# 9709/31 May/June 2011

> 9709/31 May/June 2011: 10 questions, 75 marks. Difficulty 0.47, deviation 0.16, algebraic 0.47, text density 0.31. CIE IAL Mathematics 9709 P3.

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CIE IAL · Mathematics · 2011 May/June

This Mathematics 9709 Paper 3 (Pure Mathematics 3) paper contains 10 questions worth 75 marks. The profile below reports five mark-weighted difficulty and complexity measures, and every question row repeats them. Full question text and mark scheme open in the Interactive Practice Hub.

- CIE IAL
- 9709 P3
- 10 questions
- 75 marks
- Free paper
- [Practise this paper](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0)
- [All 9709 P3 papers](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/)

Core finding

**9709/31 May/June 2011 is a past paper for CIE IAL Mathematics 9709 Paper 3 (Pure Mathematics 3), with 10 questions worth 75 marks.** Its mark-weighted Difficulty Measure is 0.47 on a 0.20–0.75 scale, with a Difficulty Deviation of 0.16 and a mark split of 13.3% Easy, 61.3% Medium and 25.4% Hard. Algebraic complexity averages 0.47; text density and modelling average 0.31. All five measures use 21 part-level tags, not candidate results. Every question row below repeats them and opens the Interactive Practice Hub for that question.

## Difficulty & complexity profile

All five measures are mark-weighted and calculated from 21 parts carrying 75 part marks. Difficulty tags cover 21 parts, algebraic tags 21 and text-density tags 21. Tags are internal metadata, not official exam-board ratings.

- **Difficulty Measure**: 0.47 — 0.20–0.75 · nearer Medium
- **Difficulty Deviation**: 0.16 — 0 = uniform; higher = more mixed
- **Difficulty Ratio**: 13.3 / 61.3 / 25.4 — % Easy / Medium / Hard
- **Algebraic Complexity Measure**: 0.47 — 0.20–0.75 · nearer Medium
- **Text Density and Modelling Measure**: 0.31 — 0.20–0.75 · nearer Low

Scale: Easy 0.20 · Medium 0.45 · Hard 0.75, and Low 0.20 · Medium 0.45 · High 0.75 for the two complexity measures. [How each band is defined and how the values are calculated →](https://www.examsci.com/guides/difficulty-and-complexity-measures/)

### Definitions and calculation

| Measure | Paper value | Formula |
| --- | --- | --- |
| **Difficulty Measure** | 0.47 | Σ(d_i × w_i) / Σw_i, with d = 0.20 / 0.45 / 0.75 |
| **Difficulty Deviation** | 0.16 | √(Σw_i(d_i − μ)^2 / Σw_i) |
| **Difficulty Ratio** | 13.3% Easy / 61.3% Medium / 25.4% Hard | marks in band / all tagged marks |
| **Algebraic Complexity Measure** | 0.47 | Σ(c_i × marks_i) / Σmarks_i, with c = 0.20 / 0.45 / 0.75 |
| **Text Density and Modelling Measure** | 0.31 | Σ(t_i × marks_i) / Σmarks_i, with t = 0.20 / 0.45 / 0.75 |

w_i equals the part marks, except for a Hard part worth 4 or more marks, where w_i = marks^0.75 to offset the accessible starter steps inside long structured questions.

## Question list

Each measure is recalculated from the parts of that question and uses the same weights as the paper profile.

