M1: Algebraic Methods and Proof

Year 1 / ASYear 2 / A-Level All Boards (AQA, Edexcel, OCR, WJEC, CCEA) AQA

Proof by deduction, contradiction, exhaustion; algebraic fractions and partial fractions.

Exam Board:

📋 Key Definitions and Core Concepts

Proof by Contradiction: Establishing truth by demonstrating that assuming the statement to be false leads to a logical contradiction.

Partial Fractions: Decomposing rational fractions P(x)/Q(x) into sums of simpler terms with linear or quadratic denominators.

To prove √2 is irrational, assume √2 = a/b in lowest terms, square to get 2b² = a², deduce a and b are both even, contradicting lowest terms.
Partial fraction forms: A/(ax+b) + B/(cx+d) for distinct linear factors; A/(ax+b) + B/(ax+b)² for repeated factors.
Improper fractions (deg numerator ≥ deg denominator) must first be divided by polynomial division.

Exam-Style Question

Question:

Express (5x - 2) / ((x - 2)(x + 1)) in partial fractions.

Model Solution & Mark Scheme:

Set 5x - 2 = A(x + 1) + B(x - 2).
x = 2: 8 = 3A => A = 8/3.
x = -1: -7 = -3B => B = 7/3.
Result: 8/(3(x - 2)) + 7/(3(x + 1)).

Q1: Prove by contradiction that there are infinitely many primes.

Show Model Answer

Answer: Assume finitely many primes p₁..pₙ. Let N = p₁p₂..pₙ + 1. N is prime or has a prime factor not in the list, contradicting finiteness.

Q2: Express (3x² + 7x - 2) / ((x - 1)(x + 2)²) in partial fractions.

Show Model Answer

Answer: A/(x-1) + B/(x+2) + C/(x+2)². A = 8/9, B = 19/9, C = -4/3.

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