Year: 2017
Paper: 2
Question Number: 4
Course: LFM Pure
Section: Integration
This year's paper was, perhaps, slightly more straightforward than usual, with more helpful guidance offered in some of the questions. Thus the mark required for a "1", a Distinction, was 80 (out of 120), around ten marks higher than that which would customarily be required to be awarded this grade. Nonetheless, a three‐figure mark is still a considerable achievement and, of the 1330 candidates sitting the paper, there were 89 who achieved this. At the other end of the scale, there were over 350 who scored 40 or below, including almost 150 who failed to exceed a total score of 25. As a general strategy for success in a STEP examination, candidates should be looking to find four "good" questions to work at (which may be chosen freely by the candidates from a total of 13 questions overall). It is unfortunately the case that so many low‐scoring candidates flit from one question to another, barely starting each one before moving on. There needs to be a willingness to persevere with a question until a measure of understanding as to the nature of the question's purpose and direction begins to emerge. Many low‐scoring candidates fail to deal with those parts of questions which cover routine mathematical processes ‐ processes that should be standard for an A‐level candidate. The significance of the "rule of four" is that four high‐scoring questions (15‐20 marks apiece) obtains you up to around the total of 70 that is usually required for a "1"; and with a couple of supporting starts to questions, such a total should not be beyond a good candidate who has prepared adequately. This year, significantly more than 10% of candidates failed to score at least half marks on any one question; and, given that Q1 (and often Q2 also) is (are) specifically set to give all candidates the opportunity to secure some marks, this indicates that these candidates are giving up too easily. Mathematics is about more than just getting to correct answers. It is about communicating clearly and precisely. Particularly with "show that" questions, candidates need to distinguish themselves from those who are just tracking back from given results. They should also be aware that convincing themselves is not sufficient, and if they are using a result from 3 pages earlier, they should make this clear in their working. A few specifics: In answers to mechanics questions, clarity of diagrams would have helped many students. If new variables or functions are introduced, it is important that students clearly define them. One area which is very important in STEP but which was very poorly done is dealing with inequalities. Although a wide range of approaches such as perturbation theory were attempted, at STEP level having a good understanding of the basics – such as changing the inequality if multiplying by a negative number – is more than enough. In fact, candidates who used more advanced methods rarely succeeded.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
The Schwarz inequality
is
\[
\left( \int_a^b \f(x)\, \g(x)\,\d x\right)^{\!\!2}
\le
\left(
\int_a^b \big( \f(x)\big)^2 \d x
\right)
\left(
\int_a^b \big( \g(x)\big)^2 \d x
\right)
.
\tag{$*$}
\]
\begin{questionparts}
\item
By setting $ \f(x)=1$ in $(*)$, and choosing $\g(x)$, $a$ and $b$ suitably, show that for $t> 0\,$,
\[
\frac {\e^t -1}{\e^t+1} \le \frac t 2
\,.
\]
\item
By setting $ \f(x)= x$ in $(*)$, and choosing $ \g(x)$ suitably, show that
\[
\int_0^1\e^{-\frac12 x^2}\d x \ge 12 \big(1-\e^{-\frac14})^2
\,.
\]
\item
Use $(*)$ to show that
\[
\frac {64}{25\pi} \le \int_0^{\frac12\pi}
\!\!
{\textstyle \sqrt{\, \sin x\, } }
\, \d x
\le \sqrt{\frac \pi 2 }
\,.
\]
\end{questionparts}\begin{questionparts}
\item Let $f(x) = 1, g(x) = e^x, a = 0, b = t$, so
\begin{align*}
&& \left ( \int_0^t e^x \d x \right)^2 &\leq \left (\int_0^t 1^2 \d x \right) \cdot \left (\int_0^t (e^x)^2 \d x \right) \\
\Rightarrow && (e^t-1)^2 &\leq t \cdot (\frac12e^{2t} - \frac12) \\
\Rightarrow && \frac{e^t-1}{e^t+1} & \leq \frac{t}{2}
\end{align*}
\item Let $f(x) = x, g(x) = e^{-\frac14 x^2}, a = 0, b = 1$
\begin{align*}
&& \left ( \int_0^1 xe^{-\frac14 x^2} \d x \right)^2 &\leq \left (\int_0^1 x^2 \d x \right) \cdot \left (\int_0^1 (e^{-\frac14x^2})^2 \d x \right) \\
\Rightarrow && \left ( \left [-2e^{-\frac14x^2} \right]_0^1 \right)^2 & \leq \frac{1}{3} \int_0^1 e^{-\frac12 x^2} \d x \\
\Rightarrow && \int_0^1 e^{-\frac12 x^2} \d x & \geq 12(1-e^{-\frac14})^2
\end{align*}
\item Let $f(x) = 1, g(x) = \sqrt{\sin x}, a = 0, b = \tfrac12 \pi$, then
\begin{align*}
&& \left ( \int_0^{\frac12 \pi} \sqrt{\sin x} \d x \right)^2 &\leq \left (\int_0^{\frac12 \pi} 1^2 \d x \right) \cdot \left (\int_0^{\frac12 \pi}|\sin x| \d x \right) \\
&&&= \frac{\pi}{2} \cdot 1 \\
\Rightarrow && \int_0^{\frac12 \pi} \sqrt{\sin x} \d x & \leq \sqrt{\frac{\pi}{2}}
\end{align*}
Let $f(x) =(\sin x)^{\frac14}, g(x) = \cos x, a = 0, b = \tfrac12 \pi$, so
\begin{align*}
&& \left ( \int_0^{\frac12 \pi} (\sin x)^{\frac14} \cos x \d x \right)^2 &\leq \left (\int_0^{\frac12 \pi} \cos^2 x \d x \right) \cdot \left (\int_0^{\frac12 \pi}\sqrt{\sin x} \d x \right) \\
\Rightarrow &&\left ( \left [\frac45 (\sin x)^{\frac54} \right]_0^{\frac12 \pi} \right)^2 & \leq \frac{\pi}{4} \int_0^{\frac12 \pi}\sqrt{\sin x} \d x \\
\Rightarrow && \frac{64}{25\pi} &\leq \int_0^{\frac12 \pi}\sqrt{\sin x} \d x
\end{align*}
\end{questionparts}
This is the first question where the difference between "attempts" and "serious attempts" arises to any significant extent: there were just over 800 of the former but well under 500 of the latter. This is also a good point at which to raise a key issue in respect of strategy for candidates sitting a STEP. Spending a few minutes of reading time, at some particular time during the examination, could be a significant asset, especially to those candidates who have particular strengths and weaknesses to play to or to avoid. A very brief analysis of this question, on first reading, should help one recognise that a result is being given (with no requirement to establish it in any way) and all that is required is to use it. Part (i) then clearly directs part of the way, and the required limits are rather obviously flagged, as is the fact that g( ) must be something to do with the exponential function. One of the two functions to be used in (ii) is also given, as are the limits; an inspection of the given should lead to the (correct) conclusion that g( ) must be e 1 x2 4 . Getting just this far takes the candidate to the 10‐mark point, a perfectly good return for a candidate who has read the question through sufficiently carefully to realise that it has decent potential for mark‐acquisition. In the final part of the question some careful thought was needed, with only the required limits obvious at first glance. Most attempts, serious or otherwise, picked up the majority of their marks in (i) and (ii) and efforts at (iii) were very varied: many candidates simply gave up and moved on; many more picked up a few extra marks by setting g( ) sin x (which is a fairly obvious candidate to try) and working towards the right‐hand half of the given result. Very few candidates indeed had the experience to realise that sin x now needed to appear as the squared term, which also meant that a cosine term had to be involved.