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Proof: OCR MEI A‑Level Maths questions

Every OCR MEI A-Level Maths exam question on P.1 Proof, 2018–2025, with its mark scheme and examiner’s report.

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560 questions

  1. Q5

    June 2025 · Paper 2

    3 marksP.1 ProofP.4 Sequences and series
    The whole paper ↗
    June 2025 Paper 2, question 5: Prove that the sum of the first n positive odd numbers is a square number. [3]

  2. Q9

    June 2025 · AS Paper 2

    3 marksP.1 Proof
    The whole paper ↗
    June 2025 AS Paper 2, question 9: n n A triangular number is any number of the form ( 1 ), + 9 where n is a positive integer. 2 Prove that the sum of any two consecutive triangular numbers is a

  3. Q1

    June 2024 · Paper 1

    2 marksP.1 Proof
    The whole paper ↗
    June 2024 Paper 1, question 1: 2 1 A student states that 1 x 1 ( 1 x ) for all values of x. + + Using a counter example, show that the student is wrong. [2]

  4. Q7

    June 2024 · Paper 3

    3 marksP.5 TrigonometryP.1 Proof
    The whole paper ↗
    June 2024 Paper 3, question 7: Prove that sin 8 tan 4 cos 8 = . 1 [3] i i + i

  5. Q3

    June 2024 · AS Paper 1

    3 marksP.1 Proof
    The whole paper ↗
    June 2024 AS Paper 1, question 3: + is a multiple of 4 but not a multiple of 8. 3 Prove that, when n is an even number, n 4 [3]

  6. Q10

    June 2023 · Paper 3

    6 marksP.5 TrigonometryP.1 Proof
    The whole paper ↗
    June 2023 Paper 3, question 10: 2 3 6 4 2 2 4 6 10 (a) You are given that ( x y ) x 3 x y 3 x y y . + = + + + 6 6 3 2 Hence, or otherwise, prove that sin cos 1 sin 2 for all values of i. [4] +

  7. Q13

    June 2023 · Paper 3

    4 marksP.4 Sequences and seriesP.8 Integration
    The whole paper ↗
    June 2023 Paper 3, question 13: n =/ 2 13 Prove that Euler’s approximate formula, as given in line 13, when applied to r gives exactly r 1 n n ( 1 2 )( n 1 ) + + . [4] 6 n

  8. Q3

    June 2023 · AS Paper 2

    2 marksP.1 Proof
    The whole paper ↗
    June 2023 AS Paper 2, question 3: A student makes the following conjecture. For all positive integers n, n 6 - is always prime. 1 Use a counter example to disprove this conjecture. [2]

  9. Q12

    June 2022 · Paper 1

    4 marksP.1 Proof
    The whole paper ↗
    June 2022 Paper 1, question 12: Prove by contradiction that 3 is the only prime number which is 1 less than a square number. [4]

  10. Q5

    June 2022 · Paper 2

    3 marksP.1 Proof
    The whole paper ↗
    June 2022 Paper 2, question 5: n - is prime. 5 Tom conjectures that if n is an odd number greater than 1, then 2 1 Find a counter example to disprove Tom’s conjecture. [3]

  11. Q1

    November 2021 · Paper 1

    2 marksP.1 Proof
    The whole paper ↗
    November 2021 Paper 1, question 1: 2 1 Beth states that for all real numbers p and q, if p 2 q then p 2 q . Prove that Beth is not correct. [2]

  12. Q15

    November 2021 · Paper 3

    4 marksP.5 TrigonometryP.1 Proof
    The whole paper ↗
    November 2021 Paper 3, question 15: Prove that arctan 1 arctan 2 arctan 3 , as given in line 41. [4] + + = r

  13. Q4

    November 2021 · AS Paper 1

    4 marksP.1 Proof
    The whole paper ↗
    November 2021 AS Paper 1, question 4: 4 (a) Show that ! 4 1 . [2] nn (b) Nina believes that the statement ! n 1 is true for all positive integers n. Prove that Nina is not correct. [2]

  14. Q9

    November 2020 · Paper 3

    3 marksP.1 Proof
    The whole paper ↗
    November 2020 Paper 3, question 9: b a 9 (a) Show that if a = 1 and b 2 1 then a 1 b . [2] (b) Find integer values of a and b with b 2 a 2 1 and ab not greater than ba (a counter example to the c

  15. Q1

    November 2020 · AS Paper 1

    2 marksP.1 Proof
    The whole paper ↗
    November 2020 AS Paper 1, question 1: + 1 Celia states that n 2 n 10 is always odd when n is a prime number. + Prove that Celia’s statement is false. [2]

  16. Q5

    June 2019 · Paper 3

    3 marksP.1 Proof
    The whole paper ↗
    June 2019 Paper 3, question 5: A student’s attempt to prove by contradiction that there is no largest prime number is shown below. If there is a largest prime, list all the primes. Multiply a

  17. Q3

    June 2019 · AS Paper 2

    3 marksP.2 Algebra and functionsP.1 Proof
    The whole paper ↗
    June 2019 AS Paper 2, question 3: Without using a calculator, prove that 3 2 [3] 2 2 3 .

