| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch two |linear| functions and solve related equation/inequality |
| Difficulty | Moderate -0.3 This is a straightforward modulus function question requiring standard techniques: sketching V-shaped graphs by finding zeros and y-intercepts, solving |f(x)| = |g(x)| by considering cases, and reading off an inequality from the sketch. While it involves multiple parts and algebraic manipulation with a parameter, these are routine A-level skills with no novel insight required, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Draw two V-shaped graphs with one vertex on negative \(x\)-axis and one vertex on positive \(x\)-axis | M1 | |
| Draw correct graphs related correctly to each other | A1 | |
| State correct coordinates \(-\frac{2}{3}a\), \(2a\), \(\frac{4}{3}a\), \(4a\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve linear equation with signs of \(3x\) different or solve non-modulus equation \((3x+2a)^2 = (3x-4a)^2\) | M1 | |
| Obtain \(x = \frac{1}{3}a\) | A1 | |
| Obtain \(y = 3a\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(x < \frac{1}{3}a\) | B1FT | FT from part (b) |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Draw two V-shaped graphs with one vertex on negative $x$-axis and one vertex on positive $x$-axis | M1 | |
| Draw correct graphs related correctly to each other | A1 | |
| State correct coordinates $-\frac{2}{3}a$, $2a$, $\frac{4}{3}a$, $4a$ | A1 | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve linear equation with signs of $3x$ different or solve non-modulus equation $(3x+2a)^2 = (3x-4a)^2$ | M1 | |
| Obtain $x = \frac{1}{3}a$ | A1 | |
| Obtain $y = 3a$ | A1 | |
---
## Question 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $x < \frac{1}{3}a$ | B1FT | FT from part (b) |
---4
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the same diagram, the graphs of $y = | 3 x + 2 a |$ and $y = | 3 x - 4 a |$, where $a$ is a positive constant.
Give the coordinates of the points where each graph meets the axes.
\item Find the coordinates of the point of intersection of the two graphs.
\item Deduce the solution of the inequality $| 3 x + 2 a | < | 3 x - 4 a |$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2020 Q4 [7]}}