| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | March |
| Marks | 5 |
| 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 a V-shaped graph and a horizontal line, followed by solving by cases (3x-5 = x+2 or 3x-5 = -(x+2)). The techniques are routine and commonly practiced, making it easier than average but not trivial since it requires understanding the modulus definition and checking solutions. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s 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 correct graph of \(y = x + 2\) with smaller positive gradient | B1 | Crossing \(y\)-axis between 0 and \(y\)-intercept of first graph |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve \(3x - 5 = x + 2\) to obtain \(x = \frac{7}{2}\) | B1 | |
| Attempt solution of linear equation where signs of \(3x\) and \(x\) are different | M1 | |
| Obtain \(x = \frac{3}{4}\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply non-modulus equation \((3x-5)^2 = (x+2)^2\) | B1 | |
| Attempt solution of 3-term quadratic equation | M1 | |
| Obtain \(\frac{3}{4}\) and \(\frac{7}{2}\) | A1 | |
| Total: 3 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw V-shaped graph with vertex on positive $x$-axis | **B1** | |
| Draw correct graph of $y = x + 2$ with smaller positive gradient | **B1** | Crossing $y$-axis between 0 and $y$-intercept of first graph |
| | **Total: 2** | |
### Part (b) — Method 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve $3x - 5 = x + 2$ to obtain $x = \frac{7}{2}$ | **B1** | |
| Attempt solution of linear equation where signs of $3x$ and $x$ are different | **M1** | |
| Obtain $x = \frac{3}{4}$ | **A1** | |
| | **Total: 3** | |
### Part (b) — Alternative Method:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply non-modulus equation $(3x-5)^2 = (x+2)^2$ | **B1** | |
| Attempt solution of 3-term quadratic equation | **M1** | |
| Obtain $\frac{3}{4}$ and $\frac{7}{2}$ | **A1** | |
| | **Total: 3** | |1
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the same diagram, the graphs of $y = | 3 x - 5 |$ and $y = x + 2$.
\item Solve the equation $| 3 x - 5 | = x + 2$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q1 [5]}}