# 9709/31 May/June 2016

> 9709/31 May/June 2016: 10 questions, 75 marks. Difficulty 0.42, deviation 0.08, algebraic 0.46, text density 0.22. CIE IAL Mathematics 9709 P3.

- Canonical HTML: https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-may-june-9709-31/
- Markdown mirror: https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-may-june-9709-31/index.md

---

CIE IAL · Mathematics · 2016 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=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2)
- [All 9709 P3 papers](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/)

Core finding

**9709/31 May/June 2016 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 0.42 on a 0.20–0.75 scale, with a Difficulty Deviation of 0.08 and a mark split of 13.3% Easy, 86.7% Medium and 0.0% Hard. Algebraic complexity averages 0.46; text density and modelling average 0.22. All five measures use 18 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 18 parts carrying 75 part marks. Difficulty tags cover 18 parts, algebraic tags 18 and text-density tags 18. Tags are internal metadata, not official exam-board ratings.

- **Difficulty Measure**: 0.42 — 0.20–0.75 · nearer Medium
- **Difficulty Deviation**: 0.08 — 0 = uniform; higher = more mixed
- **Difficulty Ratio**: 13.3 / 86.7 / 0.0 — % Easy / Medium / Hard
- **Algebraic Complexity Measure**: 0.46 — 0.20–0.75 · nearer Medium
- **Text Density and Modelling Measure**: 0.22 — 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** | 0.42 | Σ(d_i × w_i) / Σw_i, with d = 0.20 / 0.45 / 0.75 |
| **Difficulty Deviation** | 0.08 | √(Σw_i(d_i − μ)^2 / Σw_i) |
| **Difficulty Ratio** | 13.3% Easy / 86.7% Medium / 0.0% Hard | marks in band / all tagged marks |
| **Algebraic Complexity Measure** | 0.46 | Σ(c_i × marks_i) / Σmarks_i, with c = 0.20 / 0.45 / 0.75 |
| **Text Density and Modelling Measure** | 0.22 | 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/) | 5 | 0.20 Easy | 0.30 Low | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=3d4d583b-8599-4e61-944a-aade3c23d6af&source_detail=9709%2F31+May%2FJune+2016) |
| **Q2** FRQ | [5 Integration](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/integration/) | 5 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=17d3ea91-f425-4798-ab8e-5c04bb23482e&source_detail=9709%2F31+May%2FJune+2016) |
| **Q3** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 5 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=41e1a102-4b98-46e2-824b-0e6bc7fd646f&source_detail=9709%2F31+May%2FJune+2016) |
| **Q4** FRQ | [8 Differential equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differential-equations/) | 6 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=d1b05ae7-d7b0-4224-a8bb-5a73b16c72d0&source_detail=9709%2F31+May%2FJune+2016) |
| **Q5** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 6 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=805b2656-dc7b-47a2-8b00-f389f398992f&source_detail=9709%2F31+May%2FJune+2016) |
| **Q6** FRQ | [6 Numerical solution of equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/numerical-solution-of-equations/) | 7 | 0.27 Easy | 0.20 Low | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=ed4fec4c-2af3-47fe-9d2c-8fdc42ec22bf&source_detail=9709%2F31+May%2FJune+2016) |
| **Q7** FRQ | [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/) | 9 | 0.45 Medium | 0.62 High | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=16d8dd34-55c2-4ae2-95ca-462fe219a032&source_detail=9709%2F31+May%2FJune+2016) |
| **Q8** FRQ | [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/) | 10 | 0.45 Medium | 0.75 High | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=4bf75ef9-13d5-49d5-86a6-c7810771c2fb&source_detail=9709%2F31+May%2FJune+2016) |
| **Q9** FRQ | [7 Vectors](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/vectors/) | 11 | 0.45 Medium | 0.45 Medium | 0.31 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=fc77f917-d573-44d4-b11f-33cd60041af2&source_detail=9709%2F31+May%2FJune+2016) |
| **Q10** FRQ | [9 Complex numbers](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/complex-numbers/) | 11 | 0.45 Medium | 0.36 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=9d241ba1-4a50-4032-8e31-9adc8cb5d1b2&question_id=fc5cd35d-7832-42a9-82b4-a0fdf09031c6&source_detail=9709%2F31+May%2FJune+2016) |

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

## Related papers

- [9709/31](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-october-november-9709-31/) — October/November 2016 — 10 questions · 75 marks — Free
- [9709/32](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-october-november-9709-32/) — October/November 2016 — 10 questions · 75 marks — Free
- [9709/33](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-october-november-9709-33/) — October/November 2016 — 10 questions · 75 marks — Free
- [9709/32](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-may-june-9709-32/) — May/June 2016 — 10 questions · 75 marks — Free
- [9709/33](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-may-june-9709-33/) — May/June 2016 — 10 questions · 75 marks — Free
- [9709/32](https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-march-9709-32/) — March 2016 — 10 questions · 75 marks — Free

## 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 0.42 from 18 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 0.08, 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, 86.7% Medium and 0.0% 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.

---

Full page with native tables, bar charts and Practice Hub links: https://www.examsci.com/past-papers/cie-ial/mathematics/9709-p3/2016-may-june-9709-31/