A Level Mathematics A OCR

Proof

Direct and Indirect Proof

• Understand the basics of Direct Proof. This involves proving the truth of a conjecture by logical deductions from accepted facts, using theorems and principles.
• Familiarize yourself with Indirect Proof or Proof by Contradiction. This occurs when you assume the opposite of what you want to prove and then show that this assumption leads to a contradiction.

Proof in Algebra

• Learn to prove Factor Theorem. A polynomial f(x) has a factor (x-a) if and only if f(a)=0.
• Master the proof for Remainder Theorem: If a polynomial f(x) is divided by (x-a), the remainder is f(a).
• Grasp the principle and proof for Difference of Two Squares. Where any expression of the form a² - b² can be factorised as (a + b) (a - b).

Proof in Geometry

• Understand the application of Congruence and Similarity proofs in geometry, frequently used in triangle problems.
• Know and master the application of Pythagorean Theorem and its proof. If a triangle is right-angled, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.
• Get comfortable with proofs involving Circle Theorems, such as the alternate segment theorem and tangent/radius theorem.

Proof by Exhaustion and Counter-Example

• Understand how Proof by Exhaustion, also known as Proof by Case, which involves dividing the argument into a finite number of cases and proving each one individually.
• Learn to refute a conjecture by using a Counter-Example. This is a single example showing that a universal claim does not always hold.

Disproof

• Grasp the idea of Disproof by Counterexample. A single instance where a statement is false can disprove a universal truth claim.

Miscellaneous Proofs

• Get acquainted with Proof by Mathematical Induction. This is a method of proving a proposition that holds true for all natural numbers.
• Familiarize yourself with proofs involving the Binomial Theorem. This theorem provides an expression for the expansion of a binomial power in terms of its coefficients.
• Understand the proof of The Fundamental Theorem of Calculus. This theorem shows the relationship between differentiation and integration.