Year: 2011
Paper: 1
Question Number: 13
Course: LFM Stats And Pure
Section: Continuous Probability Distributions and Random Variables
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: 1484.0
Difficulty Comparisons: 1
Banger Rating: 1471.5
Banger Comparisons: 4
In this question, you may use without proof
the following
result:
\[
\int \sqrt{4-x^2}\, \d x
= 2 \arcsin (\tfrac12 x ) + \tfrac 12 x \sqrt{4-x^2} +c\,.
\]
A random variable $X$ has probability
density function $\f$ given by
\[
\f(x) =
\begin{cases}
2k & -a\le x <0 \\[3mm]
k\sqrt{4-x^2} & \phantom{-} 0\le x \le 2 \\[3mm]
0 & \phantom{-}\text{otherwise},
\end{cases}
\]
where $k$ and $a$ are positive constants.
\begin{questionparts}
\item Find, in terms of $a$, the mean of $X$.
\item Let $d$ be the value of $X$ such that $\P(X> d)=\frac1 {10}\,$. Show that $d < 0$ if $2a> 9\pi$ and find an expression for $d$ in terms of $a$ in this case.
\item Given that $d=\sqrt 2$, find $a$.
\end{questionparts}First notice that
\begin{align*}
&& 1 &= \int_{-a}^2 f(x) \d x \\
&&&= 2ka + k\pi \\
\Rightarrow && k &= (\pi + 2a)^{-1}
\end{align*}
\begin{questionparts}
\item $\,$ \begin{align*}
&& \E[X] &= \int_{-a}^2 x f(x) \d x\\
&&&= \int_{-a}^0 2kx \d x + k\int_0^{2} x\sqrt{4-x^2} \d x\\
&&&= \left [kx^2 \right]_{-a}^0 +k \left [-\frac13(4-x^2)^{\frac32} \right]_0^2 \\
&&&= -ka^2 + \frac83k \\
&&&= \frac{\frac83-a^2}{\pi + 2a}
\end{align*}
\item Consider $\mathbb{P}(X < 0)$ then $d < 0 \Leftrightarrow \mathbb{P}(X < 0) > \frac{9}{10}$
\begin{align*}
&& \frac{9}{10} &< \mathbb{P}(X < 0) \\
&&&= \int_{-a}^0 2k \d x \\
&&&= \frac{2a}{\pi+2a} \\
\Leftrightarrow && 9\pi &< 2a \\
\\
&& \frac{9}{10} &= \int_{-a}^d 2k \d x \\
&&&= \frac{2(d+a)}{\pi + 2a} \\
\Rightarrow && 9\pi &= 2a + 20d \\
\Rightarrow && d &= \frac{2a-9\pi}{20}
\end{align*}
\item Suppose $d=\sqrt 2$ then
\begin{align*}
&& \frac1{10} &= \int_{\sqrt{2}}^2 f(x) \d x \\
&&&= \int_{\sqrt{2}}^2 k\sqrt{4-x^2} \d x \\
&&&= k\left [ 2 \sin^{-1} \tfrac12 x + \tfrac12 x \sqrt{4-x^2}\right]_{\sqrt{2}}^2 \\
&&&= k\left (\pi -\frac{\pi}{2} - 1 \right) \\
\Rightarrow && \pi + 2a &= 5\pi - 10 \\
\Rightarrow && a &= 2\pi-5
\end{align*}
\end{questionparts}
This question was attempted by around 20% of candidates, but was only counted as one of the best six questions for about three-quarters of them. Even among those, it was the most poorly answered question on the whole paper, with one-third of the attempts gaining only 0 or 1 mark, a median of 4 marks and an upper quartile of 8 marks. (i) In this first part, many candidates did not even attempt to find k, and algebraic errors during the integration were common. A number of attempts to find k were obviously wrong, as they gave a negative area, but candidates did not notice this. Most were unable to integrate x√(4 − x²), with failed attempts to integrate by parts far outnumbering correct integrations of this expression. A number of candidates attempted to calculate the median rather than the mean. Other bizarre interpretations of the term "mean" were also seen. (ii) Few candidates made it this far. Of those who did, there were some very good solutions. One of the hardest parts was getting the logic correct: the phrasing of the question as "Show that X if Y" was often misinterpreted to mean "Show that if X then Y", though the majority of the marks were awarded in such cases. (iii) The handful of students who made it this far were generally successful at this part, too.