plumiProof: AQA A‑Level Maths questions
Every AQA A-Level Maths exam question on A Proof, 2018–2025, with its mark scheme and examiner’s report.
Find questions426
![June 2024 Paper 1, question 13: (a) It is given that P(x) = 4x3 + 8x2 + 11x + 4 Use the factor theorem to show that (2x + 1) is a factor of P(x) [2 marks] 13 (b) Express P(x) in the form P(x)](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun24-13-qp.webp)
![June 2024 Paper 2, question 11: (a) A student states that 3 is the smallest value of k in the interval 3 < k < 4 Explain the error in the student’s statement. [1 mark] 11 (b) The student’s tea](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun24-11-qp.webp)
![June 2023 Paper 2, question 10: (a) Expand and simplify (a − b)2 [1 mark] 10 (b) Peter thinks that the sum of any rational number and its reciprocal is always greater than 2 Peter checks two e](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun23-10-qp.webp)
![June 2022 Paper 3, question 9: Assume that a and b are integers such that a2 − 4b − 2 = 0 9 (a) Prove that a is even. [2 marks] 9 (b) Hence, prove that 2b + 1 is even and explain why this is](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun22-09-qp.webp)
![November 2021 Paper 1, question 13: (a) Given that P(x) = 125x3 + 150x2 + 55x + 6 use the factor theorem to prove that (5x + 1) is a factor of P(x). [2 marks] 13 (b) Factorise P(x) completely. [3](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-nov21-13-qp.webp)
![November 2020 Paper 1, question 5: Prove that, for integer values of n such that 0 × n < 4 2n+2 > 3n [2 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-nov20-05-qp.webp)
![June 2019 Paper 1, question 9: Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun19-09-qp.webp)
![June 2019 Paper 3, question 6: The three sides of a right-angled triangle have lengths a, b and c, where a, b, c 2 Z b c a 6 (a) State an example where a, b and c are all even. [1 mark] 6 (b)](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun19-06-qp.webp)
![June 2019 Paper 3, question 9: A curve has equation x2y2 + xy4 = 12 9 (a) Prove that the curve does not intersect the coordinate axes. [2 marks] 9 (b) (i) Show that dy dx = − 2xy + y3 2x2 + 4](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun19-09-qp.webp)
![June 2018 Paper 1, question 12: p(x) = 30x3 − 7x2 − 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) [2 marks] 12 (b) Factorise p(x) completely. [3 marks] 12 (c) Prove that there are no r](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73571-jun18-12-qp.webp)
![June 2018 Paper 2, question 1: Which of these statements is correct? Tick one box. [1 mark] x = 2 ) x2 = 4 x2 = 4 ) x = 2 x2 = 4 , x = 2 x2 = 4 ) x = −2](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun18-01-qp.webp)
![June 2018 Paper 2, question 5: Prove that 23 is a prime number. [2 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73572-jun18-05-qp.webp)
![June 2018 Paper 3, question 10: Prove by contradiction that 2 3p is an irrational number. [7 marks]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/73573-jun18-10-qp.webp)
All 19 questions on Proof, as a list
- June 2025, Paper 3, Q1 (1 mark): A student states that the product of two irrational numbers is always irrational. One of the options is a counter example that shows the student’s statement is
- June 2025, Paper 3, Q12 (3 marks): Given that the real numbers a, b, and c are such that a × b = c Use proof by contradiction to show that if c is irrational then at least one of a or b is irrati
- June 2024, Paper 1, Q13 (6 marks): (a) It is given that P(x) = 4x3 + 8x2 + 11x + 4 Use the factor theorem to show that (2x + 1) is a factor of P(x) [2 marks] 13 (b) Express P(x) in the form P(x)
- June 2024, Paper 2, Q11 (6 marks): (a) A student states that 3 is the smallest value of k in the interval 3 < k < 4 Explain the error in the student’s statement. [1 mark] 11 (b) The student’s tea
- June 2023, Paper 2, Q10 (6 marks): (a) Expand and simplify (a − b)2 [1 mark] 10 (b) Peter thinks that the sum of any rational number and its reciprocal is always greater than 2 Peter checks two e
- June 2022, Paper 2, Q6 (5 marks): (a) Asif notices that 242 = 576 and 2 + 4 = 6 gives the last digit of 576 He checks two more examples: 272 = 729 292 = 841 2 + 7 = 9 2 + 9 = 11 Last digit 9 Las
- June 2022, Paper 3, Q9 (6 marks): Assume that a and b are integers such that a2 − 4b − 2 = 0 9 (a) Prove that a is even. [2 marks] 9 (b) Hence, prove that 2b + 1 is even and explain why this is
- November 2021, Paper 1, Q4 (1 mark): Millie is attempting to use proof by contradiction to show that the result of multiplying an irrational number by a non-zero rational number is always an irrati
- November 2021, Paper 1, Q13 (8 marks): (a) Given that P(x) = 125x3 + 150x2 + 55x + 6 use the factor theorem to prove that (5x + 1) is a factor of P(x). [2 marks] 13 (b) Factorise P(x) completely. [3
- November 2021, Paper 2, Q8 (6 marks): Kai is proving that n3 − n is a multiple of 3 for all positive integer values of n. Kai begins a proof by exhaustion. Step 1 n3 − n = n(n2 − 1) Step 2 When n =
- November 2020, Paper 1, Q5 (2 marks): Prove that, for integer values of n such that 0 × n < 4 2n+2 > 3n [2 marks]
- November 2020, Paper 2, Q7 (7 marks): a and b are two positive irrational numbers. The sum of a and b is rational. The product of a and b is rational. Caroline is trying to prove 1 a + 1 b is ration
- June 2019, Paper 1, Q9 (5 marks): Prove that the sum of a rational number and an irrational number is always irrational. [5 marks]
- June 2019, Paper 3, Q6 (4 marks): The three sides of a right-angled triangle have lengths a, b and c, where a, b, c 2 Z b c a 6 (a) State an example where a, b and c are all even. [1 mark] 6 (b)
- June 2019, Paper 3, Q9 (15 marks): A curve has equation x2y2 + xy4 = 12 9 (a) Prove that the curve does not intersect the coordinate axes. [2 marks] 9 (b) (i) Show that dy dx = − 2xy + y3 2x2 + 4
- June 2018, Paper 1, Q12 (10 marks): p(x) = 30x3 − 7x2 − 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) [2 marks] 12 (b) Factorise p(x) completely. [3 marks] 12 (c) Prove that there are no r
- June 2018, Paper 2, Q1 (1 mark): Which of these statements is correct? Tick one box. [1 mark] x = 2 ) x2 = 4 x2 = 4 ) x = 2 x2 = 4 , x = 2 x2 = 4 ) x = −2
- June 2018, Paper 2, Q5 (2 marks): Prove that 23 is a prime number. [2 marks]
- June 2018, Paper 3, Q10 (7 marks): Prove by contradiction that 2 3p is an irrational number. [7 marks]
Questions, mark schemes and examiner’s reports are © AQA, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every AQA A-Level Maths paper.
Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.