| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch two |linear| functions and solve related equation/inequality |
| Difficulty | Moderate -0.8 This is a straightforward modulus question requiring standard techniques: sketching basic modulus graphs (V-shapes with known vertices), solving by considering cases (x ≥ 0 and x < 0, plus 2x - 4 ≥ 0 and 2x - 4 < 0), and reading the inequality solution from the graph or algebraic work. All steps are routine textbook exercises with no novel insight required, making it easier than average. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
| Answer | Marks | Guidance |
|---|---|---|
| Graph shown | B1 | \(y = |
| Answer | Marks |
|---|---|
| M1 A1 | 2 branches mod graph \(x > 0\) for \(y = 0\); for 2, 4; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2x - 4\), \(x = 4\); \(-x = 2x - 4\); \(x = \frac{4}{3}\) | B1 M1 A1 | OE one value only; 3 marks total |
| Alternative: \(x^2 = (2x-4)^2\); \(x = 4, \frac{4}{3}\) | M1 A1 A1 |
| Answer | Marks |
|---|---|
| M1 A1 | 4 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4}{3} < x < 4\) | M1 A1 | \(\frac{4}{3}, 4\) (ft) identified as extremes; CAO; 2 marks total |
### 4(a)(i)
Graph shown | B1 | $y = |x|$; 1 mark
### 4(a)(ii)
| M1 A1 | 2 branches mod graph $x > 0$ for $y = 0$; for 2, 4; 2 marks total
### 4(b)(i)
$x = 2x - 4$, $x = 4$; $-x = 2x - 4$; $x = \frac{4}{3}$ | B1 M1 A1 | OE one value only; 3 marks total
**Alternative:** $x^2 = (2x-4)^2$; $x = 4, \frac{4}{3}$ | M1 A1 A1 |
### 4(b)(ii)
| M1 A1 | 4 marks total
### 4(b)(iii)
$\frac{4}{3} < x < 4$ | M1 A1 | $\frac{4}{3}, 4$ (ft) identified as extremes; CAO; 2 marks total
**Total for Question 4:** 8 marks4
\begin{enumerate}[label=(\alph*)]
\item Sketch and label on the same set of axes the graphs of:
\begin{enumerate}[label=(\roman*)]
\item $y = | x |$;
\item $y = | 2 x - 4 |$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Solve the equation $| x | = | 2 x - 4 |$.
\item Hence, or otherwise, solve the inequality $| x | > | 2 x - 4 |$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q4 [8]}}