2012 Paper 2 Q5
Year: 2012
Paper: 2
Question Number: 5
Course: LFM Pure
Section: Differentiation
Difficulty: 1600.0
Banger: 1484.0
curve sketching
asymptotes
stationary points
rational function
differentiation
symmetry
partial fractions
Problem
- Sketch the curve \(y=\f(x)\), where \[ \f(x) = \frac 1 {(x-a)^2 -1} \hspace{2cm}(x\ne a\pm1), \] and \(a\) is a constant.
- The function \(\g(x)\) is defined by \[ \g(x) = \frac 1 {\big( (x-a)^2-1 \big) \big( (x-b)^2 -1\big)} \hspace{1cm}(x\ne a\pm1, \ x\ne b\pm1), \] where \(a\) and \(b\) are constants, and \(b>a\). Sketch the curves \(y=\g(x)\) in the two cases \(b>a+2\) and \(b=a+2\), finding the values of \(x\) at the stationary points.
Solution
- \(\,\)
- \(\,\) \begin{align*}
&& \frac{\d}{\d x} \left ( \frac{1}{g(x)} \right) &= \frac{\d }{\d x} \left ( \big( (x-a)^2-1 \big) \big( (x-b)^2 -1\big)\right) \\
&&&= ((x-a)^2-1)(2(x-b))+((x-b)^2-1)(2(x-a)) \\
&&&= 2(2x-a-b)(x^2-(a+b)x+ab-1) \\
\Rightarrow && x &= \frac{a+b}{2}, \frac{a+b \pm \sqrt{(a+b)^2-4ab+4}}{2} \\
&&&= \frac{a+b}{2}, \frac{a+b \pm \sqrt{(a-b)^2+4}}{2}
\end{align*}
If \(b > a+2\):
If \(b = a+2\):
Examiner's report
— 2012 STEP 2, Question 5
~70% attempted (inferred)
Inferred ~70% from intro 'around 700'; described as 'relatively low average mark overall' among popular pure questions
From the paper introduction
There were just over 1000 entries for paper II this year, almost exactly the same number as last year. Overall, the paper was found marginally easier than its predecessor, which means that it was pitched at exactly the level intended and produced the hoped-for outcomes. Almost 50 candidates scored 100 marks or more, with more than 400 gaining at least half marks on the paper. At the lower end of the scale, around a quarter of the entry failed to score more than 40 marks. It was pleasing to note that the advice of recent years, encouraging students not to make attempts at lots of early parts to questions but rather to spend their time getting to grips with the six that can count towards their paper total, was more obviously being heeded in 2012 than I can recall being the case previously. As in previous years, the pure maths questions provided the bulk of candidates' work, with relatively few efforts to be found at the applied ones. Questions 1 and 2 were the most popular questions, although each drew only around 800 "hits" – fewer than usual. Questions 3 – 5 & 8 were almost as popular (around 700), with Q6 attracting the interest of under 450 candidates and Q7 under 200. Q9 was the most popular applied question – and, as it turned out, the most successfully attempted question on the paper – with very little interest shown in the rest of Sections B or C.
Source: Cambridge STEP 2012 Examiner's Report
· 2012-full.pdf
Rating Information
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
Show LaTeX source
Problem source
\begin{questionparts}
\item Sketch the curve $y=\f(x)$, where
\[
\f(x) = \frac 1 {(x-a)^2 -1}
\hspace{2cm}(x\ne a\pm1),
\]
and $a$ is a constant.
\item The function $\g(x)$ is defined by
\[
\g(x) = \frac 1 {\big( (x-a)^2-1 \big) \big( (x-b)^2 -1\big)}
\hspace{1cm}(x\ne a\pm1, \ x\ne b\pm1),
\]
where $a$ and $b$ are constants, and $b>a$.
Sketch the curves $y=\g(x)$
in the two cases $b>a+2$ and $b=a+2$, finding the
values of $x$ at the stationary points.
\end{questionparts}Solution source
\begin{questionparts}
\item $\,$
\begin{center}
\begin{tikzpicture}
\def\a{-0.5};
\def\functionf(#1){1/(((#1)-\a)^2-1)};
\def\xl{-3};
\def\xu{3};
\def\yl{-3};
\def\yu{3};
% 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[blue, smooth, thick, domain=\xl:{\a-1-.1}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\a-1+.1}:{\a+1-0.1}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\a+1+0.1}:\xu, samples=101]
plot({\x}, {\functionf(\x)});
% \draw[red, smooth, thick, domain=1:2, samples=101]
% plot({\x}, {1+3-1});
% \draw[red, smooth, thick, domain=2:3, samples=101]
% plot({\x}, {4+6-1});
\node[blue, above, rotate=60] at (-1, {\functionf(-1)}) {\tiny $y=f(x)$};
\draw[red, dashed] ({\a-1}, \yl) -- ({\a-1}, \yu) node[pos=0.3, above, sloped] {\tiny $x = a-1$};
\draw[red, dashed] ({\a+1}, \yl) -- ({\a+1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = a-1$};
% \filldraw (-1, 0) circle (1.5pt) node[below] {$-1$};
% \filldraw ({-2/3}, {\functionf(-2/3)}) circle (1.5pt) node[below] {$(-\frac23, -\frac{4078}{81})$};
\filldraw ({\a}, {-1}) circle (1.5pt) node[above] {\tiny $(a, -1)$};
\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*}
&& \frac{\d}{\d x} \left ( \frac{1}{g(x)} \right) &= \frac{\d }{\d x} \left ( \big( (x-a)^2-1 \big) \big( (x-b)^2 -1\big)\right) \\
