| 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| = constant |
| Difficulty | Moderate -0.8 This is a straightforward modulus inequality requiring a standard technique (squaring both sides or considering cases). Part (i) is routine algebraic manipulation, and part (ii) is a simple substitution followed by solving ln n < 4. Both parts are below average difficulty for A-level, requiring only basic modulus understanding and logarithm properties with no novel problem-solving. |
| 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** | |
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Candidates answer on the Question Paper.\\
Additional Materials: List of Formulae (MF9)
\section*{READ THESE INSTRUCTIONS FIRST}
Write your centre number, candidate number and name in the spaces at the top of this page.\\
Write in dark blue or black pen.\\
You may use an HB pencil for any diagrams or graphs.\\
Do not use staples, paper clips, glue or correction fluid.\\
Answer all the questions in the space provided. If additional space is required, you should use the lined page at the end of this booklet. The question number(s) must be clearly shown.\\
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.\\
The use of an electronic calculator is expected, where appropriate.\\
You are reminded of the need for clear presentation in your answers.\\
At the end of the examination, fasten all your work securely together.\\[0pt]
The number of marks is given in brackets [ ] at the end of each question or part question.\\
The total number of marks for this paper is 50.
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]}}