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Edexcel IGCSE Maths (4MA1) · Unit 7: Vectors & Proof

IGCSE Maths Proof Questions and Answers

Proof questions with full solutions for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Read the 3 worked examples first, then try the 3 exam-style questions yourself and check your working against the mark scheme.

Practise proof in Unit 7 →Proof revision notes

What this topic covers

Proof worked examples

Worked example 1: Algebraic proof

Prove that the sum of any three consecutive integers is a multiple of 3.

Solution

  1. Let the integers be \(n\), \(n+1\), \(n+2\).
  2. Sum \(=3n+3=3(n+1)\).
  3. \(3(n+1)\) is a multiple of 3 for every integer \(n\).

Answer: \(3(n+1)\)

Tip: Use algebra, not examples, in a proof.

Worked example 2: Geometric proof

ABC is a triangle with AB = AC. M is the midpoint of BC. Prove that angle ABC = angle ACB.

Solution

  1. AB = AC (given), BM = CM (M is the midpoint), AM is common.
  2. So triangles ABM and ACM are congruent (SSS).
  3. Hence angle ABM = angle ACM (corresponding angles), i.e. angle ABC = angle ACB.

Answer: Proof via SSS

Tip: Give a reason for every statement.

Worked example 3: Harder algebraic proof

Prove that \((n+3)^2-(n-3)^2\) is a multiple of 12 for all integers \(n\).

Solution

  1. Expand: \(n^2+6n+9-(n^2-6n+9)\).
  2. \(=12n\), which is a multiple of 12.

Answer: \(12n\)

Tip: Bracket the second expansion to keep signs correct.

Proof exam-style questions

Try each question before you open the answer. The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 2 marks

Prove that the product of an even number and an odd number is always even.

Show the answer and mark scheme
  • M1 uses 2k for the even number and 2n + 1 for the odd number and multiplies: 2k(2n + 1)
  • A1 2k(2n + 1) = 2(2kn + k), which is 2 × an integer, so even

Answer: \( 2k(2n + 1) = 2(2kn + k) \), 2 × an integer, so even (proved)

Question 2 · medium · 3 marks

Prove that the sum of any three consecutive odd numbers is always 3 more than a multiple of 6.

Show the answer and mark scheme
  • M1 (2n + 1) + (2n + 3) + (2n + 5)
  • M1 6n + 9
  • A1 6n + 9 = 6(n + 1) + 3, which is 3 more than a multiple of 6

Answer: \( 6n + 9 = 6(n + 1) + 3 \), 3 more than a multiple of 6 (proved)

Question 3 · hard · 3 marks

Prove that when any odd number is squared and 7 is added, the result is always a multiple of 8.

Show the answer and mark scheme
  • M1 (2n + 1)² + 7 = 4n² + 4n + 8
  • M1 4n(n + 1) + 8
  • A1 n(n + 1) is a product of consecutive integers, so even; so 4n(n + 1) is a multiple of 8, and adding 8 gives a multiple of 8

Answer: \( (2n + 1)^2 + 7 = 4n(n + 1) + 8 \), with \( n(n + 1) \) even, so a multiple of 8 (proved)

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and each answer was re-solved independently before publishing.

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Questions students ask

Which Edexcel IGCSE Maths papers test proof?

Either Higher tier paper, 1H or 2H, can test proof: both cover the whole Pearson Edexcel International GCSE Mathematics A (4MA1) specification and both allow a calculator. Show your method, because most marks are for working.

Where can I practise more proof questions?

In the IGCSE student hub: Unit 7 (Vectors & Proof) has more exam-style questions on this topic, each with a mark scheme. Practice starts with a 7-day free trial; the worked examples and questions on this page are free.

More Unit 7 topics: Vectors & Proof