| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Find constants from sketch features |
| Difficulty | Standard +0.3 This is a straightforward C1 question requiring students to use given curve features (roots and a point) to form simultaneous equations for a cubic. The 'touches' condition gives a repeated root, making it easier. Part (b) is a standard horizontal stretch transformation. Slightly easier than average due to the helpful 'touches' clue and routine algebraic manipulation. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f(x) = (x+1)(x-2)^2\) | M1A1B1 | M1: Stating or writing that \((x \pm 1)\) or \((x \pm 2)\) is a factor. A1: Both \((x+1)\) and \((x-2)\) are factors. B1: \(y\) or \(f(x) = (x+1)(x-2)^2\) |
| \(= (x+1)(x^2 - 4x + 4) = x^3 - 3x^2 + 4\) | M1A1 | M1: Multiplying out a quadratic to get 3 terms then multiplying by the linear term. A1: \(x^3 - 3x^2 + 4\) or \(a = -3\), \(b = 0\), \(c = 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct shape with maximum on y-axis | B1 | Same shape and position, ignore coordinates |
| \(y\) intercept \(= 4\) or their \(c\) | B1ft | |
| \(x\) coordinates at \(-2\) and \(4\) marked as coordinates | B1 | Allow \((0, -2)\) and \((0, 4)\) if marked in correct position. Curve must cross or at least stop at \(x = -2\) |
## Question 9:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f(x) = (x+1)(x-2)^2$ | M1A1B1 | M1: Stating or writing that $(x \pm 1)$ or $(x \pm 2)$ is a factor. A1: Both $(x+1)$ and $(x-2)$ are factors. B1: $y$ or $f(x) = (x+1)(x-2)^2$ |
| $= (x+1)(x^2 - 4x + 4) = x^3 - 3x^2 + 4$ | M1A1 | M1: Multiplying out a quadratic to get 3 terms then multiplying by the linear term. A1: $x^3 - 3x^2 + 4$ or $a = -3$, $b = 0$, $c = 4$ |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct shape with maximum on y-axis | B1 | Same shape and position, ignore coordinates |
| $y$ intercept $= 4$ or their $c$ | B1ft | |
| $x$ coordinates at $-2$ and $4$ marked as coordinates | B1 | Allow $(0, -2)$ and $(0, 4)$ if marked in correct position. Curve must cross or at least stop at $x = -2$ |
---9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cfc23548-bf4f-4efa-9ceb-b8d03bb1f019-13_698_1413_118_280}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$.\\
The curve $C$ passes through the point $( - 1,0 )$ and touches the $x$-axis at the point $( 2,0 )$.\\
The curve $C$ has a maximum at the point ( 0,4 ).
\begin{enumerate}[label=(\alph*)]
\item The equation of the curve $C$ can be written in the form
$$y = x ^ { 3 } + a x ^ { 2 } + b x + c$$
where $a$, $b$ and $c$ are integers.\\
Calculate the values of $a , b$ and $c$.
\item Sketch the curve with equation $y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)$ in the space provided on page 24
Show clearly the coordinates of all the points where the curve crosses or meets the coordinate axes.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2013 Q9 [8]}}