plumiProof: OCR A‑Level Maths questions
Every OCR A-Level Maths exam question on P.1 Proof, 2018–2025, with its mark scheme and examiner’s report.
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![June 2024 Paper 1, question 3: (a) Find a counterexample to disprove the statement that the product of two prime numbers is always odd. [1] (b) In each of the following cases write one of the](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun24-03-qp.webp?v=3)
![June 2023 Paper 2, question 7: 7 A student wishes to prove that, for all positive integers a and b, a 4 b 2 . ! - 2 (a) Prove that a & [2] 4 b 2 a is even. - = 2 (b) Hence or otherwise prove](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun23-07-qp.webp?v=3)
![June 2022 Paper 1, question 2: (a) Given that a and b are real numbers, find a counterexample to disprove the statement that, 2 2 if a 2 b , then a 2 b . [1] (b) A student writes the statemen](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun22-02-qp.webp?v=3)
![June 2022 AS Paper 2, question 3: Give a counter example to disprove the following statement. If x and y are both irrational then x + is irrational. y [2]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23002-jun22-03-qp.webp?v=3)
![November 2021 Paper 2, question 8: 8 The number K is defined by K n + , where n is an integer greater than 2. 1 = ^h h, find the constants b and c. ^ 3 2 (a) Given that n 1 n 1 n bn c [1] + / + +](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-nov21-08-qp.webp?v=3)
![November 2021 AS Paper 1, question 8: (a) Prove that the following statement is not true. 2p 2 p is a positive integer p [1] & H (b) Prove that the following statement is true. m and n are consecuti](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-nov21-08-qp.webp?v=3)
![November 2020 Paper 1, question 4: Prove by contradiction that there is no greatest multiple of 5. [3]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-nov20-04-qp.webp?v=3)
![November 2020 Paper 2, question 5: - and n 2 + are positive integers. (a) Determine the set of values of n for which n 1 1 5 [3] 2 2 2 2 2 A ‘Pythagorean triple’ is a set of three positive intege](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-nov20-05-qp.webp?v=3)
![June 2019 Paper 2, question 6: Shona makes the following claim. n “n is an even positive integer greater than 2 2 - is not prime” 1 & Prove that Shona’s claim is true. [4]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24002-jun19-06-qp.webp?v=3)
![June 2019 AS Paper 1, question 5: (a) Prove that the following statement is not true. 2 m is an odd number greater than 1 m + is prime. 4 [1] & (b) By considering separately the case when n is o](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun19-05-qp.webp?v=3)
![June 2018 Paper 1, question 4: + 4 Prove algebraically that n n - is odd for all positive integers n. [4] 3 1](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h24001-jun18-04-qp.webp?v=3)
![June 2018 AS Paper 1, question 5: N is an integer that is not divisible by 3. Prove that N2 is of the form p 5 3 + , where p is an integer. 1 [5]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/h23001-jun18-05-qp.webp?v=3)
All 18 questions on Proof, as a list
- June 2025, Paper 1, Q11 (7 marks): A student is attempting to prove that 5 is irrational. The first three lines of their proof are shown below. a Line 1 Assume that 5 is rational, so it can be wr
- June 2024, Paper 1, Q3 (8 marks): (a) Find a counterexample to disprove the statement that the product of two prime numbers is always odd. [1] (b) In each of the following cases write one of the
- June 2023, Paper 2, Q7 (5 marks): 7 A student wishes to prove that, for all positive integers a and b, a 4 b 2 . ! - 2 (a) Prove that a & [2] 4 b 2 a is even. - = 2 (b) Hence or otherwise prove
- June 2023, AS Paper 1, Q7 (5 marks): (a) Two real numbers are denoted by a and b. (i) Write down expressions for the following. • The mean of the squares of a and b • The square of the mean of a an
- June 2022, Paper 1, Q2 (6 marks): (a) Given that a and b are real numbers, find a counterexample to disprove the statement that, 2 2 if a 2 b , then a 2 b . [1] (b) A student writes the statemen
- June 2022, Paper 2, Q7 (8 marks): It is given that any integer can be expressed in the form m 3 + , where m is an integer and r r is 0, 1 or 2. Use this fact to answer the following. (a) By cons
- June 2022, AS Paper 2, Q3 (2 marks): Give a counter example to disprove the following statement. If x and y are both irrational then x + is irrational. y [2]
- November 2021, Paper 2, Q8 (6 marks): 8 The number K is defined by K n + , where n is an integer greater than 2. 1 = ^h h, find the constants b and c. ^ 3 2 (a) Given that n 1 n 1 n bn c [1] + / + +
- November 2021, AS Paper 1, Q8 (5 marks): (a) Prove that the following statement is not true. 2p 2 p is a positive integer p [1] & H (b) Prove that the following statement is true. m and n are consecuti
- November 2020, Paper 1, Q4 (3 marks): Prove by contradiction that there is no greatest multiple of 5. [3]
- November 2020, Paper 2, Q5 (5 marks): - and n 2 + are positive integers. (a) Determine the set of values of n for which n 1 1 5 [3] 2 2 2 2 2 A ‘Pythagorean triple’ is a set of three positive intege
- June 2019, Paper 2, Q6 (4 marks): Shona makes the following claim. n “n is an even positive integer greater than 2 2 - is not prime” 1 & Prove that Shona’s claim is true. [4]
- June 2019, AS Paper 1, Q5 (5 marks): (a) Prove that the following statement is not true. 2 m is an odd number greater than 1 m + is prime. 4 [1] & (b) By considering separately the case when n is o
- June 2018, Paper 1, Q4 (4 marks): + 4 Prove algebraically that n n - is odd for all positive integers n. [4] 3 1
- June 2018, Paper 2, Q5 (6 marks): Charlie claims to have proved the following statement. “The sum of a square number and a prime number cannot be a square number.” (i) Give an example to show th
- June 2018, AS Paper 1, Q5 (5 marks): N is an integer that is not divisible by 3. Prove that N2 is of the form p 5 3 + , where p is an integer. 1 [5]
- June 2018, AS Paper 1, Q7 (7 marks): OABC is a parallelogram with OA= a and OC = c. P is the midpoint of AC. A B a P O c C (i) Find the following in terms of a and c, simplifying your answers. (a)
- June 2018, AS Paper 2, Q3 (3 marks): In each of the following cases choose one of the statements P & Q P % Q P + Q to describe the relationship between P and Q. 5 2 (i) P: y 3 x 4 x 12 x = - + d y
Questions, mark schemes and examiner’s reports are © OCR, from their published past papers. Plumi isn’t affiliated with any exam board. Want whole papers? Every OCR A-Level Maths paper.
Rather write your answers by hand on iPad? Plumi marks them against the same mark schemes.