54 p q M1 2 p q M The assumption is that and are M1 for giving the assumption about and integers. A1 q n q A1 for expressing 10p + as a multiple of 7 10p + = 7n where is also an M1 q q M1 for expressing 3p + in terms of 10p + integer. q 7p + 3p + = 7n A1 A1 for similar expression q 3p + = 7n 7p – p) = 7(n – n p n p As and are integers then is – A1 p) also an integer hence 7(n is a A1 for clear full explanation – 5 q multiple of 7 and so 3p + must be as 55 y) y) M1 2 M1 for expanding M 2(5(x 2) + = 2(5x 10 + well. – – A1 A1 cao = 10x 20 + 2y – M1 M1 for expanding …….(1) A1 A1 cao 10(x 1) + 2y 10 = 10x 10 + 2y – – – – 10 = 10x 20 + 2y – A1 A1 for explaining the two expressions are the same ….(2) 5 Equation (1) = equation (2) Hence the two expressions are equal.
54 The assumption is that p and q are integers. 10p + q = 7n where n is also an integer. 7p + 3p + q = 7n 3p + q = 7n – 7p = 7(n – p) As n and p are integers then n – p is also an integer hence 7(n – p) is a multiple of 7 and so 3p + q must be as well. M1 A1 M1 A1 A1 2 M1 for giving the assumption about p and q A1 for expressing 10p + q as a multiple of 7 M1 for expressing 3p + q in terms of 10p + q A1 for similar expression A1 for clear full explanation M 5 55 2(5(x – 2) + y) = 2(5x – 10 + y) = 10x – 20 + 2y …….(1) 10(x – 1) + 2y – 10 = 10x – 10 + 2y – 10 = 10x – 20 + 2y ….(2) Equation (1) = equation (2) Hence the two expressions are equal. M1 A1 M1 A1 A1 2 M1 for expanding A1 cao M1 for expanding A1 cao A1 for explaining the two expressions are the same M 5
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