29 a b c iv So 42 = 16 52 = 25 So is between 4 and 5. So is not less than but is equal to it. = Consider 0.6 = Hence > 0.6 False True False True M1 A1 M1 A1 M1 A1 B1 M1 A1 B1 2 M1 for finding a suitable comparison A1 for showing it is false and stating such M1 for showing the square of the limits A1 for showing it is true and stating such M1 for showing A1 for showing it is false and stating such B1 for explaining they are in fact equal M1 for showing both in similar comparable terms A1 for showing it is true and stating such B1 for clear concise communication of method H 10 30 Odd one out is. M3 A1 2 M1 for showing = M1 for showing = M1 for showing 3–1 = A1 cao H 4 31 x= y = Square both sides: = Hence √x = y Hence x2 = y Any example where x is the square of y, e.g. x = 1, y = 1 x = 4, y = 2 x = 9, y = 3 M2 A1 B1 2 M1 for squaring both sides to make information clearer M1 for considering only the denominator A1 for a correct example B1 for clear progression through the solution H 4
32 2.5 × 103 = 2500 Too small 2.5 × 104 = 25 000 2.5 × 105 = 250 000 Too big n = 4 M1 A1 2 M1 for using trial and improvement A1 for correct answer H 2 33 a b c d e f g h i j 0.6 0.9 1.3 0.3 0.1 1.2 1.5 1.4 2.1 3.5 B1 B1 B1 B1 B1 B1 B1 B1 B1 B1 2 cao cao cao cao cao cao cao cao cao cao H 10 34 Try a = 3, b = 4 = = = 5 a + b = 7 Statement is false. M1 A1 B1 2 M1 for finding an example that disproves the statement A1 for stating it is false B1 for clear communication of the solution H 3 35 For example: 2÷ = The cancels from numerator and denominator leaving 2, an integer. B1 M1 A1 2 B1 for a correct division of two surds M1 for showing how the surds will leave an integer answer A1 for clear communicating of method used H 3 36 a b For example: 3 + and 3 – For example: and B1 B1 2 B1 for a correct possible pair of surds B1 for a correct possible pair of surds H 2
39 a b c True. All terminating decimals can also be written as fractions, for example: 0.456 = True, for example: a = 0.4242……. 100a = 42.42……. 100a – a = 42.42…..– 0.42… Therefore: 99a = 42 a = For the recurring decimal with n repeating digits, replace the 100 above by 10n and this will make it possible to follow the same procedure and find the fraction. False. Irrational numbers cannot be expressed in the form a/b where a and b are integers; for example, π and are irrational numbers. B1 B1 B1 B1 B2 B2 2 2 True followed by a clear example B1 clear explanation B1 for work showing how you change a recurring decimal to fraction B1 for clear explanation B2 for clear explanation showing how all recurring decimals can be treated in this way to change to a fraction B2 Clear explanation showing definition of rational numbers and at least one example H 8 40 a b c If the prime factors of the denominator of a fraction in its simplest form are only 2 and/or 5 its decimal will terminate. So the following are terminating ,, is non-terminating because the prime factors of 6 are 2 and 3. Any multiple of where the numerator is not a factor or a multiple of 6 will also be recurring. == 2 × 0.166 666…. = = × =0.1666…. ÷ 10 Easy to convert are 0.027 272 7… and 0.272 727….. Since after multiplying by 100 in each case, you can eliminate the infinite recurrence to be left with a simple fraction. The hardest to convert are 2.727 272 and 27.272 727 only because the result will B2 B1 B2 B1 B1 2 B2 for clarity of explanation B1 for the 3 correct fractions B2 for clear explanation B1 for clear communication of method B1 for clear communication of method H