Pearson Progression Step Q Scheme Marks AOs and Progress descriptor 4 Begins the proof by assuming the opposite is true. B1 3.1 7th there exists a number such that is odd and Complete proofs ‘Assumption: n n 1 is also odd.’ using proof by n3 + contradiction. Defines an odd number. B1 2.2a 2k 1 be an odd number.’ ‘Let + Successfully calculates M1 1.1b Factors the expression and concludes that this number must be M1 1.1b even. is even. Makes a valid conclusion. B1 2.4 This contradicts the assumption that there exists a number n such that is odd and 1 is also odd, so if is odd, then n n3 + n 1 is even. n3 + (5 marks) Notes Alternative method Assume the opposite is true: there exists a number such that is odd and 1 is also odd. (B1) n n n3 + If 1 is odd, then is even. (B1) n3 + n3 So 2 is a factor of (M1) n3. This implies 2 is a factor of (M1) n. This contradicts the statement is odd. (B1) n Pearson Progression Step Q Scheme Marks AOs and Progress descriptor 5 Begins the by assuming the opposite is true. B1 3.1 there do exist integers and such that a b ‘Assumption: ’ Understands that M1 2.2a
Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 4 Begins the proof by assuming the opposite is true. ‘Assumption: there exists a number n such that n is odd and n3 + 1 is also odd.’ B1 3.1 7th Complete proofs using proof by contradiction. Defines an odd number. ‘Let 2k + 1 be an odd number.’ B1 2.2a Successfully calculates M1 1.1b Factors the expression and concludes that this number must be even. is even. M1 1.1b Makes a valid conclusion. This contradicts the assumption that there exists a number n such that n is odd and n3 + 1 is also odd, so if n is odd, then n3 + 1 is even. B1 2.4 (5 marks) Notes Alternative method Assume the opposite is true: there exists a number n such that n is odd and n3 + 1 is also odd. (B1) If n3 + 1 is odd, then n3 is even. (B1) So 2 is a factor of n3. (M1) This implies 2 is a factor of n. (M1) This contradicts the statement n is odd. (B1) Q Scheme Marks AOs Pearson Progression Step and Progress descriptor 5 Begins the by assuming the opposite is true. ‘Assumption: there do exist integers a and b such that ’ B1 3.1 Understands that M1 2.2a
al_maths_pure_unit_1_ms.docx