| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|f(x)| or y=f(|x|) for non-linear f(x) and solve |
| Difficulty | Standard +0.3 This is a straightforward modulus question requiring standard techniques: identifying transformations (stretch and translation), sketching a modulus graph by reflecting negative parts, and solving |quadratic| = constant by considering two cases. All steps are routine C3 material with no novel problem-solving required, making it slightly 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 function1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Stretch (I) | M1 | \(1 + \) (II or III) |
| Scale factor \(\frac{1}{2}\) (II) | A1 | All correct |
| parallel to \(x\)-axis (III) | ||
| (Or scale factor 4 parallel to \(y\)-axis) | ||
| Translation | M1 | |
| \(\begin{pmatrix}0\\-5\end{pmatrix}\) OE | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| translate \(\begin{pmatrix}0\\-5\\-4\end{pmatrix}\), stretch of 4 ∥ \(y\)-axis | Mark translation first. Mark stretch as above, but relative to their translation. | |
| translate \(\begin{pmatrix}0\\-5\end{pmatrix}\), stretch sf \(\frac{1}{2}\) ∥ \(x\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| [Graph showing modulus function with V-shape, vertex at (0,5)] | M1 | Modulus graph symmetrical about \(y\)-axis |
| A1 | left of \(-\frac{\sqrt{5}}{2}\) and right of \(\frac{\sqrt{5}}{2}\) | |
| A1 | \((0, 5)\), cusps drawn and no straight lines between cusps | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(4x^2 - 5 = 4\) | B1 | OE |
| \(4x^2 = 9\) | ||
| \(x = \pm\frac{3}{2}\) | ||
| \(4x^2 - 5 = -4\) | M1 | \(16x^4 - 40x^2 + 9 = 0\) |
| \(4x^2 = 1\) | ||
| \(x = \pm\frac{1}{2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x \leq -\frac{3}{2}, x \geq \frac{3}{2}\) | B1 F | 2 correct statements |
| \(-\frac{1}{2} \leq x, x \leq \frac{1}{2}\) | B1 F | 4 correct statements |
| SC c(ii): 1 mark penalty for strict inequalities |
### 7(a)
Stretch (I) | M1 | $1 + $ (II or III) |
Scale factor $\frac{1}{2}$ (II) | A1 | All correct |
parallel to $x$-axis (III) | | |
(Or scale factor 4 parallel to $y$-axis) | | |
Translation | M1 | |
$\begin{pmatrix}0\\-5\end{pmatrix}$ OE | A1 | | **Total: 4**
**Alternatives:**
translate $\begin{pmatrix}0\\-5\\-4\end{pmatrix}$, stretch of 4 ∥ $y$-axis | | Mark translation first. Mark stretch as above, but relative to their translation. |
translate $\begin{pmatrix}0\\-5\end{pmatrix}$, stretch sf $\frac{1}{2}$ ∥ $x$-axis | | |
### 7(b)
[Graph showing modulus function with V-shape, vertex at (0,5)] | M1 | Modulus graph symmetrical about $y$-axis |
| A1 | left of $-\frac{\sqrt{5}}{2}$ and right of $\frac{\sqrt{5}}{2}$ |
| A1 | $(0, 5)$, cusps drawn and no straight lines between cusps | **Total: 3**
### 7(c)(i)
$4x^2 - 5 = 4$ | B1 | OE |
$4x^2 = 9$ | | |
$x = \pm\frac{3}{2}$ | | |
$4x^2 - 5 = -4$ | M1 | $16x^4 - 40x^2 + 9 = 0$ |
$4x^2 = 1$ | | |
$x = \pm\frac{1}{2}$ | A1 | | **Total: 3**
### 7(c)(ii)
$x \leq -\frac{3}{2}, x \geq \frac{3}{2}$ | B1 F | 2 correct statements |
$-\frac{1}{2} \leq x, x \leq \frac{1}{2}$ | B1 F | 4 correct statements | **Total: 2**
| | | **SC c(ii):** 1 mark penalty for strict inequalities |7
\begin{enumerate}[label=(\alph*)]
\item Describe a sequence of two geometrical transformations that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = 4 x ^ { 2 } - 5$.
\item Sketch the graph of $y = \left| 4 x ^ { 2 } - 5 \right|$, indicating the coordinates of the point where the curve crosses the $y$-axis.
\item \begin{enumerate}[label=(\roman*)]
\item Solve the equation $\left| 4 x ^ { 2 } - 5 \right| = 4$.
\item Hence, or otherwise, solve the inequality $\left| 4 x ^ { 2 } - 5 \right| \geqslant 4$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2008 Q7 [12]}}