plumiBinary operations: AQA A‑Level Further Maths questions
Every AQA A-Level Further Maths exam question on DG Binary operations, 2019–2025, with its mark scheme and examiner’s report.
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All 21 questions on Binary operations, as a list
- June 2025, Paper 3 Discrete, Q1 (1 mark): The group A is of order 9 The group B is a proper, non-trivial subgroup of A Find the order of B Circle your answer. [1 mark] 1 3 9 18
- June 2025, Paper 3 Discrete, Q3 (1 mark): The group ( X, *) is an abelian group of order 4 The elements of X are e, a, b and c where e is the identity element of ( X, *) The following is a partially com
- June 2025, Paper 3 Discrete, Q10 (5 marks): The set S contains the elements (a, b) where a, b ∈ ℝ except (0, 0) The group G is formed by S under the binary operation ◊ where (a, b) ◊ (c, d) = (a × d + b ×
- June 2025, AS Paper 2 Discrete, Q6 (4 marks): The set S is defined as S O x x : 1 2 The binary operation * is defined as a * b = a + b – 2ab where a and b ∈ S It is given that S has exactly one identity el
- June 2024, Paper 3 Discrete, Q1 (1 mark): Which one of the following sets forms a group under the given binary operation? Tick () one box. [1 mark] Set Binary Operation {1, 2, 3} Addition modulo 4 {1,
- June 2024, Paper 3 Discrete, Q7 (12 marks): (a) By considering associativity, show that the set of integers does not form a group under the binary operation of subtraction. Fully justify your answer. [2 m
- June 2024, AS Paper 2 Discrete, Q4 (4 marks): The set S is defined as S = {1, 2, 3, 4} 4 (a) Complete the Cayley Table shown below for S under the binary operation multiplication modulo 5 [2 marks] ×5 1 2 3
- June 2024, AS Paper 2 Discrete, Q7 (5 marks): The binary operation ∇ is defined as a ∇ b = a + b + ab where a, b ∈ ℝ 7 (a) Determine if ∇ is commutative on ℝ Fully justify your answer. [2 marks] 7 (b) Prove
- June 2023, Paper 3 Discrete, Q9 (9 marks): The group (C, +4) contains the elements 0, 1, 2 and 3 9 (a) (i) Show that C is a cyclic group. [2 marks] 9 (a) (ii) State the group of symmetries of a regular p
- June 2023, AS Paper 2 Discrete, Q5 (7 marks): (a) The set S is defined as S = {0, 1, 2, 3, 4, 5} 5 (a) (i) State the identity element of S under the operation multiplication modulo 6 [1 mark] 5 (a) (ii) An
- June 2022, Paper 3 Discrete, Q7 (8 marks): The group G has binary operation * and order p, where p is a prime number. 7 (a) Determine the number of distinct subgroups of G Fully justify your answer. [2 m
- June 2022, Paper 3 Discrete, Q9 (6 marks): The binary operation × acts on the positive integers x and y such that x × y = x + y + 8 (mod k 2 − 16k + 74) where k is a positive integer. 9 (a) (i) Show that
- June 2022, AS Paper 2 Discrete, Q6 (5 marks): The set S is given by S = {A, B, C, D} where A = 1 0 0 0 B = 1 0 0 1 C = 0 0 0 1 D = 0 0 0 0 6 (a) Complete the Cayley table for S under matrix multiplication.
- November 2021, Paper 3 Discrete, Q5 (11 marks): (a) Describe the conditions necessary for a set of elements, S, under a binary operation * to form a group. [3 marks] 5 (b) In the multiplicative group of integ
- November 2021, AS Paper 2 Discrete, Q2 (4 marks): The set S is given by S = {0, 2, 4, 6} 2 (a) Construct a Cayley table, using the grid below, for S under the binary operation addition modulo 8 [3 marks] 2 (b)
- November 2021, AS Paper 2 Discrete, Q4 (5 marks): The binary operation * is defined as a * b = ab + 1 where a, b 2 R 4 (a) Prove that * is commutative on R [2 marks] 4 (b) Prove that * is not associative on R [
- November 2020, Paper 3 Discrete, Q2 (1 mark): Which of the following statements is true about the operation of matrix multiplication on the set of all 2 − 2 real matrices? Tick (3) one box. [1 mark] Matrix
- November 2020, Paper 3 Discrete, Q6 (8 marks): The group (G, ~) has the elements e, r, r2, q, qr and qr2, where r2 = r ~r , qr = q ~r , qr2 = q ~r2 and e is the identity element of G. The elements q and r ha
- November 2020, AS Paper 2 Discrete, Q8 (6 marks): The set S is defined as S = fa, b, c, dg Figure 2 shows a Cayley table for S under the commutative binary operation ≤ Figure 2 ≤ a b c d a a a a a b a d b c c a
- June 2019, Paper 3 Discrete, Q5 (12 marks): The set S is defined as S = fA, B, C, Dg where A = 1 0 0 1 B = 0 −1 1 0 C = −1 0 0 −1 D = 0 1 −1 0 The group G is formed by S under matrix multiplication. The g
- June 2019, AS Paper 2 Discrete, Q5 (5 marks): (a) Complete the Cayley table in Figure 1 for multiplication modulo 4 [2 marks] Figure 1 ×4 0 1 2 3 0 1 2 3 5 (b) The set S is defined as S = fa, b, c, dg Figur
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