2017 Paper 2 Q3
Year: 2017
Paper: 2
Question Number: 3
Course: LFM Pure
Section: Implicit equations and differentiation
Problem
- Sketch, on \(x\)-\(y\) axes, the set of all points satisfying \(\sin y = \sin x\), for \(-\pi \le x \le \pi\) and \(-\pi \le y \le \pi\). You should give the equations of all the lines on your sketch.
- Given that \[ \sin y = \tfrac12 \sin x \] obtain an expression, in terms of \(x\), for \(y'\) when \(0\le x \le \frac12 \pi\) and \(0\le y \le \frac12 \pi\), and show that \[ y'' = - \frac {3\sin x}{(4-\sin^2 x)^{\frac32}} \;. \] Use these results to sketch the set of all points satisfying \(\sin y = \tfrac12 \sin x\) for \(0 \le x \le \frac12 \pi\) and \(0 \le y \le \frac12 \pi\). Hence sketch the set of all points satisfying \(\sin y = \tfrac12 \sin x\) for \(-\pi\! \le \! x \! \le \! \pi\) and \mbox{\( -\pi \, \le\, y\, \le\, \pi\,\)}.
- Without further calculation, sketch the set of all points satisfying \(\cos y = \tfrac12 \sin x\) for \(- \pi \le x \le \pi\) and \( -\pi \le y \le \pi\).
Solution
- \(\,\)
- \(\,\) \begin{align*}
&& \sin y &= \tfrac12 \sin x \\
\Rightarrow && \frac{\d y}{\d x} \cos y &= \tfrac12 \cos x \\
\Rightarrow && \frac{\d y}{\d x} &= \frac{\cos x}{2 \cos y} \\
&&&= \frac{\cos x}{2 \sqrt{1-\sin^2 y}} \\
&&&= \frac{\cos x}{2 \sqrt{1-\frac14 \sin^2 x}} \\
&&&= \frac{\cos x}{\sqrt{4-\sin^2 x}} \\
\\
&& y'' &= \frac{-\sin x \cdot (4-\sin^2 x)^{\frac12} - \cos x \cdot (4-\sin^2 x)^{-\frac12} \cdot 2 \sin x \cos x}{(4-\sin^2 x)} \\
&&&= \frac{-\sin x \cdot (4-\sin^2 x) - \cos x \cdot 2 \sin x \cos x}{(4-\sin^2x)^{\frac32}} \\
&&&= \frac{-\sin x \cdot (4-\sin^2 x) - \sin x (1-\sin^2x)}{(4-\sin^2x)^{\frac32}} \\
&&&= \frac{-3\sin x }{(4-\sin^2x)^{\frac32}} \\
\end{align*}
- \(\,\)
Examiner's report
— 2017 STEP 2, Question 3This 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.
Rating Information
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
Show LaTeX source
Problem source
\begin{questionparts}
\item Sketch, on $x$-$y$ axes, the set of all points satisfying
$\sin y = \sin x$, for $-\pi \le x \le \pi$ and $-\pi \le y \le \pi$.
You should give the equations of all the lines on your sketch.
\item Given that
\[
\sin y = \tfrac12 \sin x
\]
obtain an expression, in terms of $x$, for $y'$ when
$0\le x \le \frac12 \pi$
and $0\le y \le \frac12 \pi$,
and show that
\[
y'' = - \frac {3\sin x}{(4-\sin^2 x)^{\frac32}}
\;.
\]
Use these results to sketch the
set of all points satisfying $\sin y = \tfrac12 \sin x$ for
$0 \le x \le \frac12 \pi$ and
$0 \le y \le \frac12 \pi$.
Hence sketch the set of all points satisfying
$\sin y = \tfrac12 \sin x$ for
$-\pi\! \le \! x \! \le \! \pi$ and
\mbox{$ -\pi \, \le\, y\, \le\, \pi\,$}.
