Past exam of the mathematics course of the University of Cambridge / 2009 / ii

Table of contents
- Paper 1 ii
  - 1G Paper 1
    - Solution 1G
  - 2F Paper 1
    - i 2F
      - Solution i
    - ii 2F
      - Solution ii
  - 3F Paper 1
    - Solution 3F
  - 4H Paper 1
    - i 4H
      - Solution i
    - ii 4H
      - Solution ii
    - iii 4H
      - Solution iii
  - 5I Paper 1
    - Solution 5I
  - 6A Paper 1
    - Solution 6A
  - 7E Paper 1
    - Solution 7E
  - 8B Paper 1
    - Solution 8B
  - 9E Paper 1
    - Solution 9E
  - 10D Paper 1
    - Solution 10D
  - 11F Paper 1
    - Solution 11F
  - 12H Paper 1
    - i 12H
      - Solution i
    - ii 12H
      - Solution ii
    - iii 12H
      - Solution iii
  - 13I Paper 1
    - Solution 13I
  - 14B Paper 1
    - Solution 14B
  - 15D Paper 1
    - i 15D
      - Solution i
    - ii 15D
      - Solution ii
  - 16G Paper 1
    - Solution 16G
  - 17F Paper 1
    - i 17F
      - Solution i
    - ii 17F
      - Solution ii
    - iii 17F
      - Solution iii
  - 18H Paper 1
    - Solution 18H
  - 19F Paper 1
    - Solution 19F
  - 20H Paper 1
    - i 20H
      - Solution i
    - ii 20H
      - Solution ii
    - iii 20H
      - Solution iii
  - 21G Paper 1
    - Solution 21G
  - 22H Paper 1
    - a 22H
      - Solution a
    - b 22H
      - Solution b
    - c 22H
      - Solution c
  - 23G Paper 1
    - a 23G
      - Solution a
    - b 23G
      - Solution b
  - 24G Paper 1
    - Solution 24G
  - 25H Paper 1
    - i 25H
      - Solution i
    - ii 25H
      - Solution ii
    - iii 25H
      - Solution iii
    - iv 25H
      - Solution iv
  - 26J Paper 1
    - Solution 26J
  - 27J Paper 1
    - a 27J
      - Solution a
    - b 27J
      - Solution b
    - c 27J
      - Solution c
  - 28I Paper 1
    - i 28I
      - Solution i
    - ii 28I
      - Solution ii
    - iii 28I
      - Solution iii
  - 29J Paper 1
    - a 29J
      - Solution a
    - b 29J
      - Solution b
    - c 29J
      - Solution c
  - 30B Paper 1
    - i 30B
      - Solution i
    - ii 30B
      - Solution ii
    - iii 30B
      - Solution iii
    - iv 30B
      - Solution iv
  - 31A Paper 1
    - Solution 31A
  - 32B Paper 1
    - Solution 32B
  - 33C Paper 1
    - Solution 33C
  - 34D Paper 1
    - Solution 34D
  - 35C Paper 1
    - i 35C
      - Solution i
    - ii 35C
      - Solution ii
    - iii 35C
      - Solution iii
  - 36D Paper 1
    - Solution 36D
  - 37E Paper 1
    - i 37E
      - Solution i
    - ii 37E
      - Solution ii
  - 38A Paper 1
    - Solution 38A
  - 39B Paper 1
    - i 39B
      - Solution i
    - ii 39B
      - Solution ii
    - iii 39B
      - Solution iii
    - iv 39B
      - Solution iv
- Paper 2 ii
  - 1G Paper 2
    - Solution 1G
  - 2F Paper 2
    - a 2F
      - Solution a
    - b 2F
      - Solution b
  - 3F Paper 2
    - Solution 3F
  - 4H Paper 2
    - Solution 4H
  - 5I Paper 2
    - Solution 5I
  - 6A Paper 2
    - Solution 6A
  - 7E Paper 2
    - i 7E
      - Solution i
    - ii 7E
      - Solution ii
  - 8B Paper 2
    - i 8B
      - Solution i
    - ii 8B
      - Solution ii
  - 9E Paper 2
    - Solution 9E
  - 10D Paper 2
    - a 10D
      - Solution a
    - b 10D
      - Solution b
  - 11F Paper 2
    - a 11F
      - Solution a
    - b 11F
      - Solution b
    - c 11F
      - Solution c
  - 12H Paper 2
    - Solution 12H
  - 13A Paper 2
    - Solution 13A
  - 14C Paper 2
    - i 14C
      - Solution i
    - ii 14C
      - Solution ii
  - 15E Paper 2
    - i 15E
      - Solution i
    - ii 15E
      - Solution ii
  - 16G Paper 2
    - i 16G
      - Solution i
    - ii 16G
      - Solution ii
  - 17F Paper 2
    - i 17F
      - Solution i
    - ii 17F
      - Solution ii
  - 18H Paper 2
    - Solution 18H
  - 19F Paper 2
    - i 19F
      - Solution i
    - ii 19F
      - Solution ii
    - iii 19F
      - Solution iii
  - 20H Paper 2
    - i 20H
      - Solution i
    - ii 20H
      - Solution ii
    - iii 20H
      - Solution iii
  - 21G Paper 2
    - Solution 21G
  - 22H Paper 2
    - a 22H
      - Solution a
    - b 22H
      - Solution b
    - c 22H
      - Solution c
  - 23G Paper 2
    - a 23G
      - Solution a
    - b 23G
      - Solution b
  - 24G Paper 2
    - Solution 24G
  - 25H Paper 2
    - a 25H
      - Solution a
    - b 25H
      - Solution b
    - c 25H
      - Solution c
  - 26J Paper 2
    - Solution 26J
  - 27J Paper 2
    - a 27J
      - Solution a
    - b 27J
      - Solution b
  - 28I Paper 2
    - Solution 28I
  - 29I Paper 2
    - Solution 29I
  - 30J Paper 2
    - Solution 30J
  - 31B Paper 2
    - a 31B
      - Solution a
    - b 31B
      - Solution b
    - c 31B
  - 32B Paper 2
    - Solution 32B
  - 33C Paper 2
    - Solution 33C
  - 34D Paper 2
    - Solution 34D
  - 35D Paper 2
    - i 35D
      - Solution i
    - ii 35D
