Year: 2014
Paper: 1
Question Number: 4
Course: LFM Pure
Section: Differentiation
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
An accurate clock has an hour hand of length $a$ and a minute hand of length $b$ (where $b>a$), both measured from the pivot at the centre of the clock face. Let $x$ be the distance between the ends of the hands when the angle between the hands is $\theta$, where $0\le\theta < \pi$.
Show that the rate of increase of $x$ is greatest when $x=(b^2-a^2)^\frac12$. In the case when $b=2a$ and the clock starts at mid-day (with both hands pointing vertically upwards), show that this occurs for the first time a little less than 11 minutes later.The position of the hands are $\begin{pmatrix} a\sin(-t) \\ a \cos(-t) \end{pmatrix}$ and $\begin{pmatrix} b\sin(-60t) \\ b \cos(-60t) \end{pmatrix}$, the distance between the hands is
\begin{align*}
x &= \sqrt{\left ( a \sin t - b \sin 60t\right)^2+\left ( a \cos t - b \cos 60t\right)^2} \\
&= \sqrt{a^2+b^2-2ab\left (\sin t \sin 60t+\cos t \cos 60t \right)} \\
&= \sqrt{a^2+b^2-2ab \cos(59t)} = \sqrt{a^2+b^2-2ab \cos \theta} \\
\\
\frac{\d x}{\d \theta} &= \frac{ab \sin \theta}{ \sqrt{a^2+b^2-2ab \cos \theta}} \\
\frac{\d^2 x}{\d \theta^2} &= \frac{ab \cos \theta\sqrt{a^2+b^2-2ab \cos \theta} - \frac{a^2b^2 \sin^2 \theta}{\sqrt{a^2+b^2-2ab \cos \theta}} }{a^2+b^2-2ab \cos \theta} \\
&= \frac{ab \cos \theta(a^2+b^2-2ab \cos \theta) - a^2b^2 \sin^2 \theta }{(a^2+b^2-2ab \cos \theta)^{3/2}} \\
&= \frac{ab \cos \theta(a^2+b^2-2ab \cos \theta) - a^2b^2(1-\cos^2 \theta)}{(a^2+b^2-2ab \cos \theta)^{3/2}} \\
&= \frac{ab(a^2+b^2) \cos \theta-a^2b^2 \cos \theta- a^2b^2}{(a^2+b^2-2ab \cos \theta)^{3/2}} \\
&= \frac{-ab(a\cos \theta -b)(b \cos \theta - a)}{(a^2+b^2-2ab \cos \theta)^{3/2}} \\
\end{align*}
So the rate of increase is largest when $\cos \theta = \frac{a}{b}$ (since $\frac{b}{a}$ is impossible. Therefore when $x = \sqrt{a^2+b^2-2ab \frac{a}{b}} = \sqrt{a^2+b^2-2a^2} = \sqrt{b^2-a^2}$
If $b = 2a$ then $\cos \theta = \frac{a}{2a} = \frac12 = \frac{\pi}{3} = 60^\circ$ The relative speed of the hands is $5.5^\circ$ per minute, so $\frac{60}{5.5} = \frac{120}{11} \approx 11$ but clearly also less than since $121 = 11^2$.
This question was very unpopular indeed, almost certainly due to the lack of given structure. It attracted the attention of less than half of all candidates, and many of these attempts got no further than the first two lines of working involving the use of the Cosine Rule. This led to a mean score of 4/20 on this question and made it the second lowest scoring question on the paper. Those candidates who did then differentiate generally overlooked the fact that they should have been differentiating with respect to time; fortunately, with dθ/dt being constant, there was very little penalty in terms of both marks lost and in straying from a profitable path of progress through the working. As mentioned already, attempts that went significantly beyond this point were few and far between, with relatively few of such candidates realising that they needed to turn the resulting equation into a quadratic in cosθ, and even fewer managing to factorise it appropriately. For the very final part of the question, quite a few managed to obtain the correct (given) answer legitimately, even though they had been unsuccessful with the bulk of the question's working until then. This, at least, demonstrates a shrewd grasp of examination technique and generally earned them 3 or 4 marks at the end.