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Proof, including contradiction

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 124

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 41 parts, 124 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 1 · Q4(b) · 2 marks
(b) Hence prove that –4 is the only real root of the equation f(x) = 0 (Solutions relying on calculator technology are not acceptable.) (2)
2. June 2025 · Paper 1 · Q8(a) · 1 marks
8. A student was asked to prove that the square of any number can be expressed in the form 5n or 5n ± 1 where n ∈ The start of the student’s proof is shown in the box below. Let m = 5k Consider m 2 = (5k) 2 = 25k 2 = 5 x 5k 2 = 5n where n = 5k 2 Let m = 5k + 1 Consider m 2 = (5k + 1) = 25k 2 + 10k + 1 = 5(5k 2 + 2k) + 1 = 5n + 1 where n = 5k 2 + 2k Let m = 5k + 2 Consider m 2 = (5k + 2) = 25k 2 + 10k + 4 = 5(5k 2 + 2k + 1) – 1 = 5n – 1 where n = 5k 2 + 2k + 1 (a) Identify and correct an algebraic error in the box above. (1)
3. June 2025 · Paper 1 · Q8(b) · 4 marks
(b) Show the calculations and statements that are required to complete the proof. (4)
4. June 2025 · AS Paper 1 · Q15(a) · 2 marks
15. 3 + 4n is prime for n ∈ ” “n (I) (a) Determine whether statement (I) is always true, sometimes true or never true. You must fully justify your answer. (2)
5. June 2025 · AS Paper 1 · Q15(b) · 2 marks
3 + 5n is prime for n ∈ ” “n (II) (b) Determine whether statement (II) is always true, sometimes true or never true. You must fully justify your answer. (2)
6. June 2024 · Paper 1 · Q4 · 3 marks
2, use differentiation from first principles to show that d y 4. Given that y = x x = 2x d (3)
7. June 2024 · AS Paper 1 · Q14 · 4 marks
14. Prove, using algebra, that 2 + 5n n is even for all n ∈ (4)
8. June 2024 · Paper 1 · Q15(i) · 2 marks
15. (i) Show that k 2 – 4k + 5 is positive for all real values of k. (2)
9. June 2024 · Paper 1 · Q15(ii) · 4 marks
(ii) A student was asked to prove by contradiction that / / O “There are no positive integers x and y such that 3 x y x y ” The start of the student’s proof is shown below. Assume that positive integers x and y exist such that / / O x y x y If 3x + 2y = 14 and 2x – 5y = 2 x y x y , Notintegers x y Show the calculations and statements needed to complete the proof. (4)
10. June 2024 · Paper 2 · Q15(c) · 5 marks
(c) Prove that the normal to C at P does not meet C again. You should use algebra for your proof and make your reasoning clear. (5)
11. June 2023 · Paper 1 · Q12 · 5 marks
12. y = sin x where x is measured in radians. Use differentiation from first principles to show that d y x = cos x d You may • use without proof the formula for sin(A ± B) assume that as h → 0, sin h → 1 and cosh -1 → 0 • h h (5)
12. June 2023 · Paper 1 · Q14 · 4 marks
14. Prove, using algebra, that 3 – n (n + 1) is odd for all n ∈ (4)
13. June 2023 · AS Paper 1 · Q17(a) · 1 marks
17. In this question p and q are positive integers with q > p Statement 1: is never a multiple of 5 (a) Show, by means of a counter example, that Statement 1 is not true. (1)
14. June 2023 · AS Paper 1 · Q17(b) · 4 marks
