From part a: is even implies that is even. a2 a Further states that if is even, then 2c. Choice of variable is M1 1.1b a a = not important. a substitution and works through to find , concluding that M1 1.1b is also even. b From part a: is even implies that is even. b2 b Makes a valid conclusion. B1 2.4 If and are even, then they have a common factor of 2, which a b contradicts the statement that and have no common factors. a b Thereforeis an irrational number. (6) (10 marks) Notes Pearson Progression Step Q Scheme Marks AOs and Progress descriptor 10 Begins the proof by assuming the opposite is true. B1 3.1 7th there is a finite amount of prime numbers.’ Complete proofs ‘Assumption: using proof by Considers what having a finite amount of prime numbers means M1 2.2a contradiction. by making an attempt to list them: Let all the prime numbers exist be Consider a new number that is one greater than the product of M1 1.1b all the existing prime numbers: Let Understands the implication of this new number is that division M1 1.1b by any of the existing prime numbers will leave a remainder of 1. So none of the existing prime numbers is a factor of N. Concludes that either is prime or has a prime factor that is B1 2.4 N N not currently listed. Recognises that either way this leads to a contradiction, and B1 2.4 therefore there is an infinite number of prime numbers.
From part a: a2 is even implies that a is even. Further states that if a is even, then a = 2c. Choice of variable is not important. M1 1.1b a substitution and works through to find , concluding that b is also even. From part a: b2 is even implies that b is even. M1 1.1b Makes a valid conclusion. If a and b are even, then they have a common factor of 2, which contradicts the statement that a and b have no common factors. Thereforeis an irrational number. B1 2.4 (6) (10 marks) Notes Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 10 Begins the proof by assuming the opposite is true. ‘Assumption: there is a finite amount of prime numbers.’ B1 3.1 7th Complete proofs using proof by contradiction. Considers what having a finite amount of prime numbers means by making an attempt to list them: Let all the prime numbers exist be M1 2.2a Consider a new number that is one greater than the product of all the existing prime numbers: Let M1 1.1b Understands the implication of this new number is that division by any of the existing prime numbers will leave a remainder of 1. So none of the existing prime numbers is a factor of N. M1 1.1b Concludes that either N is prime or N has a prime factor that is not currently listed. B1 2.4 Recognises that either way this leads to a contradiction, and therefore there is an infinite number of prime numbers. B1 2.4