      - Solution ii
    - iii 35D
      - Solution iii
  - 36D Paper 2
    - Solution 36D
  - 37E Paper 2
    - Solution 37E
  - 38A Paper 2
    - Solution 38A
  - 39B Paper 2
    - i 39B
      - Solution i
    - ii 39B
      - Solution ii
    - iii 39B
      - Solution iii
- Paper 3 ii
  - 1G Paper 3
    - Solution 1G
  - 2F Paper 3
    - a 2F
      - Solution a
    - b 2F
      - Solution b
  - 3F Paper 3
    - Solution 3F
  - 4H Paper 3
    - Solution 4H
  - 5I Paper 3
    - Solution 5I
  - 6A Paper 3
    - Solution 6A
  - 7E Paper 3
    - Solution 7E
  - 8B Paper 3
    - Solution 8B
  - 9E Paper 3
    - a 9E
      - Solution a
    - b 9E
      - Solution b
  - 10D Paper 3
    - a 10D
      - Solution a
    - b 10D
      - Solution b
  - 11G Paper 3
    - Solution 11G
  - 12F Paper 3
    - a 12F
      - Solution a
    - b 12F
      - Solution b
    - c 12F
      - Solution c
  - 13A Paper 3
    - Solution 13A
  - 14E Paper 3
    - i 14E
      - Solution i
    - ii 14E
      - Solution ii
    - iii 14E
      - Solution iii
  - 15D Paper 3
    - i 15D
      - Solution i
    - ii 15D
      - Solution ii
  - 16G Paper 3
    - Solution 16G
  - 17F Paper 3
    - a 17F
      - Solution a
    - b 17F
      - Solution b
    - c 17F
      - Solution c
  - 18H Paper 3
    - Solution 18H
  - 19F Paper 3
    - i 19F
      - Solution i
    - ii 19F
      - Solution ii
    - iii 19F
      - Solution iii
  - 20G Paper 3
    - i 20G
      - Solution i
    - ii 20G
      - Solution ii
  - 21H Paper 3
    - a 21H
      - Solution a
    - b 21H
      - Solution b
    - c 21H
      - Solution c
  - 22G Paper 3
    - i 22G
      - Solution i
    - ii 22G
      - Solution ii
  - 23G Paper 3
    - Solution 23G
  - 24H Paper 3
    - a 24H
      - Solution a
    - b 24H
      - Solution b
  - 25J Paper 3
    - Solution 25J
  - 26J Paper 3
    - a 26J
      - Solution a
    - b 26J
      - Solution b
    - c 26J
      - Solution c
  - 27I Paper 3
    - Solution 27I
  - 28I Paper 3
    - Solution 28I
  - 29J Paper 3
    - Solution 29J
  - 30B Paper 3
    - a 30B
      - Solution a
    - b 30B
      - Solution b
    - c 30B
      - Solution c
    - d 30B
      - Solution d
  - 31A Paper 3
    - Solution 31A
  - 32B Paper 3
    - i 32B
      - Solution i
    - ii 32B
      - Solution ii
  - 33C Paper 3
    - i 33C
      - Solution i
    - ii 33C
      - Solution ii
  - 34D Paper 3
    - Solution 34D
  - 35D Paper 3
    - i 35D
      - Solution i
    - ii 35D
      - Solution ii
    - iii 35D
      - Solution iii
  - 36C Paper 3
    - Solution 36C
  - 37E Paper 3
    - Solution 37E
  - 38A Paper 3
    - Solution 38A
  - 39B Paper 3
- Paper 4 ii
  - 1G Paper 4
    - Solution 1G
  - 2F Paper 4
    - Solution 2F
  - 3F Paper 4
    - Solution 3F
  - 4H Paper 4
    - i 4H
      - Solution i
    - ii 4H
      - Solution ii
    - iii 4H
      - Solution iii
    - iv 4H
      - Solution iv
  - 5I Paper 4
    - Solution 5I
  - 6A Paper 4
    - Solution 6A
  - 7E Paper 4
    - Solution 7E
  - 8D Paper 4
    - Solution 8D
  - 9E Paper 4
    - a 9E
      - Solution a
    - b 9E
      - Solution b
  - 10D Paper 4
    - a 10D
      - Solution a
    - b 10D
      - Solution b
  - 11G Paper 4
    - Solution 11G
  - 12F Paper 4
    - ii 12F
      - Solution ii
  - 13I Paper 4
    - Solution 13I
  - 14E Paper 4
    - Solution 14E
  - 15E Paper 4
    - Solution 15E
  - 16G Paper 4
    - Solution 16G
  - 17F Paper 4
    - Solution 17F
  - 18H Paper 4
    - a 18H
      - Solution a
    - b 18H
      - Solution b
    - c 18H
      - Solution c
  - 19F Paper 4
    - a 19F
      - Solution a
    - b 19F
      - Solution b
    - c 19F
      - Solution c
  - 20H Paper 4
    - Solution 20H
  - 21G Paper 4
    - Solution 21G
  - 22H Paper 4
    - a 22H
      - Solution a
    - b 22H
      - Solution b
    - c 22H
      - Solution c
    - d 22H
      - Solution d
  - 23G Paper 4
    - Solution 23G
  - 24H Paper 4
    - a 24H
      - Solution a
    - b 24H
      - Solution b
    - c 24H
      - Solution c
  - 25J Paper 4
    - Solution 25J
  - 26J Paper 4
    - a 26J
      - Solution a
    - b 26J
      - Solution b
  - 27I Paper 4
    - Solution 27I
  - 28I Paper 4
    - a 28I
      - Solution a
    - b 28I
      - Solution b
  - 29J Paper 4
    - Solution 29J
  - 30B Paper 4
    - Solution 30B
  - 31A Paper 4
    - Solution 31A
  - 32C Paper 4
    - Solution 32C
  - 33D Paper 4
    - Solution 33D
  - 34D Paper 4
    - i 34D
      - Solution i
    - ii 34D
      - Solution ii
    - iii 34D
      - Solution iii
    - iv 34D
      - Solution iv
  - 35C Paper 4
    - Solution 35C
  - 36D Paper 4
    - Solution 36D
  - 37E Paper 4
    - Solution 37E
  - 38A Paper 4
    - Solution 38A
  - 39B Paper 4
    - a 39B
      - Solution a
    - b 39B
      - Solution b

