Year: 2014
Paper: 1
Question Number: 8
Course: LFM Pure and Mechanics
Section: Differentiation from first principles
More than 1800 candidates sat this paper, which represents another increase in uptake for this STEP paper. The impression given, however, is that many of these extra candidates are just not sufficiently well prepared for questions which are not structured in the same way as are the A-level questions that they are, perhaps, more accustomed to seeing. Although STEP questions try to give all able candidates "a bit of an intro." into each question, they are not intended to be easy, and (at some point) imagination and real flair (as well as determination) are required if one is to score well on them. In general, it is simply not possible to get very far into a question without making some attempt to think about what is actually going on in the situation presented therein; and those students who expect to be told exactly what to do at each stage of a process are in for a shock. Too many candidates only attempt the first parts of many questions, restricting themselves to 3-6 marks on each, rather than trying to get to grips with substantial portions of work – the readiness to give up and try to find something else that is "easy pickings" seldom allows such candidates to acquire more than 40 marks (as was the case with almost half of this year's candidature, in fact). Poor preparation was strongly in evidence – curve-sketching skills were weak, inequalities very poorly handled, algebraic capabilities (especially in non-standard settings) were often pretty poor, and the ability to get to grips with extended bits of working lacking in the extreme; also, an unwillingness to be imaginative and creative, allied with a lack of thoroughness and attention to detail, made this a disappointing (and, possibly, very uncomfortable) experience for many of those students who took the paper. On the other side of the coin, there was a very pleasing number of candidates who produced exceptional pieces of work on 5 or 6 questions, and thus scored very highly indeed on the paper overall. Around 100 of them scored 90+ marks of the 120 available, and they should be very proud of their performance – it is a significant and noteworthy achievement.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
Let $L_a$ denote the line joining the points $(a,0)$ and $(0, 1-a)$, where $0< a < 1$. The line $L_b$ is defined similarly.
\begin{questionparts}
\item Determine the point of intersection of $L_a$ and $L_b$, where $a\ne b$.
\item Show that this point of intersection, in the limit as $b\to a$, lies on the curve $C$ given by
\[
y=(1-\sqrt x)^2\, \ \ \ \ (0< x < 1)\,.
\]
\item Show that every tangent to $C$ is of the form $L_a$ for some $a$.
\end{questionparts}\begin{questionparts}
\item $L_a : \frac{y}{x-a} = \frac{1-a-0}{0-a} = \frac{a-1}{a} \Rightarrow ay+(1-a)x = a(1-a)$
\begin{align*}
&& ay + (1-a)x &= a(1-a) \\
&& by + (1-b)x &= b(1-b) \\
\Rightarrow && aby + b(1-a)x &= ba(1-a) \\
&& aby + a(1-b)x &= ab(1-b) \\
\Rightarrow && (b-a)x &= ab(b-a) \\
\Rightarrow && x &= ab \\
&& y &= \frac{a-1}{a} \cdot a(b-1) \\
&&&= (1-a)(1-b)
\end{align*}
\item As $a \to b$, $x \to a^2, y \to 1-2a+a^2 =(1-a)^2 = (1-\sqrt{x})^2$
\item $\frac{\d y}{\d x} = 2(1-\sqrt{x})\cdot \left (-\tfrac12 \frac{1}{\sqrt{x}} \right) = 1 - \frac{1}{\sqrt{x}}$.
Therefore the tangent when $x = c^2, y = (1-c)^2$ is
\begin{align*}
&& \frac{y-(1-c)^2}{x-c^2} &= 1 - \frac{1}{c} \\
\Rightarrow && cy + (1-c)x &= c(c-1)+c(1-c)^2 \\
&&&= c(1-c)
\end{align*}
Which is an equation of the form $L_c$
\end{questionparts}
In hindsight, this question was a little too straightforward, and could well have been placed earlier on in the paper. Nonetheless, around two-thirds of all candidates attempted it, and marks were generally very high, making it the second highest-scoring question on the paper (and only marginally behind Q2) at just under 10/20. Finding equations of lines and intersections in the coordinate geometry setting was clearly much more in candidates' comfort zone than the vector setting of Q7, although there were problems caused by the surfeit of minus signs, and many repeated their working for La when finding Lb rather than simply changing the a's into b's. Parts (ii) and (iii) were also handled well, though slightly less confidently than (i). Part of this was due to the lack of clear explanations given by candidates as to what had been done or found, or a failure to realise that there was a need to justify that "c" satisfied the same conditions as the a from earlier on. Sadly, some failed to give the x a new label (c here), and persisted to substitute x's as part of a gradient into what then became a non-linear formula. Overall, however, this was a good question for candidates and most managed to make substantial progress most of the way through it.