| Question | Topic | Marks | Difficulty Measure | Algebraic Complexity Measure | Text Density and Modelling Measure | Practice |
| --- | --- | --- | --- | --- | --- | --- |
| **Q1** FRQ | [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/) | 4 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=865c49a3-394b-478b-86ce-60d6a9150709&source_detail=9709%2F31+May%2FJune+2011) |
| **Q2** FRQ | [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/) | 4 | 0.20 Easy | 0.20 Low | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=9536c827-0880-4a7b-a6f1-c237c5a3b0cb&source_detail=9709%2F31+May%2FJune+2011) |
| **Q3** FRQ | [7 Vectors](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/vectors/) | 7 | 0.34 Medium | 0.34 Medium | 0.45 Medium | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=c0ead671-2f36-40d6-97d2-5d4fbb74e7f9&source_detail=9709%2F31+May%2FJune+2011) |
| **Q4** FRQ | [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/) | 7 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=a9e48979-af1f-4229-823b-c9ee95e4b783&source_detail=9709%2F31+May%2FJune+2011) |
| **Q5** FRQ | [2 Logarithmic and exponential functions](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/logarithmic-and-exponential-functions/) | 7 | 0.38 Medium | 0.38 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=e794e6f3-8d0f-4e4e-ad39-b1d37e33f1fe&source_detail=9709%2F31+May%2FJune+2011) |
| **Q6** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 8 | 0.45 Medium | 0.45 Medium | 0.29 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=7254f964-06af-48e1-b9d4-893c28fc468c&source_detail=9709%2F31+May%2FJune+2011) |
| **Q7** FRQ | [5 Integration](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/integration/) | 8 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=c2297557-6022-4748-9f5f-d5b24906d34f&source_detail=9709%2F31+May%2FJune+2011) |
| **Q8** FRQ | [9 Complex numbers](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/complex-numbers/) | 10 | 0.63 Hard | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=8191aeff-7906-4f6d-9884-2f55d3b5e697&source_detail=9709%2F31+May%2FJune+2011) |
| **Q9** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 10 | 0.55 Medium | 0.57 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=d22e353f-cb6b-4eab-804d-803fa0d99802&source_detail=9709%2F31+May%2FJune+2011) |
| **Q10** FRQ | [5 Integration](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/integration/) | 10 | 0.66 Hard | 0.70 High | 0.75 High | [Go to Practice Hub](https://www.examsci.com/practice?set_id=88421e9f-1492-466f-879d-08767a2d9eb0&question_id=a180364a-5ca9-4f6b-bd9b-112b1e9b6466&source_detail=9709%2F31+May%2FJune+2011) |

## Topics in this paper

- [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/)
- [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/)
- [7 Vectors](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/vectors/)
- [2 Logarithmic and exponential functions](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/logarithmic-and-exponential-functions/)
- [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/)
- [6 Numerical solution of equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/numerical-solution-of-equations/)
- [5 Integration](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/integration/)
- [9 Complex numbers](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/complex-numbers/)
- [8 Differential equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differential-equations/)

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## Paper FAQ

**How are the difficulty and complexity measures calculated?**

Each part is weighted by its official marks. Difficulty uses Easy 0.20, Medium 0.45 and Hard 0.75; algebraic complexity and text density and modelling use Low 0.20, Medium 0.45 and High 0.75. A Hard part worth 4 or more marks is down-weighted to marks^0.75, so the accessible starter steps inside a long structured question do not inflate the difficulty result. This paper's Difficulty Measure is 0.47 from 21 parts.

**What does the Difficulty Deviation show?**

It is the mark-weighted standard deviation of the part difficulty values around the paper mean. A value of 0 would mean every tagged part has the same difficulty; this paper's deviation is 0.16, so its parts are more mixed than a single-level paper.

**Is the Difficulty Ratio based on question counts or marks?**

It is based on official marks, not question counts. This paper's shares are 13.3% Easy, 61.3% Medium and 25.3% Hard; the three shares sum to 100% before rounding.

**Are these official difficulty ratings?**

No. They are derived from the platform's part-level difficulty and static-question-profile tags. They are not exam-board grade boundaries and are not calculated from candidate outcomes. Use them to compare the demand profile of papers, not to predict a grade.

**Why does each question row show its own measures?**

Each row recalculates the same mark-weighted formula from that question's parts. A single-part MCQ therefore shows the tag value directly, while a multi-part structured question blends the tags of its parts using the same Hard-part decay as the paper profile.

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Full page with native tables, bar charts and Practice Hub links: https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2011-may-june-9709-31/