\item
Without further calculation,
sketch the set of all points satisfying
$\cos y = \tfrac12 \sin x$ for $- \pi \le x \le \pi$ and
$ -\pi \le y \le \pi$.
\end{questionparts}Solution source
\begin{questionparts}
\item $\,$
\begin{center}
\begin{tikzpicture}
\def\functionf(#1){((#1)+.5)*((#1)-1)*((#1)-2.1)};
\def\xl{-4};
\def\xu{4};
\def\yl{-4};
\def\yu{4};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{10/\xrange}
\pgfmathsetmacro{\yscale}{10/\yrange}
% Define the styles for the axes and grid
\tikzset{
axis/.style={very thick, ->},
grid/.style={thin, gray!30},
x=\xscale cm,
y=\yscale cm
}
% Define the bounding region with clip
\begin{scope}
% You can modify these values to change your plotting region
\clip (\xl,\yl) rectangle (\xu,\yu);
% Draw a grid (optional)
% \draw[grid] (-5,-3) grid (5,3);
% \filldraw (-0.5, 0) circle (1.5pt) node[below]{$p$};
% \filldraw (1, 0) circle (1.5pt) node[below]{$q$};
% \filldraw (2.1, 0) circle (1.5pt) node[below]{$r$};
% \draw[thick, blue, smooth, domain=\xl:\xu, samples=100]
% plot (\x, {\functionf(\x)});
\draw ({-pi}, {-pi}) -- ({pi}, {pi}) node[pos=0.3, sloped, below] {$y=x$};
\draw ({0}, {pi}) -- ({pi}, {0}) node[pos=0.3, sloped, below] {$x+y = \pi$};
\draw ({0}, {-pi}) -- ({-pi}, {0}) node[pos=0.3, sloped, below] {$x+y = -\pi$};
\filldraw ({pi}, {-pi}) circle (1pt) node[above, left] {$(\pi, -\pi)$};
\filldraw ({-pi}, pi) circle (1pt) node[below, right] {$(-\pi, \pi)$};
% \node[blue, above, rotate=70] at (-1, {\functionf(-1)}) {\tiny $y = (x-p)(x-q)(x-r)$};
\end{scope}
% Set up axes
\draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
\draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
\end{tikzpicture}
\end{center}
\item $\,$ \begin{align*}
&& \sin y &= \tfrac12 \sin x \\
\Rightarrow && \frac{\d y}{\d x} \cos y &= \tfrac12 \cos x \\
\Rightarrow && \frac{\d y}{\d x} &= \frac{\cos x}{2 \cos y} \\
&&&= \frac{\cos x}{2 \sqrt{1-\sin^2 y}} \\
&&&= \frac{\cos x}{2 \sqrt{1-\frac14 \sin^2 x}} \\
&&&= \frac{\cos x}{\sqrt{4-\sin^2 x}} \\
\\
&& y'' &= \frac{-\sin x \cdot (4-\sin^2 x)^{\frac12} - \cos x \cdot (4-\sin^2 x)^{-\frac12} \cdot 2 \sin x \cos x}{(4-\sin^2 x)} \\
&&&= \frac{-\sin x \cdot (4-\sin^2 x) - \cos x \cdot 2 \sin x \cos x}{(4-\sin^2x)^{\frac32}} \\
&&&= \frac{-\sin x \cdot (4-\sin^2 x) - \sin x (1-\sin^2x)}{(4-\sin^2x)^{\frac32}} \\
&&&= \frac{-3\sin x }{(4-\sin^2x)^{\frac32}} \\
\end{align*}
\begin{center}
\begin{tikzpicture}
\def\functionf(#1){((#1)+.5)*((#1)-1)*((#1)-2.1)};
\def\xl{-4};
\def\xu{4};
\def\yl{-4};
\def\yu{4};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{10/\xrange}
\pgfmathsetmacro{\yscale}{10/\yrange}
% Define the styles for the axes and grid
\tikzset{
axis/.style={very thick, ->},
grid/.style={thin, gray!30},
x=\xscale cm,
y=\yscale cm
}
% Define the bounding region with clip
\begin{scope}
% You can modify these values to change your plotting region
\clip (\xl,\yl) rectangle (\xu,\yu);
% Draw a grid (optional)
% \draw[grid] (-5,-3) grid (5,3);
% \filldraw (-0.5, 0) circle (1.5pt) node[below]{$p$};