Statement 2: When p and q are consecutive even integers is a multiple of 8 (b) Prove, using algebra, that Statement 2 is true. (4)
15. June 2023 · Paper 2 · Q15 · 3 marks
15. A student attempts to answer the following question: Given that x is an obtuse angle, use algebra to prove by contradiction that sin x – cos x 1 The student starts the proof with: Assume that sin x – cos x < 1 when x is an obtuse angle Þ(sin x – cos x) 2 < 1 Þ... The start of the student’s proof is reprinted below. Complete the proof. (3)
16. June 2022 · Paper 1 · Q7(i) · 3 marks
7. (i) Given that p and q are integers such that pq is even use algebra to prove by contradiction that at least one of p or q is even. (3)
17. June 2022 · Paper 1 · Q7(ii) · 2 marks
(ii) Given that x and y are integers such that • x < 0 • (x + y)2 < 9x2 + y2 show that y > 4x (2)
18. June 2022 · Paper 1 · Q13(i) · 3 marks
13. (i) In an arithmetic series, the first term is a and the common difference is d. Show that Sn = n 2 [2a + (n - 1)d] (3)
19. June 2022 · AS Paper 1 · Q14(i) · 1 marks
14. (i) A student states “if x2 is greater than 9 then x must be greater than 3” Determine whether or not this statement is true, giving a reason for your answer. (1)
20. June 2022 · AS Paper 1 · Q14(ii) · 3 marks
(ii) Prove that for all positive integers n, n3 + 3n2 + 2n is divisible by 6 (3)
21. June 2022 · Paper 2 · Q4 · 3 marks
4. Given that y = 2x use differentiation from first principles to show that d y x = 4x d (3)
22. June 2022 · Paper 2 · Q11 · 4 marks
11. Prove, using algebra, that 2 + 5) n(n is even for all n . (4)
23. Oct 2021 · AS Paper 1 · Q10(a) · 4 marks
10. A student is investigating the following statement about natural numbers. 3 – n is a multiple of 4 ” “ n (a) Prove, using algebra, that the statement is true for all odd numbers. (4)
24. Oct 2021 · AS Paper 1 · Q10(b) · 1 marks
(b) Use a counterexample to show that the statement is not always true. (1)
25. Oct 2021 · Paper 1 · Q15(i) · 2 marks
15. (i) Use proof by exhaustion to show that for n ∈ , n 4 n 3 > 3 (n + 1) (2)
26. Oct 2021 · Paper 1 · Q15(ii) · 4 marks
(ii) Given that m3 + 5 is odd, use proof by contradiction to show, using algebra, that m is even. (4)
27. Oct 2020 · AS Paper 1 · Q13(a) · 4 marks
13. (a) Prove that for all positive values of a and b 4a b 4 + b a (4)
28. Oct 2020 · AS Paper 1 · Q13(b) · 1 marks
(b) Prove, by counter example, that this is not true for all values of a and b. (1)
29. Oct 2020 · Paper 1 · Q16 · 4 marks
16. Prove by contradiction that there are no positive integers p and q such that 4 p2 - q2 = 25 (4)
30. Oct 2020 · Paper 2 · Q15(a) · 4 marks
15. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. A geometric series has common ratio r and first term a. Given r ≠ 1 and a ≠ 0 (a) prove that ( ) Sn = a - r n - r (4)
31. Oct 2020 · Paper 2 · Q16 · 4 marks
16. Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3 (4)
32. June 2019 · Paper 1 · Q10(i) · 4 marks
10. (i) Prove that for all n  8, n2 + 2 is not divisible by 4 (4)
33. June 2019 · Paper 1 · Q10(ii) · 2 marks
(ii) “Given x  <, the value of mu 3x – 28 mu is greater than or equal to the value of (x – 9).” State, giving a reason, if the above statement is always true, sometimes true or never true. (2)
