Maths Pure Edexcel A-Level :T1 Proofs

• What is a proof in mathematics?

  Logical argument
• A mathematical proof must show that something is true in every case.

  True
• A statement that has been proven is called a theorem
• Steps to prove a mathematical statement:

  1️⃣ State assumptions

  2️⃣ Show every step clearly

  3️⃣ Follow steps logically

  4️⃣ Cover all possible cases

  5️⃣ Write a statement of proof
• What must you state at each stage of a proof?

  Assumptions
• 𝑛! − 𝑛 is always an even number for all values of 𝑛.

  True
• Proof by deduction starts from known facts or definitions
• What is the result of multiplying an odd number by an even number?

  Even
• If 𝑛 is even, then 𝑛 − 1 is odd
• When 𝑘 = 0, the equation 𝑘𝑥! + 3𝑘𝑥 + 2 = 0 has no real roots.

  True
• The symbol ≡ means ‘is always equal to’ and shows that two expressions are mathematically identical
• What condition on 𝑘 ensures that the equation 𝑘𝑥! + 3𝑘𝑥 + 2 = 0 has no real roots?

  0 ≤ 𝑘 < 8/9
• Steps to prove an identity:

  1️⃣ Start with one side of the identity

  2️⃣ Manipulate algebraically

  3️⃣ Match the other side

  4️⃣ Show every step of working
• What method can be used to prove that (3𝑥 + 2)(𝑥 − 5)(𝑥 + 7) ≡ 3𝑥³ + 8𝑥² − 101𝑥 − 70?

  Algebraic manipulation
• Proof by exhaustion is suitable for statements with a large number of cases.

  False
• Any odd number can be written in the form 2𝑛 + 1
• What is one example enough to prove a statement is true?

  No
• Match the type of number with its form:

  Odd number ↔️ 2𝑛 + 1

  Even number ↔️ 2𝑛
• The inequality 𝑝 + 𝑞 > R4𝑝𝑞 holds true when both 𝑝 and 𝑞 are negative.

  False
• What can be used to disprove a mathematical statement?

  Counter-example
• 2 and 3 are consecutive prime numbers.
• What is the sum of 2 and 3?

  5