2007 Paper 1 Q8
Year: 2007
Paper: 1
Question Number: 8
Course: LFM Stats And Pure
Section: Polynomials
Problem
- Sketch on the same axes the curves \((*)\) and \((**)\) when \(a = 2\).
- Sketch on the same axes the curves \((*)\) and \((**)\) when \(a = 1\).
- Sketch on the same axes the curves \((*)\) and \((**)\) when \(a = -2\).
Solution
Examiner's report
— 2007 STEP 1, Question 8There were significantly more candidates attempting this paper this year (an increase of nearly 50%), but many found it to be very difficult and only achieved low scores. In particular, the level of algebraic skill required by the questions was often lacking. The examiners' express their concern that this was the case despite a conscious effort to make the paper more accessible than last year's. At this level, the fluent, confident and correct handling of mathematical symbols (and numbers) is necessary and is expected; many good starts to questions soon became unstuck after a simple slip. Graph sketching was usually poor: if future candidates wanted to improve one particular skill, they would be well advised to develop this. There were of course some excellent scripts, full of logical clarity and perceptive insight. It was pleasing to note that the applied questions were more popular this year, and many candidates scored well on at least one of these. It was however surprising how rarely answers to questions such as 5, 9, 10, 11 and 12 began with a diagram. However, the examiners were left with the overall feeling that some candidates had not prepared themselves well for the examination. The use of past papers to ensure adequate preparation is strongly recommended. A student's first exposure to STEP questions can be a daunting, demanding experience; it is a shame if that takes place during a public examination on which so much rides. Further, and fuller, discussion of the solutions to these questions can be found in the Hints and Answers document.
Rating Information
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1516.0
Banger Comparisons: 1
Show LaTeX source
Problem source
A curve is given by the equation
\[
y = ax^3 - 6ax^2+ \left( 12a + 12 \right)x - \left( 8a + 16 \right)\,,
\tag{$*$}
\]
where $a$ is a real number. Show that this curve touches the curve with equation
\[
y=x^3
\tag{$**$}
\]
at $\left( 2 \, , \, 8 \right)$.
Determine the coordinates of any other point of intersection of the two curves.
\begin{questionparts}
\item Sketch on the same axes the curves $(*)$ and $(**)$ when $a = 2$.
\item Sketch on the same axes the curves $(*)$ and $(**)$ when $a = 1$.
\item Sketch on the same axes the curves $(*)$ and $(**)$ when $a = -2$.
\end{questionparts}Solution source
\begin{align*}
&& y &= ax^3 - 6ax^2+ \left( 12a + 12 \right)x - \left( 8a + 16 \right) \\
&& y(2) &= 8a-24a+24a+24-8a-16 \\
&&&= 8 \\
&& y'(x) &= 3ax^2-12ax+(12a+12) \\
&& y'(0) &= 12a-24a+12a+12 \\
&&&= 12
\end{align*}
Therefore since our curve has the same value and gradient at $(2,8)$ as $y = x^3$ they must touch at this point.
Therefore
\begin{align*}
&& ax^3 - 6ax^2+ \left( 12a + 12 \right)x - \left( 8a + 16 \right) - x^3 &= (x-2)^2((a-1)x-(2a+4))
\end{align*}
Therefore if $a \neq 1$, they touch again when $x = \frac{2a+4}{a-1}$.
\begin{questionparts}
\item \begin{center}
\begin{tikzpicture}
\def\a{2};
\def\functionf(#1){ \a*(#1)*(#1)*(#1)-6*\a*(#1)*(#1)+ (12*\a + 12)*(#1) -(8*\a + 16)};
\def\xl{-10};
\def\xu{10};
\def\yl{-50};
\def\yu{600};
% 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);
\draw[thick, blue, smooth, domain=\xl:\xu, samples=100]
plot (\x, {\functionf(\x)});
\draw[thick, red, smooth, domain=\xl:\xu, samples=100]
plot (\x, {\x*\x*\x});
\filldraw (2,8) circle (1.5pt) node[above] {$(2,8)$};
\filldraw (8,512) circle (1.5pt) node[right] {$(8,8^3)$};
\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\a{1};
\def\functionf(#1){ \a*(#1)*(#1)*(#1)-6*\a*(#1)*(#1)+ (12*\a + 12)*(#1) -(8*\a + 16)};
\def\xl{-10};
\def\xu{10};
\def\yl{-200};
\def\yu{600};
% 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);
\draw[thick, blue, smooth, domain=\xl:\xu, samples=100]
plot (\x, {\functionf(\x)});
\draw[thick, red, smooth, domain=\xl:\xu, samples=100]
plot (\x, {\x*\x*\x});
\filldraw (2,8) circle (1.5pt) node[above] {$(2,8)$};
% \filldraw (8,512) circle (1.5pt) node[right] {$(8,8^3)$};
\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\a{-2};
\def\functionf(#1){ \a*(#1)*(#1)*(#1)-6*\a*(#1)*(#1)+ (12*\a + 12)*(#1) -(8*\a + 16)};
\def\xl{-5};
\def\xu{8};
\def\yl{-50};
\def\yu{200};
% 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);
\draw[thick, blue, smooth, domain=\xl:\xu, samples=100]
plot (\x, {\functionf(\x)});
\draw[thick, red, smooth, domain=\xl:\xu, samples=100]
plot (\x, {\x*\x*\x});
\filldraw (2,8) circle (1.5pt) node[above] {$(2,8)$};
\filldraw (0,0) circle (1.5pt) node[above] {$(0,0)$};
% \filldraw (8,512) circle (1.5pt) node[right] {$(8,8^3)$};
\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}
This question was answered poorly; many candidates were unable to sketch the graphs correctly, even given the results derived earlier in the question. For example, many graphs did not touch at (2, 8). Also, many graphs were drawn with turning points, when a simple check of the derivative would have revealed that there were none. In part (iii), the effect of the negative coefficient of x3 was often ignored. Graph sketching is a very important skill in all mathematical subjects – from Economics to Engineering. STEP candidates are strongly advised to practise this skill as much as possible.