| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Transformations of modulus graphs from given f(x) sketch |
| Difficulty | Moderate -0.3 This question involves standard transformations of graphs (horizontal translation and composition with |x|) and solving a modulus equation. While it requires understanding of transformations and careful sketching, these are routine C3 techniques with no novel problem-solving required. Part (d) involves a straightforward case-by-case solution of |x-1| - 2 = 5x, which is slightly below average difficulty for A-level. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Translation \(\leftarrow\) by 1 | M1 | Method mark for translation |
| Intercepts correct (graph showing intercepts at \(-2\), \(0\), \(2\)) | A1 | (2 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x \geq 0\), correct "shape" | B1 | Provided graph is not original graph |
| Reflection in \(y\)-axis | B1\(\checkmark\) | Follow-through mark |
| Intercepts correct (at \(-3\), \(0\), \(3\)) | B1 | (3 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = -2\), \(b = -1\) | B1 B1 | (2 marks total) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Intersection of \(y = 5x\) with \(y = -x - 1\) | M1 A1 | |
| Solving to give \(x = -\frac{1}{6}\) | M1 A1 | (4 marks total) |
| Notes: If both values found for \(5x = -x-1\) and \(5x = x-3\), can score 3/4 for \(x = -\frac{1}{6}\) and \(x = -\frac{3}{4}\); must eliminate \(x = -\frac{3}{4}\) for final mark. Squaring approach: M1 correct method, \(24x^2 + 22x + 3 = 0\) (correct 3-term quadratic) A1, Solving M1, correct answer A1 | [Total: 11] |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Translation $\leftarrow$ by 1 | M1 | Method mark for translation |
| Intercepts correct (graph showing intercepts at $-2$, $0$, $2$) | A1 | (2 marks total) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x \geq 0$, correct "shape" | B1 | Provided graph is not original graph |
| Reflection in $y$-axis | B1$\checkmark$ | Follow-through mark |
| Intercepts correct (at $-3$, $0$, $3$) | B1 | (3 marks total) |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = -2$, $b = -1$ | B1 B1 | (2 marks total) |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Intersection of $y = 5x$ with $y = -x - 1$ | M1 A1 | |
| Solving to give $x = -\frac{1}{6}$ | M1 A1 | (4 marks total) |
| **Notes:** If both values found for $5x = -x-1$ and $5x = x-3$, can score 3/4 for $x = -\frac{1}{6}$ and $x = -\frac{3}{4}$; must eliminate $x = -\frac{3}{4}$ for final mark. Squaring approach: M1 correct method, $24x^2 + 22x + 3 = 0$ (correct 3-term quadratic) A1, Solving M1, correct answer A1 | | **[Total: 11]** |
---6.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{5af2eea6-bac1-455b-b25a-487d113e44ca-08_458_876_285_539}
\end{center}
\end{figure}
Figure 1 shows part of the graph of $y = \mathrm { f } ( x ) , x \in \mathbb { R }$. The graph consists of two line segments that meet at the point $( 1 , a ) , a < 0$. One line meets the $x$-axis at $( 3,0 )$. The other line meets the $x$-axis at $( - 1,0 )$ and the $y$-axis at $( 0 , b ) , b < 0$.
In separate diagrams, sketch the graph with equation
\begin{enumerate}[label=(\alph*)]
\item $y = \mathrm { f } ( x + 1 )$,
\item $y = \mathrm { f } ( | x | )$.
Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
Given that $\mathrm { f } ( x ) = | x - 1 | - 2$, find
\item the value of $a$ and the value of $b$,
\item the value of $x$ for which $\mathrm { f } ( x ) = 5 x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2005 Q6 [11]}}