% \filldraw (1, 0) circle (1.5pt) node[below]{$q$};
% \filldraw (2.1, 0) circle (1.5pt) node[below]{$r$};
\draw[ultra thick, blue, smooth, domain={0}:{pi/2}, samples=400]
plot (\x, {asin(0.5*sin(\x*180/pi))*pi/180});
\draw[thick, blue, smooth, domain={-pi}:{pi}, samples=400]
plot (\x, {asin(0.5*sin(\x*180/pi))*pi/180});
\draw[thick, blue, smooth, domain={0}:{pi}, samples=400]
plot (\x, {pi-asin(0.5*sin(\x*180/pi))*pi/180});
\draw[thick, blue, smooth, domain={-pi}:{0}, samples=400]
plot (\x, {-pi-asin(0.5*sin(\x*180/pi))*pi/180});
\filldraw[blue] ({pi}, {-pi}) circle (1pt) node[above, left] {$(\pi, -\pi)$};
\filldraw[blue] ({-pi}, pi) circle (1pt) node[below, right] {$(-\pi, \pi)$};
\end{scope}
% Set up axes
\draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
\draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
\end{tikzpicture}
\end{center}
\item $\,$
\begin{center}
\begin{tikzpicture}
\def\functionf(#1){((#1)+.5)*((#1)-1)*((#1)-2.1)};
\def\xl{-4};
\def\xu{4};
\def\yl{-4};
\def\yu{4};
% Calculate scaling factors to make the plot square
\pgfmathsetmacro{\xrange}{\xu-\xl}
\pgfmathsetmacro{\yrange}{\yu-\yl}
\pgfmathsetmacro{\xscale}{10/\xrange}
\pgfmathsetmacro{\yscale}{10/\yrange}
% Define the styles for the axes and grid
\tikzset{
axis/.style={very thick, ->},
grid/.style={thin, gray!30},
x=\xscale cm,
y=\yscale cm
}
% Define the bounding region with clip
\begin{scope}
% You can modify these values to change your plotting region
\clip (\xl,\yl) rectangle (\xu,\yu);
% Draw a grid (optional)
% \draw[grid] (-5,-3) grid (5,3);
% \filldraw (-0.5, 0) circle (1.5pt) node[below]{$p$};
% \filldraw (1, 0) circle (1.5pt) node[below]{$q$};
% \filldraw (2.1, 0) circle (1.5pt) node[below]{$r$};
\draw[thick, blue, smooth, domain={-pi}:{pi}, samples=400]
plot (\x, {acos(0.5*sin(\x*180/pi))*pi/180});
\draw[thick, blue, smooth, domain={-pi}:{pi}, samples=400]
plot (\x, {-acos(0.5*sin(\x*180/pi))*pi/180});
\end{scope}
% Set up axes
\draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
\draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};
\end{tikzpicture}
\end{center}
\end{questionparts}
Attempts fell to around the 50% figure with marks scored by those who attempted the question averaging about 10 out of 20. There is not much to this question beyond the baseline realisation that sin sin does not necessarily imply that . In essence, it is all about "quadrants" work, where candidates need to consider the two solutions, and – in one period of the sine function, and then adding or subtracting multiples of 2 as necessary. Once one has done this, the accompanying straight‐line segments are straightforward marks in the last part of (i). A lot of marks were gained in (ii), as candidates were clearly attracted by the familiar "differentiate this couple of times" demand; most of them were quite happy with the differentiation, performed either implicitly or directly using arcsines. The drawings required in (ii) and (iii) then relied on an appreciation of the symmetries of the sine function, along with the use of the identity cos ≡ sin 21 y .