| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 4 |
| 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 linear function, followed by solving by considering two cases (2x-9 ≥ 0 and 2x-9 < 0). The algebraic manipulation is routine and the question follows a predictable template commonly seen in textbooks, making it easier than average. |
| Spec | 1.02s Modulus graphs: sketch graph of |ax+b|1.02t Solve modulus equations: graphically with modulus function1.03a Straight lines: equation forms y=mx+c, ax+by+c=0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw V-shaped graph with vertex on positive \(x\)-axis | \*B1 | |
| Draw (more or less) correct graph of \(y = 5x - 3\) with greater gradient | DB1 | crossing \(x\)-axis between origin and vertex of first graph |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt solution of linear equation where signs of \(2x\) and \(5x\) are different | M1 | |
| Solve \(-2x + 9 = 5x - 3\) to obtain \(\dfrac{12}{7}\), 1.71 or better | A1 | and no second answer |
| Alternative method: | ||
| Attempt solution of 3-term quadratic equation \((2x-9)^2 = (5x-3)^2\) to obtain at least one value of \(x\) | M1 | \(7x^2 + 2x - 24 = 0\) |
| Obtain \(\dfrac{12}{7}\), 1.71 or better | A1 | and no second answer |
| Total | 2 |
**Question 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw V-shaped graph with vertex on positive $x$-axis | \*B1 | |
| Draw (more or less) correct graph of $y = 5x - 3$ with greater gradient | DB1 | crossing $x$-axis between origin and vertex of first graph |
| **Total** | **2** | |
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**Question 2(b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt solution of linear equation where signs of $2x$ and $5x$ are different | M1 | |
| Solve $-2x + 9 = 5x - 3$ to obtain $\dfrac{12}{7}$, 1.71 or better | A1 | and no second answer |
| **Alternative method:** | | |
| Attempt solution of 3-term quadratic equation $(2x-9)^2 = (5x-3)^2$ to obtain at least one value of $x$ | M1 | $7x^2 + 2x - 24 = 0$ |
| Obtain $\dfrac{12}{7}$, 1.71 or better | A1 | and no second answer |
| **Total** | **2** | |2
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the same diagram, the graphs of $y = | 2 x - 9 |$ and $y = 5 x - 3$.
\item Solve the equation $| 2 x - 9 | = 5 x - 3$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q2 [4]}}