| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|linear| and y=linear with unknown constants, then solve |
| Difficulty | Moderate -0.8 Part (a) is a routine sketch of a V-shaped modulus graph with vertex at (2a, 0), requiring only basic understanding of translations. Part (b) involves solving a linear-modulus inequality by considering two cases, which is a standard textbook exercise with straightforward algebraic manipulation and no novel insight required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct V-shape graph, roughly symmetrical, both sections solid straight lines. \(2a\) marked on each axis. Graph extends into negative \(x\). | B1 | Correct shape, roughly symmetrical. Both sections should be solid straight lines. Allow construction lines if dashed or clearly fainter. \(2a\) marked on each axis (must be \(2a\), not just 2). Needs to extend into negative \(x\). If \(a\) is given a value, then B0. Ignore \(y = 2x - 3a\) if seen. |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve linear equation or inequality to obtain critical value \(x = \frac{5}{3}a\) or exact equivalent | B1 | Ignore \(x = a\) if seen |
| Obtain \(x < \frac{5}{3}a\) or exact equivalent | B1 | Accept \(x < \frac{10}{6}a\) or \(\left(-\infty, \frac{5}{3}a\right)\). Must be strict inequality. Need a clear final solution: \(x > a\) or \(x < a\) must be rejected if seen as part of working. Rejection can be implied, e.g. if only the correct inequality is underlined. B0 B0 if \(a\) is given a value. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve quadratic equation \(\left(2x-3a\right)^2 = \left(x-2a\right)^2\) to obtain critical value \(x = \frac{5}{3}a\) or exact equivalent | (B1) | \(\left(3x^2 - 8ax + 5a^2 = 0\right)\). Ignore \(x = a\) if seen. |
| Obtain \(x < \frac{5}{3}a\) or exact equivalent | (B1) | Accept \(x < \frac{10}{6}a\) or \(\left(-\infty, \frac{5}{3}a\right)\). Must be strict inequality. Need a clear final solution: \(x > a\) or \(x < a\) must be rejected if seen as part of working. B0 B0 if \(a\) is given a value. |
| Total: 2 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct V-shape graph, roughly symmetrical, both sections solid straight lines. $2a$ marked on each axis. Graph extends into negative $x$. | **B1** | Correct shape, roughly symmetrical. Both sections should be solid straight lines. Allow construction lines if dashed or clearly fainter. $2a$ marked on each axis (must be $2a$, not just 2). Needs to extend into negative $x$. If $a$ is given a value, then B0. Ignore $y = 2x - 3a$ if seen. |
| | **Total: 1** | |
---
### Part (b):
**Main Method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve linear equation or inequality to obtain critical value $x = \frac{5}{3}a$ or exact equivalent | **B1** | Ignore $x = a$ if seen |
| Obtain $x < \frac{5}{3}a$ or exact equivalent | **B1** | Accept $x < \frac{10}{6}a$ or $\left(-\infty, \frac{5}{3}a\right)$. Must be strict inequality. Need a clear final solution: $x > a$ or $x < a$ must be rejected if seen as part of working. Rejection can be implied, e.g. if only the correct inequality is underlined. B0 B0 if $a$ is given a value. |
**Alternative Method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve quadratic equation $\left(2x-3a\right)^2 = \left(x-2a\right)^2$ to obtain critical value $x = \frac{5}{3}a$ or exact equivalent | **(B1)** | $\left(3x^2 - 8ax + 5a^2 = 0\right)$. Ignore $x = a$ if seen. |
| Obtain $x < \frac{5}{3}a$ or exact equivalent | **(B1)** | Accept $x < \frac{10}{6}a$ or $\left(-\infty, \frac{5}{3}a\right)$. Must be strict inequality. Need a clear final solution: $x > a$ or $x < a$ must be rejected if seen as part of working. B0 B0 if $a$ is given a value. |
| | **Total: 2** | |1
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $\mathrm { y } = | \mathrm { x } - 2 \mathrm { a } |$, where $a$ is a positive constant.
\item Solve the inequality $2 \mathrm { x } - 3 \mathrm { a } < | \mathrm { x } - 2 \mathrm { a } |$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q1 [3]}}