# 9709/31 May/June 2012

> 9709/31 May/June 2012: 10 questions, 75 marks. Difficulty 0.46, deviation 0.19, algebraic 0.51, text density 0.23. CIE IAL Mathematics 9709 P3.

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CIE IAL · Mathematics · 2012 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=d20cbcab-be46-4e40-a78e-5e486bc36b4f)
- [All 9709 P3 papers](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/)

Core finding

**9709/31 May/June 2012 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 0.46 on a 0.20–0.75 scale, with a Difficulty Deviation of 0.19 and a mark split of 21.3% Easy, 44.0% Medium and 34.7% Hard. Algebraic complexity averages 0.51; text density and modelling average 0.23. All five measures use 20 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 20 parts carrying 75 part marks. Difficulty tags cover 20 parts, algebraic tags 20 and text-density tags 20. Tags are internal metadata, not official exam-board ratings.

- **Difficulty Measure**: 0.46 — 0.20–0.75 · nearer Medium
- **Difficulty Deviation**: 0.19 — 0 = uniform; higher = more mixed
- **Difficulty Ratio**: 21.3 / 44.0 / 34.7 — % Easy / Medium / Hard
- **Algebraic Complexity Measure**: 0.51 — 0.20–0.75 · nearer Medium
- **Text Density and Modelling Measure**: 0.23 — 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** | 0.46 | Σ(d_i × w_i) / Σw_i, with d = 0.20 / 0.45 / 0.75 |
| **Difficulty Deviation** | 0.19 | √(Σw_i(d_i − μ)^2 / Σw_i) |
| **Difficulty Ratio** | 21.3% Easy / 44.0% Medium / 34.7% Hard | marks in band / all tagged marks |
| **Algebraic Complexity Measure** | 0.51 | Σ(c_i × marks_i) / Σmarks_i, with c = 0.20 / 0.45 / 0.75 |
| **Text Density and Modelling Measure** | 0.23 | 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/) | 3 | 0.20 Easy | 0.20 Low | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=28ecadf3-3989-4b9e-a3d7-e7d5f97a1484&source_detail=9709%2F31+May%2FJune+2012) |
| **Q2** FRQ | [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/) | 5 | 0.30 Easy | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=dfdd84b2-9379-4bc7-9d3f-9d2885ddc811&source_detail=9709%2F31+May%2FJune+2012) |
| **Q3** FRQ | [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/) | 7 | 0.27 Easy | 0.38 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=e42e50fd-c578-487b-8eca-bebfbfb5c8bb&source_detail=9709%2F31+May%2FJune+2012) |
| **Q4** FRQ | [9 Complex numbers](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/complex-numbers/) | 7 | 0.45 Medium | 0.34 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=fcfec024-5d89-465b-be57-61cb3486c7dd&source_detail=9709%2F31+May%2FJune+2012) |
| **Q5** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 7 | 0.60 Medium | 0.62 High | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=97f74c0b-77ea-42b9-892e-6735d4ae3908&source_detail=9709%2F31+May%2FJune+2012) |
| **Q6** FRQ | [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/) | 7 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=4bc5bd78-1a03-4394-8345-9775781ac7db&source_detail=9709%2F31+May%2FJune+2012) |
| **Q7** FRQ | [5 Integration](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/integration/) | 8 | 0.75 Hard | 0.75 High | 0.45 Medium | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=571301e7-ab62-430b-95f4-fcdca4b15eed&source_detail=9709%2F31+May%2FJune+2012) |
| **Q8** FRQ | [7 Vectors](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/vectors/) | 9 | 0.56 Medium | 0.58 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=fc424ee8-7f17-4fe3-bad7-89d43a82d482&source_detail=9709%2F31+May%2FJune+2012) |
| **Q9** FRQ | [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/) | 10 | 0.75 Hard | 0.75 High | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=b0c84626-7b6f-4566-939c-9395a7efb9df&source_detail=9709%2F31+May%2FJune+2012) |
| **Q10** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 12 | 0.35 Medium | 0.35 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=d20cbcab-be46-4e40-a78e-5e486bc36b4f&question_id=eb1adf4d-e11c-4089-954f-7045b8ce8bec&source_detail=9709%2F31+May%2FJune+2012) |

## Topics in this paper

- [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/)
- [2 Logarithmic and exponential functions](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/logarithmic-and-exponential-functions/)
- [9 Complex numbers](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/complex-numbers/)
- [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/)
- [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/)
- [5 Integration](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/integration/)
- [8 Differential equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differential-equations/)
- [7 Vectors](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/vectors/)
- [6 Numerical solution of equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/numerical-solution-of-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 0.46 from 20 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 0.19, 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 21.3% Easy, 44.0% Medium and 34.7% 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-p3/2012-may-june-9709-31/