) circle (1.5pt) node[above, right] {$(\tfrac{b}6,y(\tfrac{b}{6}))$};
\end{scope}
% \node[below] at ({pi/2}, 0) {$\frac{\pi}2$};
% \node[below] at (0.5,0) {$\frac12$};
% \node[below] at (1,0) {$1$};
% Set up axes
\draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
\draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
\end{tikzpicture}
\end{center}
\item \begin{align*}
&& m-n &= 2(p^3-q^3)-b(p^2-q^2)+c(p-q) \\
&&&= (p-q)(2(p^2+qp+q^2)-b(p+q)+c) \\
&&&= (p-q)(2(p^2+qp+q^2)-3(p+q)^2+6pq) \\
&&&= (p-q)(-p^2-q^2+2pq) \\
&&&= (q-p)^3
\end{align*}
\item The area of $R$ is \begin{align*}
A &= \frac12 bh \\
&= \frac12 (q-p)(m-n) = \frac12(q-p)^4
\end{align*} as required.
\end{questionparts}
This question was also a very popular one, although many candidates gave up their attempt when the algebra started to get a little too tough for them, which generally happened later if not sooner. With this in mind, it has to be said that when candidates did get stuck at some stage of this question, the principal cause was (again!) an unwillingness or inability to simplify algebraic expressions before attempting to work with them. This was particularly important when factorising otherwise lengthy expressions with lots of (q – p)s involved in them. The sketch required in (ii) was intended to be a gentle prod in the right direction for later use in the question, and should have been four easy marks for the taking. Strangely, however, it was often not very well attempted at all. A surprising number of candidates couldn't even manage to draw their cubic through O; and many others seemed unable to make good use of the given conditions, which – despite looking complicated – actually just ensured that all the fun was going on in the first quadrant in an attempt to make life easy. Even more surprising still was the number of sketches that had non-cubic-like kinks, bumps and extra inflection points in them. I was particularly baffled by this widespread lack of grasp as to what a cubic should actually look like! I was equally baffled by the extraordinarily large body of candidates who failed to do what the question explicitly told them they were required to do, by not marking the point of inflection on their sketch and, in many cases, not even attempting to describe the symmetry of it either. An apology has to be made at this point, since the region R in the question was insufficiently clearly defined and there were, in fact, two possibilities. Candidates were not penalised for choosing the "wrong" one at any stage of the proceedings, although the choice of the "left-hand" R would have prevented such candidates from using the short-cut for the following attempt at the area. ALL scripts where candidates made the "wrong" choice were passed to the Principal Examiner and given careful individual consideration. Only about 25 candidates made such a choice: of these, over half had failed to make any attempt at all at the area, and most of the rest had started work on the area and, to all intents and purposes, given up immediately. Two more had found the intended area anyway, despite their previous working (and were not penalised for having switched regions), and (I think) only three had pursued the "left-hand" area almost to a conclusion. Of course, they were unable to get the given answer, but they did get 7 of the 8 marks available. In each of these cases, it was fortunate (for us and them) that this was their last question, so it was safe to say that they hadn't been unduly penalised for time in any way. It is, of course, impossible to say whether they might have seen the intended short-cut approach. In this respect, however, it has to be said that remarkably few candidates saw the symmetry approach anyhow. Partly, I suspect, due to not having picked up the hint at the diagram stage (see earlier)! On the plus side, for us, I imagine that the reference to the point of inflection on the diagram had at least ensured that most candidates chose the intended region R. Only 2 of the 25 or so candidates scored an overall mark that fell just below a grade boundary, and both of these were given the benefit of the doubt by the Chief Examiner.