Year: 2011
Paper: 1
Question Number: 4
Course: LFM Pure and Mechanics
Section: Parametric equations
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: 1524.2
Banger Comparisons: 8
The distinct points $P$ and $Q$, with coordinates $(ap^2,2ap)$
and $(aq^2,2aq)$ respectively, lie on the curve $y^2=4ax$.
The tangents to the curve at $P$ and $Q$ meet at the point $T$.
Show that $T$ has coordinates $\big(apq, a(p+q)\big)$.
You may assume that $p\ne0$ and $q\ne0$.
The point $F$ has coordinates $(a,0)$ and $\phi$ is
the angle $TFP$. Show that
\[
\cos\phi = \frac{pq+1}{\sqrt{(p^2+1)(q^2+1)}\ }
\]
and deduce that the line $FT$ bisects the angle $PFQ$.The tangent at $(at^2, 2at)$ can be found
\begin{align*}
&& \frac{\d y}{\d x} &= \frac{\dot{y}}{\dot{x}} \\
&&&= \frac{2a}{2at} = \frac1t \\
\Rightarrow && \frac{y-2at}{x-at^2} &= \frac1t \\
\Rightarrow && ty -x &= at^2 \\
\\
PT: && py - x &= ap^2 \\
QT: && qy - x &= aq^2 \\
\Rightarrow && (p-q)y &= a(p^2-q^2) \\
\Rightarrow && y &= a(p+q) \\
&& x &= aq(p+q) - aq^2 \\
&&&= apq
\end{align*}
By the cosine rule:
\begin{align*}
&& TP^2 &= FT^2 + FP^2 - 2 \cdot FT \cdot FP \cdot \cos \phi \\
&& (apq - ap^2)^2 + (a(p+q)-2ap)^2 &= (a-apq)^2+(a(p+q))^2 + \\
&&&\quad + (a-ap^2) + (2ap)^2 - 2 \cdot FT \cdot FP \cdot \cos \phi \\
\Rightarrow && a^2p^2(q-p)^2 + a^2(q-p)^2 &= a^2(1-pq)^2+a^2(p+q)^2 + \\
&&&\quad + a^2(1-p^2)^2+4a^2p^2 - 2 \cdot FT \cdot FP \cdot \cos \phi \\
&& a^2(p^2+1)(q-p)^2 &= a^2(1+p^2)(1+q^2) + a^2(1+p^2)^2 + \\
&&&\quad - 2 \cdot a^2(1+p^2)\sqrt{(1+p^2)(1+q^2)} \cos \phi \\
\Rightarrow && \cos \phi &= \frac{a^2(1+p^2)(2+q^2+p^2-(q-p)^2)}{2 a^2 (1+p^2)\sqrt{(1+p^2)(1+q^2)}} \\
&&&= \frac{1+pq}{\sqrt{(1+p^2)(1+q^2)}}
\end{align*}
As required.
Notice that by symmetry, $\cos \angle TFQ = \frac{1+qp}{\sqrt{(1+q^2)(1+p^2)}} = \cos \phi$. Therefore they have the same angle and $FT$ bisects $PFQ$
This was a popular question, attempted by two-thirds of candidates. It was also one of the most successfully answered, with a median mark of 11. Candidates were very good at differentiating to find the coordinates of T, though there were some issues. Those who rearranged to find y = √(4ax) generally did not handle the possibility that y could be negative. There were also a number of candidates who are still confused when trying to find the equation of a tangent: they used the general expression for dy/dx rather than substituting in the values of x and y at the point of tangency. This gave them a "straight line" with equation y − 2ap = (2ay)(x − ap²) which was then sometimes rearranged to give a quadratic. The vast majority were fine with this step, though, and went on to successfully find the coordinates of T. Some used the symmetry of the situation to simply write down the equation of the second tangent, while others determined it from scratch. There was one sticking point, though: at this level of work, candidates are expected to take care when dividing to ensure that they are not dividing by zero. A mark was therefore awarded for stating that p − q ≠ 0 or p ≠ q when dividing by it, but very few candidates did so. When it came to deducing the given formula for cos φ, most candidates made a good start, with the dot-product approach more popular than the cosine rule. However, there was a need for some fluent algebraic manipulations, in particular the ability to factorise. This should have been made a little easier by knowing the desired final result, but most candidates became bogged down at this point and were unable to deduce the given expression. The dot-product approach, with its slightly simpler algebra, was generally more successful. The final part of the question, requiring candidates to deduce that the line FT bisects the angle PFQ, produced many spurious attempts. Few candidates appreciated the symmetry of the situation, and so went on to calculate cos(∠TFQ) from scratch. Others attempted to find cos(∠PFQ), presumably hoping to use a double angle formula or similar. These approaches were sometimes successful. There were also candidates who attempted to answer this part by using right-angled trigonometry in one of the triangles, or by identifying similar or congruent triangles, even though none of these approaches made sense in this situation.