  18. Q11

    June 2018 · Paper 3

    2 marksP.1 Proof
    The whole paper ↗
    June 2018 Paper 3, question 11: Line 8 states that a b + 11 ab for a, b 0 . Explain why the result cannot be extended to apply in each of H H 2 the following cases. (i) One of the numbers a an

  19. Q12

    June 2018 · Paper 3

    3 marksP.1 Proof
    The whole paper ↗
    June 2018 Paper 3, question 12: a b + 2 H 12 ab a b Lines 5 and 6 outline the stages in a proof that . Starting from ( ) 0 , give a detailed H - 2 [3] proof of the inequality of arithmetic and

  20. Q13

    June 2018 · Paper 3

    3 marksP.4 Sequences and seriesP.1 Proof
    The whole paper ↗
    June 2018 Paper 3, question 13: Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three [3] consecutive terms of this sequence, the middle one
All 23 questions on Proof, as a list
  1. June 2025, Paper 2, Q5 (3 marks): Prove that the sum of the first n positive odd numbers is a square number. [3]
  2. June 2025, AS Paper 2, Q9 (3 marks): n n A triangular number is any number of the form ( 1 ), + 9 where n is a positive integer. 2 Prove that the sum of any two consecutive triangular numbers is a
  3. June 2024, Paper 1, Q1 (2 marks): 2 1 A student states that 1 x 1 ( 1 x ) for all values of x. + + Using a counter example, show that the student is wrong. [2]
  4. June 2024, Paper 3, Q7 (3 marks): Prove that sin 8 tan 4 cos 8 = . 1 [3] i i + i
  5. June 2024, AS Paper 1, Q3 (3 marks): + is a multiple of 4 but not a multiple of 8. 3 Prove that, when n is an even number, n 4 [3]
  6. June 2023, Paper 3, Q10 (6 marks): 2 3 6 4 2 2 4 6 10 (a) You are given that ( x y ) x 3 x y 3 x y y . + = + + + 6 6 3 2 Hence, or otherwise, prove that sin cos 1 sin 2 for all values of i. [4] +
  7. June 2023, Paper 3, Q13 (4 marks): n =/ 2 13 Prove that Euler’s approximate formula, as given in line 13, when applied to r gives exactly r 1 n n ( 1 2 )( n 1 ) + + . [4] 6 n
  8. June 2023, AS Paper 2, Q3 (2 marks): A student makes the following conjecture. For all positive integers n, n 6 - is always prime. 1 Use a counter example to disprove this conjecture. [2]
  9. June 2022, Paper 1, Q12 (4 marks): Prove by contradiction that 3 is the only prime number which is 1 less than a square number. [4]
  10. June 2022, Paper 2, Q5 (3 marks): n - is prime. 5 Tom conjectures that if n is an odd number greater than 1, then 2 1 Find a counter example to disprove Tom’s conjecture. [3]
  11. November 2021, Paper 1, Q1 (2 marks): 2 1 Beth states that for all real numbers p and q, if p 2 q then p 2 q . Prove that Beth is not correct. [2]
  12. November 2021, Paper 3, Q15 (4 marks): Prove that arctan 1 arctan 2 arctan 3 , as given in line 41. [4] + + = r
  13. November 2021, AS Paper 1, Q4 (4 marks): 4 (a) Show that ! 4 1 . [2] nn (b) Nina believes that the statement ! n 1 is true for all positive integers n. Prove that Nina is not correct. [2]
  14. November 2020, Paper 3, Q9 (3 marks): b a 9 (a) Show that if a = 1 and b 2 1 then a 1 b . [2] (b) Find integer values of a and b with b 2 a 2 1 and ab not greater than ba (a counter example to the c
  15. November 2020, AS Paper 1, Q1 (2 marks): + 1 Celia states that n 2 n 10 is always odd when n is a prime number. + Prove that Celia’s statement is false. [2]
  16. June 2019, Paper 3, Q5 (3 marks): A student’s attempt to prove by contradiction that there is no largest prime number is shown below. If there is a largest prime, list all the primes. Multiply a
  17. June 2019, AS Paper 2, Q3 (3 marks): Without using a calculator, prove that 3 2 [3] 2 2 3 .
  18. June 2018, Paper 3, Q11 (2 marks): Line 8 states that a b + 11 ab for a, b 0 . Explain why the result cannot be extended to apply in each of H H 2 the following cases. (i) One of the numbers a an
  19. June 2018, Paper 3, Q12 (3 marks): a b + 2 H 12 ab a b Lines 5 and 6 outline the stages in a proof that . Starting from ( ) 0 , give a detailed H - 2 [3] proof of the inequality of arithmetic and
  20. June 2018, Paper 3, Q13 (3 marks): Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three [3] consecutive terms of this sequence, the middle one
  21. June 2018, Paper 3, Q14 (4 marks): (i) In Fig. C1.3, angle CBD = i. Show that angle CDA is also i, as given in line 23. [2] (ii) Prove that h ab [2] , as given in line 24. =
  22. June 2018, Paper 3, Q15 (3 marks): It is given in lines 31–32 that the square has the smallest perimeter of all rectangles with the same area. Using this fact, prove by contradiction that among r
  23. June 2018, AS Paper 2, Q3 (3 marks): P and Q are consecutive odd positive integers such that P > Q. Prove that P 2 ‒ Q 2 is a multiple of 8. [3]

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 MEI A-Level Maths paper.

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