Proof — A-Level Mathematics Revision
Revise Proof for A-Level Mathematics. Step-by-step explanation, worked examples, common mistakes and exam-style practice aligned to AQA, Edexcel, OCR, WJEC, Eduqas, CCEA, Cambridge International (CIE), SQA, IB, AP.
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Go to Algebra & FunctionsWhat is Proof?
Proof in A-Level Mathematics involves using logical deduction to demonstrate the truth of a mathematical statement. This can include proof by deduction, proof by exhaustion, and disproof by counter-example, which are fundamental methods for establishing mathematical certainty.
Board notes: Proof by induction is a key component of the A-Level Further Maths specification for all major exam boards (AQA, Edexcel, OCR), but the fundamental methods of proof are covered in the standard A-Level Maths course.
Step-by-step explanationWorked example
Prove that the sum of two consecutive odd numbers is always a multiple of 4. Let the two consecutive odd numbers be 2n+1 and 2n+3. Their sum is (2n+1) + (2n+3) = 4n+4. This can be factorised as 4(n+1). Since n is an integer, n+1 is also an integer, and therefore 4(n+1) is a multiple of 4.
Mini lesson for Proof
1. Understand the core idea
Proof in A-Level Mathematics involves using logical deduction to demonstrate the truth of a mathematical statement. This can include proof by deduction, proof by exhaustion, and disproof by counter-example, which are fundamental methods for establishing mathematical certainty.
Can you explain Proof without copying the notes?
2. Turn it into marks
Prove that the sum of two consecutive odd numbers is always a multiple of 4. Let the two consecutive odd numbers be 2n+1 and 2n+3.
Underline the method, evidence, or command-word move that would earn credit in A-Level Pure Mathematics.
3. Fix the likely mark leak
Watch for this mistake: Assuming what you are trying to prove. For example, when proving an identity, starting with the assumption that the two sides are already equal.
Write one correction rule before doing another practice question.
Practise this topic
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Proof practice questions
These are original StudyVector questions for revision practice. They are not official exam-board questions.
Question 1
In one A-Level sentence, explain what Proof is testing.
Answer: Proof in A-Level Mathematics involves using logical deduction to demonstrate the truth of a mathematical statement. This can include proof by deduction, proof by exhaustion, and disproof by counter-example, which are fundamental methods for establishing mathematical certainty.
Mark focus: Precise definition and topic focus.
Question 2
A student sees a Proof question but is not sure how to start. What should the first method line establish?
Answer: It should identify the rule, equation, diagram feature, or transformation before any calculation. That protects method marks and makes later checking easier.
Mark focus: Method selection and command-word control.
Question 3
A student makes this mistake: "Assuming what you are trying to prove. For example, when proving an identity, starting with the assumption that the two sides are already equal." What should their next repair task be?
Answer: Do one Proof question and review the mistake type.
Mark focus: Error correction and next-step practice.
Proof flashcards
Core idea
What is the main idea in Proof?
Proof in A-Level Mathematics involves using logical deduction to demonstrate the truth of a mathematical statement. This can include proof by deduction, proof by exhaustion, and disproof by counter-example, which are...
Common mistake
What mistake should you avoid in Proof?
Assuming what you are trying to prove. For example, when proving an identity, starting with the assumption that the two sides are already equal.
Practice
What is one useful practice task for Proof?
Answer one Proof question and review the mistake type.
Exam board
How should you use board notes for Proof?
Proof by induction is a key component of the A-Level Further Maths specification for all major exam boards (AQA, Edexcel, OCR), but the fundamental methods of proof are covered in the standard A-Level Maths course.
Common mistakes
- 1Assuming what you are trying to prove. For example, when proving an identity, starting with the assumption that the two sides are already equal.
- 2Making a leap in logic without justification. Every step in a proof must be a clear consequence of the previous steps or a known mathematical fact.
- 3Using a single example to prove a general statement. A proof must hold for all possible cases, not just a specific one.
Proof exam questions
Exam-style questions for Proof with mark-scheme style solutions and timing practice. Aligned to AQA, Edexcel, OCR, WJEC, Eduqas, CCEA, Cambridge International (CIE), SQA, IB, AP specifications.
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Step-by-step method
Step-by-step explanation
4 steps · Worked method for Proof
Core concept
Proof in A-Level Mathematics involves using logical deduction to demonstrate the truth of a mathematical statement. This can include proof by deduction, proof by exhaustion, and disproof by counter-ex…
Frequently asked questions
What is the difference between proof by deduction and proof by exhaustion?
Proof by deduction uses a series of logical steps to arrive at a conclusion from a set of premises. Proof by exhaustion involves checking every possible case to show that a statement is true.
How do I disprove a statement?
To disprove a mathematical statement, you only need to find one single case where the statement is false. This is called a counter-example.