34. June 2019 · AS Paper 1 · Q15 · 4 marks
15. Given n  `, prove that n3 + 2 is not divisible by 8 (4)
35. June 2018 · AS Paper 1 · Q2(i) · 3 marks
2. (i) Show that [࣠2 í 8x + 17 < 0 for all real values of x (3)
36. June 2018 · AS Paper 1 · Q2(ii) · 2 marks
(ii) “If I add 3 to a number and square the sum, the result is greater than the square of the original number.” State, giving a reason, if the above statement is always true, sometimes true or never true. (2)
37. June 2018 · AS Paper 1 · Q10 · 4 marks
10. Prove, from first principles, that the derivative of x3 is 3x2 (4)
38. June 2018 · Paper 2 · Q3(a) · 2 marks
3. (a) “If m and n are irrational numbers, where m ≠ n, then mn is also irrational.” Disprove this statement by means of a counter example. (2)
39. June 2018 · Paper 2 · Q3(b) · 3 marks
(b) (i) Sketch the graph of y = |x| + 3 (ii) Explain why |x| + 3 |x + 3| for all real values of x. (3)
40. June 2018 · Paper 2 · Q6(b) · 2 marks
Using the answer to (a)(ii), (b) prove that there are exactly two real solutions to the equation -3y 6 + 8y 4 - 9y 2 + 10 = 0 (2)
41. June 2018 · Paper 2 · Q9 · 5 marks
9. Given that theta is measured in radians, prove, from first principles, that d ( ) cos theta = - sin theta d theta You may assume the formula for cos (A ± B) and that as h → 0, sin h → 1 and cosh -1 → 0 h h (5)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · Paper 1 · Q4
Mark scheme June 2025 Paper 1 Q4(b)
June 2025 · Paper 1 · Q8
Mark scheme June 2025 Paper 1 Q8(b)
June 2025 · AS Paper 1 · Q15
Mark scheme June 2025 Paper 1 Q15(b)
June 2024 · Paper 1 · Q4
Mark scheme June 2024 Paper 1 Q4
June 2024 · AS Paper 1 · Q14
Mark scheme June 2024 Paper 1 Q14
June 2024 · Paper 1 · Q15
Mark scheme June 2024 Paper 1 Q15(ii)
June 2024 · Paper 2 · Q15
Mark scheme June 2024 Paper 2 Q15(c)
June 2023 · Paper 1 · Q12
Mark scheme June 2023 Paper 1 Q12
June 2023 · Paper 1 · Q14
Mark scheme June 2023 Paper 1 Q14
June 2023 · AS Paper 1 · Q17
Mark scheme June 2023 Paper 1 Q17(b)
June 2023 · Paper 2 · Q15
Mark scheme June 2023 Paper 2 Q15
June 2022 · Paper 1 · Q7
Mark scheme June 2022 Paper 1 Q7(ii)
June 2022 · Paper 1 · Q13
Mark scheme June 2022 Paper 1 Q13(i)
June 2022 · AS Paper 1 · Q14
Mark scheme June 2022 Paper 1 Q14(ii)
June 2022 · Paper 2 · Q4
Mark scheme June 2022 Paper 2 Q4
June 2022 · Paper 2 · Q11
Mark scheme June 2022 Paper 2 Q11
Oct 2021 · AS Paper 1 · Q10
Mark scheme Oct 2021 Paper 1 Q10(b)
Oct 2021 · Paper 1 · Q15
Mark scheme Oct 2021 Paper 1 Q15(ii)
Oct 2020 · AS Paper 1 · Q13
Mark scheme Oct 2020 Paper 1 Q13(b)
Oct 2020 · Paper 1 · Q16
Mark scheme Oct 2020 Paper 1 Q16
Oct 2020 · Paper 2 · Q15
Mark scheme Oct 2020 Paper 2 Q15(a)
Oct 2020 · Paper 2 · Q16
Mark scheme Oct 2020 Paper 2 Q16
June 2019 · Paper 1 · Q10
Mark scheme June 2019 Paper 1 Q10(ii)
June 2019 · AS Paper 1 · Q15
Mark scheme June 2019 Paper 1 Q15
June 2018 · AS Paper 1 · Q2
Mark scheme June 2018 Paper 1 Q2(ii)
June 2018 · AS Paper 1 · Q10
Mark scheme June 2018 Paper 1 Q10
June 2018 · Paper 2 · Q3
Mark scheme June 2018 Paper 2 Q3(b)
June 2018 · Paper 2 · Q6
Mark scheme June 2018 Paper 2 Q6(b)
June 2018 · Paper 2 · Q9
Mark scheme June 2018 Paper 2 Q9