| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | November |
| Marks | 5 |
| 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 identification of the vertex at x=0.5 and reflection of negative values. Part (b) involves solving a linear-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 | 1.02g Inequalities: linear and quadratic in single variable1.02s Modulus graphs: sketch graph of |ax+b| |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sketch of \(y = \ | 4x - 2\ | \) — V-shaped graph, roughly symmetrical, extending into second quadrant, with vertex at \(\left(\frac{1}{2}, 0\right)\) and passing through \((0, 2)\); scale indicated on both axes; graph must extend beyond \((0, 2)\) and \((1, 2)\) |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain critical value \(x = 3\) | B1 | Allow incorrect inequality. Allow if later rejected. Allow \(\frac{21}{7}\) |
| Solve the linear equation \(1 + 3x = 2 - 4x\) | M1 | Or corresponding linear inequality |
| Obtain critical value \(\frac{1}{7}\) | A1 | Allow 0.143 or better. Allow incorrect inequality. Allow if later rejected |
| Obtain final answer \(x < \frac{1}{7}\) [or] \(x > 3\) | A1 | Or equivalent. Allow with a comma, or nothing between. Strict inequalities only. Exact values. A0 for \(\frac{1}{7} > x > 3\). A0 for \(x < \frac{1}{7}\) and \(x > 3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Solve the quadratic inequality \((4x-2)^2 > (1+3x)^2\), or corresponding quadratic equation | M1 | e.g. \(7x^2 - 22x + 3 = 0\). Available if they start with correct equation/inequality, have correct method for squaring (i.e. not \((a+b)^2 = a^2 + b^2\)) and a correct method for solving. Need to obtain at least one critical value |
| Obtain critical value \(x = 3\) | A1 | Allow incorrect inequality. Allow if later rejected. Allow \(\frac{21}{7}\) |
| Obtain critical value \(\frac{1}{7}\) | A1 | Allow 0.143 or better. Allow incorrect inequality. Allow if later rejected |
| Obtain final answer \(x < \frac{1}{7}\) [or] \(x > 3\) | A1 | Or equivalent. Strict inequalities only. Allow with a comma, or nothing between. Exact values. A0 for \(\frac{1}{7} > x > 3\). A0 for \(x < \frac{1}{7}\) and \(x > 3\) |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sketch of $y = \|4x - 2\|$ — V-shaped graph, roughly symmetrical, extending into second quadrant, with vertex at $\left(\frac{1}{2}, 0\right)$ and passing through $(0, 2)$; scale indicated on both axes; graph must extend beyond $(0, 2)$ and $(1, 2)$ | B1 | Show a recognisable sketch of $y = \|4x-2\|$. Roughly symmetrical. Should extend into the second quadrant. Ignore $y = 4x-2$ below the axis if intention is clear e.g. dashed or the required lines are clearly bolder. Some indication of scale on **both** axes — accept dashes. Must go beyond $(0, 2)$ and $(1, 2)$. Ignore any attempt to sketch $y = 1 + 3x$. |
| | **Total: 1** | |
## Question 1(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain critical value $x = 3$ | B1 | Allow incorrect inequality. Allow if later rejected. Allow $\frac{21}{7}$ |
| Solve the linear equation $1 + 3x = 2 - 4x$ | M1 | Or corresponding linear inequality |
| Obtain critical value $\frac{1}{7}$ | A1 | Allow 0.143 or better. Allow incorrect inequality. Allow if later rejected |
| Obtain final answer $x < \frac{1}{7}$ [or] $x > 3$ | A1 | Or equivalent. Allow with a comma, or nothing between. Strict inequalities only. Exact values. A0 for $\frac{1}{7} > x > 3$. A0 for $x < \frac{1}{7}$ **and** $x > 3$ |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Solve the quadratic inequality $(4x-2)^2 > (1+3x)^2$, or corresponding quadratic equation | M1 | e.g. $7x^2 - 22x + 3 = 0$. Available if they start with correct equation/inequality, have correct method for squaring (i.e. not $(a+b)^2 = a^2 + b^2$) and a correct method for solving. Need to obtain at least one critical value |
| Obtain critical value $x = 3$ | A1 | Allow incorrect inequality. Allow if later rejected. Allow $\frac{21}{7}$ |
| Obtain critical value $\frac{1}{7}$ | A1 | Allow 0.143 or better. Allow incorrect inequality. Allow if later rejected |
| Obtain final answer $x < \frac{1}{7}$ [or] $x > 3$ | A1 | Or equivalent. Strict inequalities only. Allow with a comma, or nothing between. Exact values. A0 for $\frac{1}{7} > x > 3$. A0 for $x < \frac{1}{7}$ and $x > 3$ |
---1
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = | 4 x - 2 |$.
\item Solve the inequality $1 + 3 x < | 4 x - 2 |$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q1 [5]}}