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IGCSE Maths · Edexcel 4XMA1 · Unit 2 Higher · Algebra

Algebraic Proof — Practice Questions

12 exam-style questions (40 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Section A — Warm-up (grades 4–5)
  1. (2 marks)Give a counterexample to show that this statement is false: “The sum of two prime numbers is always even.”
  2. (3 marks)Prove that the sum of any three consecutive integers is a multiple of 3.
  3. (3 marks)Prove that the difference between the squares of two consecutive integers is always odd.
Section B — Core (grades 5–7)
  1. (3 marks)Prove that the product of any two odd numbers is odd.
  2. (3 marks)Prove that the sum of the squares of two consecutive even numbers is a multiple of 4.
  3. (3 marks)Prove that (x + 2)² − (x − 2)² ≡ 8x.
  4. (3 marks)Prove that x² − 8x + 19 > 0 for all values of x.
  5. (4 marks)n is a positive integer. Prove that (n + 1)² + (n − 1)² is always even, and is 2 more than twice a square number.
Section C — Stretch (grades 8–9)
  1. (4 marks)Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8.
  2. (4 marks)Prove that the square of any odd number is one more than a multiple of 8.
    HintWrite the odd number as 2n + 1 and factorise 4n² + 4n.
  3. (4 marks)Prove that n³ − n is a multiple of 6 for all integers n.
  4. (4 marks)(a) Write 2x² + 4x + 7 in the form a(x + b)² + c.
    (b) Hence prove that 2x² + 4x + 7 is always greater than 4.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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