| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Sketch modulus of linear and non-modulus linear, find intersection |
| Difficulty | Standard +0.3 Part (a) is routine sketching of a V-shaped modulus graph and a linear function. Part (b) requires standard technique of splitting into cases or squaring, yielding a straightforward inequality. Part (c) adds a mild substitution step (x = e^{0.1N}) and taking logarithms, but follows directly from part (b). Overall slightly easier than average due to structured scaffolding and standard methods. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02s Modulus graphs: sketch graph of |ax+b|1.06a Exponential function: a^x and e^x graphs and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Draw V-shaped graph with vertex on positive \(x\)-axis in the first quadrant | B1 | |
| Draw correct graph of \(y = 5 - x\) correctly positioned with respect to modulus graph | B1 | Two points of intersection |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Solve \(3x - 8 = 5 - x\) to obtain \(\frac{13}{4}\) | B1 | Or inequality |
| Solve linear equation or inequality with signs of \(3x\) and \(x\) the same | M1 | |
| Obtain \(\frac{3}{2}\) | A1 | |
| Conclude \(\frac{3}{2} < x < \frac{13}{4}\) or \(x > \frac{3}{2}\) and \(x < \frac{13}{4}\) | A1 | Allow alternative notation e.g. \(\left(\frac{3}{2}, \frac{13}{4}\right)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \((3x-8)^2 = (5-x)^2\) | (B1) | |
| Attempt solution of three-term equation (or inequality) | (M1) | |
| Obtain \(\frac{3}{2}\) and \(\frac{13}{4}\) | (A1) | |
| Conclude \(\frac{3}{2} < x < \frac{13}{4}\) or \(x > \frac{3}{2}\) and \(x < \frac{13}{4}\) | (A1) | Allow alternative notation e.g. \(\left(\frac{3}{2}, \frac{13}{4}\right)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt value of \(N\) (maybe non-integer) for \(e^{0.1N} < \textit{their } \frac{13}{4}\) | M1 | Allow \(e^{0.1N} = \textit{their } \frac{13}{4}\) (or inequality) |
| Conclude with single integer 11 | A1 |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw V-shaped graph with vertex on positive $x$-axis in the first quadrant | B1 | |
| Draw correct graph of $y = 5 - x$ correctly positioned with respect to modulus graph | B1 | Two points of intersection |
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Solve $3x - 8 = 5 - x$ to obtain $\frac{13}{4}$ | B1 | Or inequality |
| Solve linear equation or inequality with signs of $3x$ and $x$ the same | M1 | |
| Obtain $\frac{3}{2}$ | A1 | |
| Conclude $\frac{3}{2} < x < \frac{13}{4}$ or $x > \frac{3}{2}$ **and** $x < \frac{13}{4}$ | A1 | Allow alternative notation e.g. $\left(\frac{3}{2}, \frac{13}{4}\right)$ |
**Alternative Method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $(3x-8)^2 = (5-x)^2$ | (B1) | |
| Attempt solution of three-term equation (or inequality) | (M1) | |
| Obtain $\frac{3}{2}$ and $\frac{13}{4}$ | (A1) | |
| Conclude $\frac{3}{2} < x < \frac{13}{4}$ or $x > \frac{3}{2}$ **and** $x < \frac{13}{4}$ | (A1) | Allow alternative notation e.g. $\left(\frac{3}{2}, \frac{13}{4}\right)$ |
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt value of $N$ (maybe non-integer) for $e^{0.1N} < \textit{their } \frac{13}{4}$ | M1 | Allow $e^{0.1N} = \textit{their } \frac{13}{4}$ (or inequality) |
| Conclude with single integer 11 | A1 | |
---3
\begin{enumerate}[label=(\alph*)]
\item Sketch on the same diagram the graphs of $y = | 3 x - 8 |$ and $y = 5 - x$.
\item Solve the inequality $| 3 x - 8 | < 5 - x$.
\item Hence determine the largest integer $N$ satisfying the inequality $\left| 3 e ^ { 0.1 N } - 8 \right| < 5 - e ^ { 0.1 N }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q3 [8]}}