# 9709/31 October/November 2016

> 9709/31 October/November 2016: 10 questions, 75 marks. Difficulty 0.46, deviation 0.17, algebraic 0.46, text density 0.30. CIE IAL Mathematics 9709 P3.

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CIE IAL · Mathematics · 2016 October/November

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=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3)
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Core finding

**9709/31 October/November 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 0.46 on a 0.20–0.75 scale, with a Difficulty Deviation of 0.17 and a mark split of 16.0% Easy, 56.0% Medium and 28.0% Hard. Algebraic complexity averages 0.46; text density and modelling average 0.30. 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**: 0.46 — 0.20–0.75 · nearer Medium
- **Difficulty Deviation**: 0.17 — 0 = uniform; higher = more mixed
- **Difficulty Ratio**: 16.0 / 56.0 / 28.0 — % Easy / Medium / Hard
- **Algebraic Complexity Measure**: 0.46 — 0.20–0.75 · nearer Medium
- **Text Density and Modelling Measure**: 0.30 — 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** | 0.17 | √(Σw_i(d_i − μ)^2 / Σw_i) |
| **Difficulty Ratio** | 16.0% Easy / 56.0% Medium / 28.0% 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** | 0.30 | 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 | [2 Logarithmic and exponential functions](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/logarithmic-and-exponential-functions/) | 3 | 0.20 Easy | 0.20 Low | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=12304258-d145-4792-8bf1-1d17bfff9c08&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q2** 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=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=b28b2840-70a0-4f35-bdb4-9a34aaf8202f&source_detail=9709%2F31+October%2FNovember+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=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=de704961-c09a-4535-9af3-36373854bfb0&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q4** FRQ | [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/) | 7 | 0.75 Hard | 0.75 High | 0.45 Medium | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=af44f99a-5956-4e05-b670-e8d94552e02e&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q5** FRQ | [3 Trigonometry](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/trigonometry/) | 8 | 0.45 Medium | 0.45 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=8d3413d8-9753-4463-a8fe-2b2e5ec5dd2b&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q6** FRQ | [6 Numerical solution of equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/numerical-solution-of-equations/) | 9 | 0.28 Easy | 0.20 Low | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=a1d8b2c3-533f-43dc-92ed-e79d1c2a4721&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q7** FRQ | [4 Differentiation](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differentiation/) | 9 | 0.59 Medium | 0.62 High | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=f9f19e5d-0205-413a-b3d0-525f39bad76f&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q8** FRQ | [7 Vectors](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/vectors/) | 9 | 0.37 Medium | 0.37 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=5a1a48c6-f1a3-40a7-ac7d-d2216a04f2cb&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q9** FRQ | [9 Complex numbers](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/complex-numbers/) | 10 | 0.45 Medium | 0.33 Medium | 0.20 Low | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=169cd031-189e-463f-bc4b-4d9956210251&source_detail=9709%2F31+October%2FNovember+2016) |
| **Q10** FRQ | [8 Differential equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differential-equations/) | 11 | 0.67 Hard | 0.65 High | 0.75 High | [Go to Practice Hub](https://www.examsci.com/practice?set_id=4e4c93d8-7ea2-43b9-a205-8cbb9bf1aea3&question_id=f2f2886e-2879-4d24-9d41-80137a7f21a8&source_detail=9709%2F31+October%2FNovember+2016) |

## Topics in this paper

- [2 Logarithmic and exponential functions](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/logarithmic-and-exponential-functions/)
- [1 Algebra](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/algebra/)
- [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/)
- [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/)
- [8 Differential equations](https://www.examsci.com/topics/cie-ial/mathematics/9709-p3/differential-equations/)

## Related papers

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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 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 0.17, 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 16.0% Easy, 56.0% Medium and 28.0% 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.

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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-p3/2016-october-november-9709-31/