| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Modulus function |
| Type | Solve |linear| < |linear| |
| Difficulty | Standard +0.3 This is a straightforward modulus inequality requiring squaring both sides to eliminate the modulus signs, leading to a simple linear inequality. Part (ii) adds a logarithmic substitution but is routine once part (i) is solved. Slightly above average due to the two-part structure and logarithmic application, but still a standard textbook exercise. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply non-modular inequality \((2x-7)^2 < (2x-9)^2\) or corresponding equation or linear equation (with signs of \(2x\) different) | M1 | |
| Obtain critical value 4 | A1 | |
| State \(x < 4\) only | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt to find \(n\) from \(\ln n =\) *their* critical value from part (i) | M1 | |
| Obtain or imply \(n < e^4\) and hence 54 | A1 | |
| Total | 2 |
## Question 1(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply non-modular inequality $(2x-7)^2 < (2x-9)^2$ or corresponding equation or linear equation (with signs of $2x$ different) | M1 | |
| Obtain critical value 4 | A1 | |
| State $x < 4$ only | A1 | |
| **Total** | **3** | |
## Question 1(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt to find $n$ from $\ln n =$ *their* critical value from part (i) | M1 | |
| Obtain or imply $n < e^4$ and hence 54 | A1 | |
| **Total** | **2** | |1 (i) Solve the inequality $| 2 x - 7 | < | 2 x - 9 |$.\\
(ii) Hence find the largest integer $n$ satisfying the inequality $| 2 \ln n - 7 | < | 2 \ln n - 9 |$.\\
\hfill \mbox{\textit{CAIE P2 2019 Q1 [5]}}