Year: 2011
Paper: 1
Question Number: 6
Course: LFM Stats And Pure
Section: Generalised Binomial Theorem
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: 1500.0
Banger Comparisons: 0
Use the binomial expansion to show that the
coefficient of $x^r$ in the expansion of
$(1-x)^{-3}$ is $\frac12 (r+1)(r+2)\,$.
\begin{questionparts}
\item
Show that the coefficient of $x^r$ in the expansion of
\[
\frac{1-x+2x^2}{(1-x)^3}
\]
is $r^2+1$ and hence find the sum of the series
\[
1+\frac22+\frac54+\frac{10}8+\frac{17}{16}+\frac{26}{32}+\frac{37}{64}
+\frac{50}{128}+ \cdots \,.
\]
\item Find the sum of the series
\[
1+2+\frac94+2+\frac{25}{16}+\frac{9}{8}+\frac{49}{64}
+ \cdots \,.
\]
\end{questionparts}Notice that the coefficient of $x^r$ is $(-1)^r\frac{(-3) \cdot (-3-1) \cdots (-3-r+1)}{r!} = (-1)^r \frac{(-1)(-2)(-3)(-4) \cdots (-(r+2))}{(-1)(-2)r!} = (-1)^r(-1)^{r+2}\frac{(r+2)!}{2r!} = \frac{(r+2)(r+1)}2$.
\begin{questionparts}
\item The coefficient of $x^r$ is
\begin{align*}
&& c_r &=\frac{(r+1)(r+2)}{2} - \frac{(r-1+1)(r-1+2)}{2} + 2 \frac{((r-2+1)(r-2+2)}{2} \\
&&&= \frac{r^2+3r+2}{r} - \frac{r^2+r}{2} + \frac{2r^2-2r}{2}\\
&&&= \frac{2r^2+2}{2} = r^2+1
\end{align*}
\begin{align*}
&& S & = 1+\frac22+\frac54+\frac{10}8+\frac{17}{16}+\frac{26}{32}+\frac{37}{64}
+\frac{50}{128}+ \cdots \\
&&&= \sum_{r=0}^{\infty} \frac{r^2+1}{2^r} \\
&&&= \frac{1-\tfrac12+2 \cdot \tfrac14}{(1-\tfrac12)^3} \\
&&&= 8
\end{align*}
\item $\,$
\begin{align*}
&& S &= 1+2+\frac94+2+\frac{25}{16}+\frac{9}{8}+\frac{49}{64}
+ \cdots \\
&&&= \sum_{r=0}^{\infty} \frac{(r+1)^2}{2^r} \\
&&&= 2 \sum_{r=0}^{\infty} \frac{(r+1)^2}{2^{r+1}} \\
&&&= 2 \sum_{r=1}^{\infty} \frac{r^2}{2^{r}} \\
&&&= 2 \left (\sum_{r=0}^{\infty} \frac{r^2+1}{2^{r}} - \sum_{r=0}^{\infty} \frac{1}{2^{r}} \right) \\
&&&= 2 (8 - 1) = 14
\end{align*}
\end{questionparts}
This was another fairly popular question, but there were many very weak attempts; the median mark was 5. The first part was answered very poorly. A significant number of candidates only worked out the first few terms and showed that they satisfied the given formula, without making any attempt to justify the formula in general. The majority attempted to write down the general term, with varying degrees of success: many forgot to take account of the minus sign, and so did not include (−x)^r. Another common error was to write expressions involving (−n)!. Few candidates gave any justification for removal of the minus signs, and solutions which correctly dealt with the case r < 2 were rare indeed. (i) This part was answered very well by the majority of candidates. A number attempted to factorise the numerator as 1 − x + 2x² = (1 + x)(1 − 2x) or other incorrect ways. Very few answers were careful about the boundary cases, namely where r = 0 and r = 1, and so most candidates only achieved 4/5 on this part. (ii) Only a minority of candidates made any progress on this part: most candidates were unable to correctly identify the rule r²/2^(r−1) for the terms of the sequence. Of those who did, the most common approach was to relate the sequence to that of part (i) as in the first approach described above. Some candidates were able to do the manipulations correctly, but there were a significant number who made slips along the way (for example, using r²/2^r or leaving out the initial term). Some used the second approach or a variant of it. It was very pleasing to see some students use the third approach described, many of whom were successful.