| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|linear| then solve equation or inequality (numeric coefficients) |
| Difficulty | Moderate -0.8 Part (a) is a routine sketch of a basic translated modulus function requiring only recall of the V-shape centered at x=2. Part (b) involves solving a modulus inequality by considering cases (x≥2 and x<2), leading to straightforward linear inequalities—this is a standard textbook exercise with minimal problem-solving demand. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Make a recognisable sketch graph of \(y = | x - 2 | \) |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Find \(x\)-coordinate of intersection with \(y = 3x - 4\) | M1 | |
| Obtain \(x = \dfrac{3}{2}\) | A1 | |
| State final answer \(x > \dfrac{3}{2}\) only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve the linear inequality \(3x - 4 > 2 - x\), or corresponding equation | M1 | |
| Obtain critical value \(x = \dfrac{3}{2}\) | A1 | |
| State final answer \(x > \dfrac{3}{2}\) only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve the quadratic inequality \((x-2)^2 < (3x-4)^2\), or corresponding equation | M1 | |
| Obtain critical value \(x = \dfrac{3}{2}\) | A1 | |
| State final answer \(x > \dfrac{3}{2}\) only | A1 | |
| Total | 3 |
## Question 1:
### Part 1(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Make a recognisable sketch graph of $y = |x - 2|$ | **B1** | |
| **Total** | **1** | |
---
### Part 1(b):
**Method 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Find $x$-coordinate of intersection with $y = 3x - 4$ | **M1** | |
| Obtain $x = \dfrac{3}{2}$ | **A1** | |
| State final answer $x > \dfrac{3}{2}$ only | **A1** | |
**Alternative Method 1:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve the linear inequality $3x - 4 > 2 - x$, or corresponding equation | **M1** | |
| Obtain critical value $x = \dfrac{3}{2}$ | **A1** | |
| State final answer $x > \dfrac{3}{2}$ only | **A1** | |
**Alternative Method 2:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve the quadratic inequality $(x-2)^2 < (3x-4)^2$, or corresponding equation | **M1** | |
| Obtain critical value $x = \dfrac{3}{2}$ | **A1** | |
| State final answer $x > \dfrac{3}{2}$ only | **A1** | |
| **Total** | **3** | |1
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = | x - 2 |$.
\item Solve the inequality $| x - 2 | < 3 x - 4$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q1 [4]}}