| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |linear| < |linear| |
| Difficulty | Standard +0.3 Part (i) is a standard modulus inequality requiring squaring both sides or considering cases, yielding a straightforward quadratic inequality. Part (ii) adds a simple substitution step (u = ln y) then exponentiating to find y, but remains routine application of technique with no novel insight required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02l Modulus function: notation, relations, equations and inequalities1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply non-modulus inequality \((2x-5)^2 < (x+3)^2\) or corresponding equation or pair of linear equations | B1 | |
| Attempt solution of 3-term quadratic inequality or equation or of 2 linear equations | M1 | |
| Obtain critical values \(\frac{2}{3}\) and \(8\) | A1 | |
| State answer \(\frac{2}{3} < x < 8\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt to find \(y\) from \(\ln y =\) upper limit of answer to part (i) | M1 | |
| Obtain \(2980\) | A1 |
## Question 3(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply non-modulus inequality $(2x-5)^2 < (x+3)^2$ or corresponding equation or pair of linear equations | B1 | |
| Attempt solution of 3-term quadratic inequality or equation or of 2 linear equations | M1 | |
| Obtain critical values $\frac{2}{3}$ and $8$ | A1 | |
| State answer $\frac{2}{3} < x < 8$ | A1 | |
## Question 3(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to find $y$ from $\ln y =$ upper limit of answer to part (i) | M1 | |
| Obtain $2980$ | A1 | |
---3 (i) Solve the inequality $| 2 x - 5 | < | x + 3 |$.\\
(ii) Hence find the largest integer $y$ satisfying the inequality $| 2 \ln y - 5 | < | \ln y + 3 |$.\\
\hfill \mbox{\textit{CAIE P2 2017 Q3 [6]}}