| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch modulus of linear and non-modulus linear, find intersection |
| Difficulty | Moderate -0.8 This is a straightforward modulus question requiring a standard sketch of two simple functions, solving a basic equation by cases (|x-3| = 3x), and reading off an inequality from the graph. All techniques are routine for P2 level with no conceptual challenges or novel problem-solving required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw V-shaped graph with vertex on positive \(x\)-axis | B1 | Must be straight lines. |
| Draw straight line through origin with positive gradient greater than gradient of first graph, together with a V shaped graph | B1 | Must have the first B1. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve linear equation with signs of \(3x\) and \(x\) different or solve non-modulus equation \((3x)^2 = (x-3)^2\) | M1 | |
| Obtain \(x = \frac{3}{4}\) | A1 | |
| Obtain \(y = \frac{9}{4}\) | A1 | And no other point. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(x < \frac{3}{4}\) | B1 FT | Following *their* (single) \(x\)-coordinate from part (b). |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw V-shaped graph with vertex on positive $x$-axis | **B1** | Must be straight lines. |
| Draw straight line through origin with positive gradient greater than gradient of first graph, together with a V shaped graph | **B1** | Must have the first **B1**. |
---
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve linear equation with signs of $3x$ and $x$ different or solve non-modulus equation $(3x)^2 = (x-3)^2$ | **M1** | |
| Obtain $x = \frac{3}{4}$ | **A1** | |
| Obtain $y = \frac{9}{4}$ | **A1** | And no other point. |
---
## Question 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $x < \frac{3}{4}$ | **B1 FT** | Following *their* (single) $x$-coordinate from part **(b)**. |2
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the same diagram, the graphs of $y = 3 x$ and $y = | x - 3 |$.
\item Find the coordinates of the point where the two graphs intersect.
\item Deduce the solution of the inequality $3 x < | x - 3 |$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q2 [6]}}