&&&= ((x-a)^2-1)(2(x-b))+((x-b)^2-1)(2(x-a)) \\
&&&= 2(2x-a-b)(x^2-(a+b)x+ab-1) \\
\Rightarrow && x &= \frac{a+b}{2}, \frac{a+b \pm \sqrt{(a+b)^2-4ab+4}}{2} \\
&&&= \frac{a+b}{2}, \frac{a+b \pm \sqrt{(a-b)^2+4}}{2}
\end{align*}
If $b > a+2$:
\begin{center}
\begin{tikzpicture}
\def\a{-0.5};
\def\b{3};
\def\functionf(#1){1/(((#1)-\a)^2-1)*1/(((#1)-\b)^2-1)};
\def\xl{-3};
\def\xu{6};
\def\yl{-1};
\def\yu{1};
% 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[blue, smooth, thick, domain=\xl:{\a-1-.01}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\a-1+.01}:{\a+1-0.01}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\a+1+0.1}:{\b-1-0.01}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\b-1+0.01}:{\b+1-0.01}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\b+1+0.01}:\xu, samples=101]
plot({\x}, {\functionf(\x)});
% \draw[red, smooth, thick, domain=1:2, samples=101]
% plot({\x}, {1+3-1});
% \draw[red, smooth, thick, domain=2:3, samples=101]
% plot({\x}, {4+6-1});
\node[blue, above, rotate=30] at (-2, {\functionf(-2)}) {\tiny $y=f(x)$};
\draw[red, dashed] ({\a-1}, \yl) -- ({\a-1}, \yu) node[pos=0.3, above, sloped] {\tiny $x = a-1$};
\draw[red, dashed] ({\a+1}, \yl) -- ({\a+1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = a+1$};
\draw[red, dashed] ({\b+1}, \yl) -- ({\b+1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = b+1$};
\draw[red, dashed] ({\b-1}, \yl) -- ({\b-1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = b-1$};
% \filldraw (-1, 0) circle (1.5pt) node[below] {$-1$};
% \filldraw ({-2/3}, {\functionf(-2/3)}) circle (1.5pt) node[below] {$(-\frac23, -\frac{4078}{81})$};
\filldraw ({(\a+\b)/2}, 0) circle (1.5pt) node[below] {\tiny $\frac{a+b}2$};
\filldraw ({((\a+\b)+sqrt((\a-\b)^2+4))/2}, 0) circle (1.5pt) node[above] {\tiny $\frac{a+b + \sqrt{(b-a)^2+4}}2$};
\filldraw ({((\a+\b)-sqrt((\a-\b)^2+4))/2}, 0) circle (1.5pt) node[above] {\tiny $\frac{a+b - \sqrt{(b-a)^2+4}}2$};
\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}
If $b = a+2$:
\begin{center}
\begin{tikzpicture}
\def\a{-0.5};
\def\b{1.5};
\def\functionf(#1){1/(((#1)-\a)^2-1)*1/(((#1)-\b)^2-1)};
\def\xl{-3};
\def\xu{6};
\def\yl{-1};
\def\yu{1};
% 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[blue, smooth, thick, domain=\xl:{\a-1-.01}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\a-1+.01}:{\a+1-0.1}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\b-1+0.1}:{\b+1-0.01}, samples=101]
plot({\x}, {\functionf(\x)});
\draw[blue, smooth, thick, domain={\b+1+0.01}:\xu, samples=101]
plot({\x}, {\functionf(\x)});
% \draw[red, smooth, thick, domain=1:2, samples=101]
% plot({\x}, {1+3-1});
% \draw[red, smooth, thick, domain=2:3, samples=101]
% plot({\x}, {4+6-1});
\node[blue, above, rotate=30] at (-2, {\functionf(-2)}) {\tiny $y=f(x)$};
\draw[red, dashed] ({\a-1}, \yl) -- ({\a-1}, \yu) node[pos=0.3, above, sloped] {\tiny $x = a-1$};
\draw[red, dashed] ({\a+1}, \yl) -- ({\a+1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = a+1=b-1$};
\draw[red, dashed] ({\b+1}, \yl) -- ({\b+1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = b+1$};
% \draw[red, dashed] ({\b-1}, \yl) -- ({\b-1}, \yu) node[pos=0.3, below, sloped] {\tiny $x = b-1$};
% \filldraw (-1, 0) circle (1.5pt) node[below] {$-1$};
% \filldraw ({-2/3}, {\functionf(-2/3)}) circle (1.5pt) node[below] {$(-\frac23, -\frac{4078}{81})$};
% \filldraw ({(\a+\b)/2}, 0) circle (1.5pt) node[below] {\tiny $\frac{a+b}2$};
\filldraw ({((\a+\b)+sqrt((\a-\b)^2+4))/2}, 0) circle (1.5pt) node[above] {\tiny $\frac{a+b + \sqrt{(b-a)^2+4}}2$};
\filldraw ({((\a+\b)-sqrt((\a-\b)^2+4))/2}, 0) circle (1.5pt) node[above] {\tiny $\frac{a+b - \sqrt{(b-a)^2+4}}2$};
\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}
Another popular question, but scoring a relatively low average mark overall; however, this was partly due to the high number of partial attempts, and good efforts usually scored around 14 marks. Surprisingly, the greatest difficulty was found in the differentiation in part (ii). There were lots of marks for the curve-sketching, and several easy features to work with: principally the symmetry and the asymptotes (and the behaviour of the functions on either side of these). For many who struggled, the biggest problem lay in where to put the y-axis, which was largely immaterial. As mentioned already, differentiation attempts were rather poor on the whole, with muddling of the Chain, Product and Quotient Rules. Even for those who differentiated correctly, extracting the factor (2x – a – b) proved too tough, despite the fact it might have been obvious with a bit of thought.