| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Deduce inequality solutions from sketch |
| Difficulty | Moderate -0.8 This is a straightforward transformation question requiring knowledge of horizontal translations (f(x+2) shifts left by 2), identifying new x-intercepts by subtracting 2 from original intercepts, and substituting x=0 to find the y-intercept. All steps are routine applications of standard C1 transformation rules with no problem-solving insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Horizontal translation (graph) | B1 | Does not have to cross the \(y\)-axis on the right but must at least reach the \(x\)-axis |
| Touching at \((-5, 0)\) | B1 | Could be stated anywhere or \(-5\) marked on \(x\)-axis. Or \((0,-5)\) marked in correct place. Be fairly generous with 'touching' if intention is clear |
| Right hand tail crossing at \((-1, 0)\) | B1 | Could be stated anywhere or \(-1\) marked on \(x\)-axis. Or \((0,-1)\) marked in correct place. Curve must cross \(x\)-axis and not stop at \(-1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x+5)^2(x+1)\) | B1 | Allow \((x+3+2)^2(x-1+2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| When \(x = 0\), \(y = 25\) | M1 A1 | M1: Substitutes \(x=0\) into their expression in part (b) which is not \(f(x)\). A1: \(y=25\) (coordinates not needed) |
## Question 8:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Horizontal translation (graph) | B1 | Does **not** have to cross the $y$-axis on the right but must at least reach the $x$-axis |
| Touching at $(-5, 0)$ | B1 | Could be stated anywhere or $-5$ marked on $x$-axis. Or $(0,-5)$ **marked in correct place**. Be fairly generous with 'touching' if intention is clear |
| Right hand tail crossing at $(-1, 0)$ | B1 | Could be stated anywhere or $-1$ marked on $x$-axis. Or $(0,-1)$ **marked in correct place**. Curve must **cross** $x$-axis and not stop at $-1$ |
**(3 marks)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x+5)^2(x+1)$ | B1 | Allow $(x+3+2)^2(x-1+2)$ |
**(1 mark)**
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| When $x = 0$, $y = 25$ | M1 A1 | M1: Substitutes $x=0$ into their expression in part (b) which is not $f(x)$. A1: $y=25$ (coordinates not needed) |
**Note:** If they expand incorrectly prior to substituting $x=0$, score M1 A0. $f(x+2) = x^3 + 11x^2 + 35x + 25$ **(2 marks) [6 marks total]**
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5cee336b-d9c9-4b18-ab82-52fdca1eb1e7-09_369_956_287_504}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$ where
$$\mathrm { f } ( x ) = ( x + 3 ) ^ { 2 } ( x - 1 ) , \quad x \in \mathbb { R }$$
The curve crosses the $x$-axis at $( 1,0 )$, touches it at $( - 3,0 )$ and crosses the $y$-axis at $( 0 , - 9 )$
\begin{enumerate}[label=(\alph*)]
\item In the space below, sketch the curve $C$ with equation $y = \mathrm { f } ( x + 2 )$ and state the coordinates of the points where the curve $C$ meets the $x$-axis.
\item Write down an equation of the curve $C$.
\item Use your answer to part (b) to find the coordinates of the point where the curve $C$ meets the $y$-axis.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2013 Q8 [6]}}