AS LEVEL FURTHER MATHEMATICS: Actual May 2025 PAST Further Pure Mathematics 2:
Further Pure Mathematics 2. All Assessment Questions & Mark Scheme [Edexcel 8FM0/22]
Exam Summary
The AS Level Further Mathematics Paper 22: Further Pure Mathematics 2 (Edexcel 8FM0/22) tests
advanced pure mathematics topics including complex numbers, matrices and eigenvalues, proof by
induction, vectors in 3D, polar coordinates, and polynomial roots. Students are expected to demonstrate
strong algebraic manipulation, logical reasoning in proofs, and problem-solving using abstract mathematical
concepts. The paper emphasizes both method and accuracy, requiring clear presentation of working and
justification of results. This content forms the core foundation for Further Pure Mathematics, making it
essential for effective revision and preparation for the 2026 exams, where similar advanced pure topics
and reasoning skills will be further developed and assessed.
Turn over
,1. In this question you must show all stages of your working.
Solutions based entirely on calculator technology are not acceptable.
DO NOT WRITE IN THIS AREA
(i) (a) Use the Euclidean algorithm to determine the highest common factor h of
105 and 24
(3)
(b) Hence determine integers a and b such that
105a + 24b = h
(3)
5
(ii) Determine the remainder when 179 is divided by 11
(2)
DO NOT WRITE IN THIS AREA
DO NOT WRITE IN THIS AREA
2
■■■■
, 2. (i) Using a suitable algorithm and without performing any division, determine whether
13 306 617 is divisible by 9
(2)
DO NOT WRITE IN THIS AREA
(ii) The group G = {1, 3, 7, 9, 11, 13, 17, 19} has multiplication modulo 20 as
its operation.
(a) Complete the following Cayley table for G
× 1 3 7 9 11 13 17 19
20
1 1 3 7 9 11 13 17 19
3 3 1 19 11
7 7 1 17 13
9 9 3 1 17 13
11 11 19 1
DO NOT WRITE IN THIS AREA
13 13 11 9 1 7
17 17 11 3
19 19 13 9 3
A copy of this table is given on page 9 if you need to rewrite your Cayley table.
(3)
(b) State the inverse of the element 7
(1)
(c) Determine the order of the element 13
(1)
(d) Write down a subgroup of G of order 4
DO NOT WRITE IN THIS AREA
(1)
6
■■■■
Further Pure Mathematics 2. All Assessment Questions & Mark Scheme [Edexcel 8FM0/22]
Exam Summary
The AS Level Further Mathematics Paper 22: Further Pure Mathematics 2 (Edexcel 8FM0/22) tests
advanced pure mathematics topics including complex numbers, matrices and eigenvalues, proof by
induction, vectors in 3D, polar coordinates, and polynomial roots. Students are expected to demonstrate
strong algebraic manipulation, logical reasoning in proofs, and problem-solving using abstract mathematical
concepts. The paper emphasizes both method and accuracy, requiring clear presentation of working and
justification of results. This content forms the core foundation for Further Pure Mathematics, making it
essential for effective revision and preparation for the 2026 exams, where similar advanced pure topics
and reasoning skills will be further developed and assessed.
Turn over
,1. In this question you must show all stages of your working.
Solutions based entirely on calculator technology are not acceptable.
DO NOT WRITE IN THIS AREA
(i) (a) Use the Euclidean algorithm to determine the highest common factor h of
105 and 24
(3)
(b) Hence determine integers a and b such that
105a + 24b = h
(3)
5
(ii) Determine the remainder when 179 is divided by 11
(2)
DO NOT WRITE IN THIS AREA
DO NOT WRITE IN THIS AREA
2
■■■■
, 2. (i) Using a suitable algorithm and without performing any division, determine whether
13 306 617 is divisible by 9
(2)
DO NOT WRITE IN THIS AREA
(ii) The group G = {1, 3, 7, 9, 11, 13, 17, 19} has multiplication modulo 20 as
its operation.
(a) Complete the following Cayley table for G
× 1 3 7 9 11 13 17 19
20
1 1 3 7 9 11 13 17 19
3 3 1 19 11
7 7 1 17 13
9 9 3 1 17 13
11 11 19 1
DO NOT WRITE IN THIS AREA
13 13 11 9 1 7
17 17 11 3
19 19 13 9 3
A copy of this table is given on page 9 if you need to rewrite your Cayley table.
(3)
(b) State the inverse of the element 7
(1)
(c) Determine the order of the element 13
(1)
(d) Write down a subgroup of G of order 4
DO NOT WRITE IN THIS AREA
(1)
6
■■■■
