| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| 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 -0.3 This is a standard modulus function question requiring sketching a V-shaped graph and a straight line, then finding their intersection algebraically by considering cases. The inequality in part (c) follows directly from the sketch. While it involves multiple steps, each component is routine for A-level students who have covered modulus functions, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw V-shaped graph with vertex on positive \(x\)-axis | B1 | |
| Draw straight line graph correctly positioned with greater gradient | B1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve linear equation with signs of \(\frac{1}{2}x\) and \(\frac{3}{2}x\) different, or solve non-modulus equation \(\left(\frac{1}{2}x - a\right)^2 = \left(\frac{3}{2}x - \frac{1}{2}a\right)^2\) to obtain \(x =\) | M1 | |
| Obtain \(x = \frac{3}{4}a\) | A1 | |
| Obtain \(y = \frac{5}{8}a\) | A1 | And no other point |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(x < \frac{3}{4}a\) | B1 FT | Following *their* (single) \(x\)-coordinate from part (b) |
| Total | 1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw V-shaped graph with vertex on positive $x$-axis | B1 | |
| Draw straight line graph correctly positioned with greater gradient | B1 | |
| **Total** | **2** | |
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve linear equation with signs of $\frac{1}{2}x$ and $\frac{3}{2}x$ different, or solve non-modulus equation $\left(\frac{1}{2}x - a\right)^2 = \left(\frac{3}{2}x - \frac{1}{2}a\right)^2$ to obtain $x =$ | M1 | |
| Obtain $x = \frac{3}{4}a$ | A1 | |
| Obtain $y = \frac{5}{8}a$ | A1 | And no other point |
| **Total** | **3** | |
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $x < \frac{3}{4}a$ | B1 FT | Following *their* (single) $x$-coordinate from part **(b)** |
| **Total** | **1** | |
---3
\begin{enumerate}[label=(\alph*)]
\item Sketch, on a single diagram, the graphs of $y = \left| \frac { 1 } { 2 } x - a \right|$ and $y = \frac { 3 } { 2 } x - \frac { 1 } { 2 } a$, where $a$ is a positive constant.
\item Find the coordinates of the point of intersection of the two graphs.
\item Deduce the solution of the inequality $\left| \frac { 1 } { 2 } x - a \right| > \frac { 3 } { 2 } x - \frac { 1 } { 2 } a$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q3 [6]}}