| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch y=|linear| and y=linear, solve inequality: numeric coefficients |
| Difficulty | Moderate -0.8 Part (a) is a routine sketch of a single modulus function requiring only knowledge of the V-shape transformation (reflecting negative parts). Part (b) involves solving a modulus inequality by considering two cases, which is a standard technique. Both parts are straightforward applications of basic modulus concepts with minimal problem-solving required. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show a recognisable sketch graph of \(y = | 2x + 1 | \) — V-shaped graph with vertex at \(\left(-\dfrac{1}{2}, 0\right)\), crossing y-axis at \((0, 1)\), both arms pointing upward |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Find \(x\)-coordinate of intersection with \(y = 3x + 5\) | M1 | |
| Obtain \(x = -\frac{6}{5}\) | A1 | |
| State final answer \(x < -\frac{6}{5}\) only | A1 | Do not condone \(\leq\) for \(<\) in the final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve the linear inequality \(3x + 5 < -(2x+1)\), or corresponding equation | M1 | Must solve the relevant equation |
| Obtain critical value \(x = -\frac{6}{5}\) | A1 | Ignore \(-4\) if seen |
| State final answer \(x < -\frac{6}{5}\) only | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve the quadratic inequality \((3x+5)^2 < (2x+1)^2\), or corresponding equation | M1 | \(5x^2 + 26x + 24 < 0\) |
| Obtain critical value \(x = -\frac{6}{5}\) | A1 | Ignore \(-4\) if seen |
| State final answer \(x < -\frac{6}{5}\) only | A1 | |
| Total | 3 |
## Question 1:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show a recognisable sketch graph of $y = |2x + 1|$ — V-shaped graph with vertex at $\left(-\dfrac{1}{2}, 0\right)$, crossing y-axis at $(0, 1)$, both arms pointing upward | **B1** | Ignore $y = 3x + 5$ if also drawn on the sketch |
| | **Total: 1** | |
## Question 1(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Find $x$-coordinate of intersection with $y = 3x + 5$ | M1 | |
| Obtain $x = -\frac{6}{5}$ | A1 | |
| State final answer $x < -\frac{6}{5}$ only | A1 | Do not condone $\leq$ for $<$ in the final answer |
**Alternative method 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve the linear inequality $3x + 5 < -(2x+1)$, or corresponding equation | M1 | Must solve the relevant equation |
| Obtain critical value $x = -\frac{6}{5}$ | A1 | Ignore $-4$ if seen |
| State final answer $x < -\frac{6}{5}$ only | A1 | |
**Alternative method 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve the quadratic inequality $(3x+5)^2 < (2x+1)^2$, or corresponding equation | M1 | $5x^2 + 26x + 24 < 0$ |
| Obtain critical value $x = -\frac{6}{5}$ | A1 | Ignore $-4$ if seen |
| State final answer $x < -\frac{6}{5}$ only | A1 | |
| **Total** | **3** | |
---1
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = | 2 x + 1 |$.
\item Solve the inequality $3 x + 5 < | 2 x + 1 |$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q1 [4]}}