Year: 2009
Paper: 1
Question Number: 3
Course: LFM Stats And Pure
Section: Quadratics & Inequalities
There were significantly more candidates attempting this paper again this year (over 900 in total), and the scores were pleasing: fewer than 5% of candidates failed to get at least 20 marks, and the median mark was 48. The pure questions were the most popular as usual; about two-thirds of candidates attempted each of the pure questions, with the exceptions of question 2 (attempted by about 90%) and question 5 (attempted by about one third). The mechanics questions were only marginally more popular than the probability and statistics questions this year; about one quarter of the candidates attempted each of the mechanics questions, while the statistics questions were attempted by about one fifth of the candidates. A significant number of candidates ignored the advice on the front cover and attempted more than six questions. In general, those candidates who submitted answers to eight or more questions did fairly poorly; very few people who tackled nine or more questions gained more than 60 marks overall (as only the best six questions are taken for the final mark). This suggests that a skill lacking in many students attempting STEP is the ability to pick questions effectively. This is not required for A-levels, so must become an important part of STEP preparation. Another "rubric"-type error was failing to follow the instructions in the question. In particular, when a question says "Hence", the candidate must make (significant) use of the preceding result(s) in their answer if they wish to gain any credit. In some questions (such as question 2), many candidates gained no marks for the final part (which was worth 10 marks) as they simply quoted an answer without using any of their earlier work. There were a number of common errors which appeared across the whole paper. These included a noticeable weakness in algebraic manipulations, sometimes indicating a serious lack of understanding of the mathematics involved. As examples, one candidate tried to use the misremembered identity cos β = sin √(1 − β²), while numerous candidates made deductions of the form "if a² + b² = c², then a + b = c" at some point in their work. Fraction manipulations are also notorious in the school classroom; the effects of this weakness were felt here, too. Another common problem was a lack of direction; writing a whole page of algebraic manipulations with no sense of purpose was unlikely to either reach the requested answer or gain the candidate any marks. It is a good idea when faced with a STEP question to ask oneself, "What is the point of this (part of the) question?" or "Why has this (part of the) question been asked?" Thinking about this can be a helpful guide. One aspect of this is evidenced by pages of formulæ and equations with no explanation. It is very good practice to explain why you are doing the calculation you are, and to write sentences in English to achieve this. It also forces one to focus on the purpose of the calculations, and may help avoid some dead ends. Finally, there is a tendency among some candidates when short of time to write what they would do at this point, rather than using the limited time to actually try doing it. Such comments gain no credit; marks are only awarded for making progress in a question. 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.
Difficulty Rating: 1516.0
Difficulty Comparisons: 1
Banger Rating: 1468.7
Banger Comparisons: 2
\begin{questionparts}
\item
By considering the equation $x^2+x-a=0\,$,
show that the equation
$x={(a-x)\vphantom M}^{\frac12}$ has one real solution when $a\ge0$ and no
real solutions when $a<0\,$.
Find the number of distinct real solutions of the equation
\[
x={\big((1+a)x-a\big)}^{\!\frac13}
\]
in the cases that arise according to the value of $a$.
\item Find the number of distinct real solutions of the equation
\[
x={(b+x)\vphantom M}^{\frac12}
\]
in the cases that arise according to the value of $b\,$.
\end{questionparts}\begin{questionparts}
\item $\,$
\begin{align*}
&& x &= (a-x)^{\frac12} \\
\Rightarrow && x^2 &= a - x \\
\Rightarrow && 0 &= x^2 + x - a
\end{align*}
This has a roots if $\Delta = 1 + 4a \geq 0 \Rightarrow a \geq -\frac14$. These roots also need to be positive (since $x \geq 0$). Since $f(0) = -a$ we have one positive root if $a \geq 0$. If $a \leq 0$ then since the roots are symmetric about $x = -\frac12$, both roots are negative and there are no positive roots. Therefore we have on real solution if $a \geq 0$ and non otherwise.
\begin{align*}
&& x & = \left ( (1+a)x - a \right)^{\frac13} \\
\Leftrightarrow && x^ 3 &= (1+a)x - a \\
\Leftrightarrow && 0 &= x^3- (1+a)x + a \\
\Leftrightarrow && 0 &= (x-1)(x^2+x-a) \\
\end{align*}
Since every solution to the first equation is a solution to the second, we have $x = 1$ always works, and there is an additional two solutions if $a > -\frac14$ and a single extra solution if $a = -\frac14$. We can also repeat solutions if $1$ is a root of $x^2+x -a$, ie when $a = 2$ Therefore:
One solution if $a < -\frac14$
Two solutions if $a = -\frac14, 2$
Three solutions if $a > -\frac14, a \neq 2$
\item $\,$ \begin{align*}
&& x &= (b+x)^{\frac12} \\
\Rightarrow && x^2 &= b + x \\
\Rightarrow && 0 &= x^2 - x - b
\end{align*}
This has a positive root if $\frac14 - \frac12 - b \leq 0 \rightarrow b \geq \frac14$. It has two positive roots if $b \geq 0$.
Therefore two solutions if $b > \frac14$ and one solution if $b = \frac14$
\end{questionparts}
This was found to be one of the hardest questions on the whole paper, with a median mark of 2 and a mean mark of 3.9. A relatively small number of candidates appreciated the subtleties involved and rest consequently produced fairly nonsensical answers. For the first half of part (i), relatively few candidates successfully attempted to relate the two given equations. It was encouraging that the majority of candidates were able to correctly quote the quadratic formula (especially as it is given in the formula book!), but there was considerable difficulty in evaluating it in this case. It seemed that candidates were also confused by the fact that the formula is given for the equation ax² + bx + c = 0, whereas here the constant term involves the variable a. Other candidates became confused about the distinction between negative roots and non-real (complex) roots; negative numbers are real! Of those who related the two equations, very few appreciated that the solutions of an equation involving a square root may be different from the corresponding squared equation. This was compounded by many students asserting that 1 + 4a < 0 if and only if a < 0 (where the "if and only if" may have been implied). Few understood that proving that there was no real solution when a < 0 was insufficient to show that there was one real solution when a > 0. A small number of candidates proved this part of the question using an effective graphical method, which was very impressive. Interestingly, the second half of part (i) proved to be more straightforward for a number of candidates, possibly because there were no square roots involved. However, some were confused and thought that the conditions on a in the first half of part (i) would be inherited in the second half. Only a handful of students made any significant attempt at part (ii), and of those, some used algebraic methods following the ideas of part (i) while others used graph-sketching approaches, often quite successfully.