Year: 2011
Paper: 1
Question Number: 1
Course: LFM Pure
Section: Implicit equations and differentiation
There were again significantly more candidates attempting this paper than last year (just over 1100), but the scores were significantly lower than last year: fewer than 2% of candidates scored above 100 marks, and the median mark was only 44, compared to 61 last year. It is not clear why this was the case. One possibility is that Questions 2 and 3, which superficially looked straightforward, turned out to be both popular and far harder than candidates anticipated. The only popular and well-answered questions were 1 and 4. The pure questions were the most popular as usual, though there was noticeable variation: questions 1–4 were the most popular, while question 7 (on differential equations) was fairly unpopular. Just over half of all candidates attempted at least one mechanics question, which one-third attempted at least one probability question, an increase on last year. The marks were surprising, though: the two best-answered questions were the pure questions 1 and 4, but the next best were statistics question 12 and mechanics question 9. The remainder of the questions were fairly similar in their marks. A number of candidates ignored the advice on the front cover and attempted more than six questions, with a fifth of candidates trying eight or more questions. A good number of those extra attempts were little more than failed starts, but still suggest that some candidates are not very effective at question-picking. This is an important skill to develop during STEP preparation. Nevertheless, the good marks and the paucity of candidates who attempted the questions in numerical order does suggest that the majority are being wise in their choices. Because of the abortive starts, I have generally restricted my attention to those attempts which counted as one of the six highest-scoring answers in the detailed comments. On occasions, candidates spent far longer on some questions than was wise. Often, this was due to an algebraic slip early on, and they then used time which could have been far better spent tackling another question. It is important to balance the desire to finish a question with an appreciation of when it is better to stop and move on. Many candidates realised that for some questions, it was possible to attempt a later part without a complete (or any) solution to an earlier part. An awareness of this could have helped some of the weaker students to gain vital marks when they were stuck; it is generally better to do more of one question than to start another question, in particular if one has already attempted six questions. It is also fine to write "continued later" at the end of a partial attempt and then to continue the answer later in the answer booklet. As usual, though, some candidates ignored explicit instructions to use the previous work, such as "Hence", or "Deduce". They will get no credit if they do not do what they are asked to! (Of course, a question which has the phrase "or otherwise" gives them the freedom to use any method of their choosing; often the "hence" will be the easiest, though.) It is wise to remember that STEP questions do require a greater facility with mathematics and algebraic manipulation than the A-level examinations, as well as a depth of understanding which goes beyond that expected in a typical sixth-form classroom. There were a number of common errors and issues which appeared across the whole paper. The first was a lack of fluency in algebraic manipulations. STEP questions often use more variables than A-level questions (which tend to be more numerical), and therefore require candidates to be comfortable engaging in extended sequences of algebraic manipulations with determination and, crucially, accuracy. This is a skill which requires plenty of practice to master. Another area of weakness is logic. A lack of confidence in this area showed up several times. In particular, a candidate cannot possibly gain full marks on a question which reads "Show that X if and only if Y" unless they provide an argument which shows that Y follows from X and vice versa. Along with this comes the need for explanations in English: a sequence of formulæ or equations with no explicit connections between them can leave the reader (and writer) confused as to the meaning. Brief connectives or explanations would help, and sometimes longer sentences are necessary. Another related issue continues to be legibility. Many candidates at some point in the paper lost marks through misreading their own writing. One frequent error was dividing by zero. On several occasions, an equation of the form xy = xz appeared, and candidates blithely divided by x to reach the conclusion y = z. Again, I give a strong reminder that it is vital to draw appropriate, clear, accurate diagrams when attempting some questions, mechanics questions in particular.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1479.0
Banger Comparisons: 4
\begin{questionparts}
\item Show that the gradient of the curve $\; \dfrac a x + \dfrac by =1$, where $b\ne0$, is $\; -\dfrac{ay^2}{bx^2}\,$.
The point $(p,q)$ lies on both the straight line $ax+by=1$ and the curve $\dfrac a x + \dfrac by =1\,$, where $ab\ne0$.
Given that, at this point, the line and the curve have the same gradient, show that $ p=\pm q\,$. Show further that either $(a-b)^2 =1\,$ or $(a+b)^2 =1\,$.
\item Show that if the straight line $ax+by=1$, where $ab\ne0$, is a normal to the curve $\dfrac a x - \dfrac by =1$, then
$a^2-b^2 = \frac12\,$.
\end{questionparts}\begin{questionparts}
\item $\,$ \begin{align*}
&& 1 &= \frac{a}{x} + \frac{b}{y} \\
\frac{\d}{\d x}: && 0 &= -\frac{a}{x^2} - \frac{b}{y^2} \frac{\d y}{\d x} \\
\Rightarrow && \frac{\d y}{\d x} &= -\frac{ay^2}{bx^2} \\
\\
(p,q): && -\frac{aq^2}{bp^2} &= -\frac{a}{b} \\
\Rightarrow && p^2 &= q^2 \\
\Rightarrow && p &= \pm q \\
\\
\Rightarrow && ap \pm b p &= 1 \\
\Rightarrow && (a\pm b)p &= 1 \\
\Rightarrow && \frac{a}{p} \pm \frac{b}{p} &= 1 \\
\Rightarrow && (a \pm b)\frac{1}{p} &= 1 \\
\Rightarrow && (a \pm b)^2 &= 1
\end{align*}
\item $\,$
\begin{align*}
&& 1 &= \frac{a}{x} - \frac{b}{y} \\
\Rightarrow && \frac{\d y}{\d x} &= \frac{ay^2}{bx^2} \\
\Rightarrow && \frac{aq^2}{bp^2} &= \frac{b}{a} \\
\Rightarrow && aq &= \pm bp \\
\Rightarrow && 1 &= \frac{a}{p} - \frac{b}{q} \\
&&&= \frac{aq-bp}{pq} \\
\Rightarrow && aq &= -bp \\
\Rightarrow && 1 &= \frac{2aq}{pq} \\
\Rightarrow && p &= 2a \\
\Rightarrow && q &= -2b \\
\Rightarrow && 1 &= 2a^2-2b^2 \\
\Rightarrow && \frac12 &= a^2-b^2
\end{align*}
\end{questionparts}
This was an easy and very popular first question, attempted by almost all of the candidates. It was also the most successfully answered, with a median mark of 14. (i) Most candidates differentiated the equation implicitly, reaching the specified gradient with ease. Some decided to rearrange the equation into the form y = · · · before differentiating; this was frequently unsuccessful as the final manipulations required were relatively tricky. The next step, showing that p = ±q, was also generally answered well, even by candidates who had become stuck on the first part of the question. It was surprising to see many solutions using implicit differentiation to find the gradient of the straight line ax + by = 1, rather than rearranging the equation to get y = mx + c and then reading off the answer. The final step was also generally answered well, though a number of candidates could not see how to use the previous result (p = ±q) to progress. Furthermore, there were a few candidates who managed to deduce one of the possibilities but not the other; this was a little strange, as the argument was essentially identical. (ii) While the majority of candidates used the result that the products of normal gradients is −1, relatively few paid enough attention to notice that the equation of the curve was different to that of part (i). This led them to conclude that a²q² = −b²p², from which they were not able to make any progress. (Note here that the left hand side is always positive and the right hand side is always negative.) Of those who overcame this hurdle and attempted the question posed, some succeeded in reaching the specified conclusion, while most deduced that a²q² = b²p² and then became stuck, unable to see how to progress. This is presumably because these equations were more complex than in the first part of the question. Also, some candidates who gave otherwise good answers did not consider both possible cases aq = +bp and aq = −bp.