Paper 1

www.maths.cam.ac.uk/undergrad/pastpapers/files/2009/PaperII_1.pdf

1G (Number Theory)

Solution

2F (Topics in Analysis)

i

Solution

ii

Solution

3F (Geometry of Group Actions)

Solution

4H (Coding and Cryptography)

i

Solution

ii

Solution

iii

Solution

5I (Statistical Modelling)

Solution

6A (Mathematical Biology)

Solution

7E (Dynamical Systems)

Solution

8B (Further Complex Methods)

Solution

9E (Classical Dynamics)

Solution

10D (Cosmology)

Solution

11F (Geometry of Group Actions)

Solution

12H (Coding and Cryptography)

i

Solution

ii

Solution

iii

Solution

13I (Statistical Modelling)

Solution

14B (Further Complex Methods)

Solution

15D (Cosmology)

i

Solution

ii

Solution

16G (Logic and Set Theory)

Solution

17F (Graph Theory)

i

Solution

ii

Solution

iii

Solution

18H (Galois Theory)

Solution

19F (Representation Theory)

Solution

20H (Number Fields)

i

Solution

ii

Solution

iii

Solution

21G (Algebraic Topology)

Solution

22H (Linear Analysis)

a

Solution

b

Solution

c

Solution

23G (Riemann Surfaces)

a

Solution

b

Solution

24G (Algebraic Geometry)

Solution

25H (Differential Geometry)

i

Solution

ii

Solution

iii

Solution

iv

Solution

26J (Probability and Measure)

Solution

27J (Applied Probability)

a

Solution

b

Solution

c

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