Question Working Answer Mark AO Notes Grade 1 a b c d 1 : 6 ≠ 6 : 1 because 1 : 6 = 6 : 36 (×6) Or 6:1 = 1: (÷6) 19 : 95 = 1 × 19 : 5 × 19 = 1 : 5 B: G 2 : 5 4 : 10 6 : 15 (21 students) 7 : 17.5 (not possible) 8 : 20 (28 students) No 19 : 95 (÷19) 1 : 5 No, because the units must be the same in order to compare No, to retain this ratio requires 2 boys and 5 girls each time, so 7 students. This means that there can only be multiples of 7 students in the club. 24 is not a multiple of 7. M1 B1 B1 B1 2 M1 for demonstrating an understanding of each ratio in its unitary form as a method of comparison, oe B1 for calculation showing cancelling B1 for an understanding of scale and equivalence of units B1 for reference to multiples of 7 B 4 2 a b Packs of 3: 90 ÷ 3 = 30 30 packs cost 30 × £1.50 = £45 Packs of 15: 90 ÷ 15 = 6 6 packs cost 6 × £5= £30 Packs of 25: Not possible, because 90 is not divisible by 25. Buy 2 get one free on packs of 15. Buy two packs of 15 for £10 Get a pack of 15 free. 45 will cost £20. So new cost = £20 Or (3 × 15) + (3 × 15) = 90 £10 + £10 = £20 6 packs of 15. 6 packs of 15 for £30. No, still buy 6 packs of 15 but now for the lower price of £20. B1 B1 B1 B1 2 B1 for correct combination to 90 B1 for correct cost B1 for a method for calculating of the cost B1 for correct justification of choice B 4
b 400 ÷ 4.00 = 100 minutes per £1 There are more CDs than are needed. A recording time of 80 minutes is not long enough. £6.50 is too expensive at time of purchase (prefer just to spend £4). A1 A1 for explanation of possible reasons not to choose the best buy 4 10 £800 × 1.19 gives €952 £800 × 1.22 gives €976 €976 – €952 = €24 They will get €24 more. M1 M1 A1 2 M1 for multiplications M1 for subtraction ft A1 cao M 3 11 a i ii iii b By expressing this as: ‘How many….. in ….’ Use chosen method from part a to explain correctly how to divide, using fractions. B2 B1 B1 B1 M1 2 B1 for correct justification B1 for showing diagram oe B1 for correct justification showing diagram oe 1 for correct justification showing diagram oe B1 for correct explanation M1 for process showing that dividing by doubles the number of pieces, so is the same as multiplying by 2 M 6
12 a b 8 kg = 8000 g 8000 ÷ 250 = 32 3 kg = 3000 g 3000 ÷ 85 = 35 (to nearest whole number) 2 kg = 2000 g 2000 ÷ 20 = 100 7 kg = 7000 g 7000 ÷ 250 = 28 So the limiting value is the amount of icing sugar. Therefore she can make 24 × 28 = 672 biscuits. 672 ÷ 15 = 44.8 44 × = 33 33 × £2.99 = £98.67 44 – 33 = 11 discounted £2.99 × 0.85 = £2.54 to 2 dp 11 × 2.54 = £27.94 Total sales = £98.67 + £27.94 = £126.61 Total costs = £59 + £26 = £85 To calculate percentage profit: profit = = 0.489 529 412 and percentage profit = 0.489 529 412 × 100% = 48.95% She can make 44 complete packs of 15 biscuits. 49% profit to the nearest integer. M1 B1 M1 A1 M1 A1 M1 A1 2 3 M1 for process of division to see how many batches of 15 biscuits can be made with each ingredient B1 for 32, 35, 100 and 28 M1 for correct identification of limiting value A1 for correct cost of of biscuits M1 for use of 0.85 multiplier A1 for cao M1 for division of total sales by total cost (ft) A1 for correct percentage with rounding M 8 13 Price including VAT = £595 × 1.20 = £714 With a 20% discount: £714 × 0.8 = £571.20 £571.20 – £595 = £23.80 OR £595 × 0.8 = £476 £476 × 1.2 = £571.20 He is overpaying by £23.80 Disagree. He would pay the shop more than he needs to. M1 M1 M1 A1 2 M1 for process of multiplying by 1.2 to find cost with VAT M1 for multiplying by 0.8 to find 20% reduced price (ft) M1 for subtracting to find overpayment A1 for demonstrating overpayment with explanation M 4
29 B × 0.8n < Divide both sides by B. 0.8n < 0.83 = 0.512 0.84 = 0.4096 OR £100 × 0.8 =£80 £80 × 0.8 = £64 £64 × 0.8 = £51.20 £51.20 × 0.8 = £40.96 4 weeks M1 M1 M1 A1 2 M1 for choosing a starting a position, either a variable such as B or a specific amount such as £100 M1 for working through the weeks in some way M1 for the process of finding amounts for weeks 3 and 4 to show the point at which the bank account first dips below 50% of the original balance cao H 4 30 graph d graph e graph b graph c graph f graph a B6 2 B1 for each correctly identified graph with reference to why, for example: f(x) x2 is graph d as points are (–2, 4), (–1, 1), (0, 0), (1, 1), (2, 4) and it is a parabola f(x) = 2x , x >0 f(x) = –2x , x < 0 is graph e as it is linear and has no negative f(x) values; the gradient is 2 and –2 H 6 31 a b c y = xy = k where k is the constant of proportionality Inverse proportion describes the relationship between two variables such that as one increases the other decreases. xy = k or y = Own problem, for example: It takes 5 men 10 days to dig a hole. The number of men, y, is inversely proportional to the number of days, x. How long would it take for ten men to dig the same hole? (5 days) B1 M1 A1 2 B1 for clear explanation of inverse proportion M1 for correct equation A1 for clear question H 3
A1 A1 for gradient from points on the straight line 5 37 a b £28 000 × 1.053 = £32 413.50 £14 500 × 1.05n > £ 20 000 = 1.4 Try n = 7 years. £14 500 × 1.057 = £20 402.96 £32 413.50 7 years B1 M1 A1 2 B1 for identification and use of multiplier M1 for trial and improvement or reasoning to try 7 years A1 cao H 3 38 Sycamore: 4 × 1.0811 = 9.327 4 × 1.0812 = 10.073 Conifer: 2 × 1.1511 = 9.305 2 × 1.1512 = 10.7 12 years After 11 years, the sycamore is 9.326 m tall and the conifer is 9.305 m tall. After 12 years, the sycamore is 10.073 m tall and the conifer is 10.7 m tall. M1 M1 A1 2 M1 for correct calculation method to find heights of trees M1 for clarity of final reasoning A1 for finding all four heights after 11 and 12 years H 3
41 I ii iii F(x) = a(b)x b < 1 the population decreases. b = 1 the population stays the same. b > 1 the population increases. B3 2 B1 for each correct explanation of the impact on the population as b varies H 3 42 a b Epidemic started by a single carrier so x0 = 1. Considering infection after 10 days so t = 10. xn + 1 = Rtx0 x10 = R10 Newspaper headline to engage readers with the story of this epidemic e.g. how long before x people are infected. M1 A1 B1 2 M1 for correct interpretation of , and use of, the iterative formula A1 for cao B1 for relevant , informative headline H 3 43 0.43 bar M1 A1 2 M1 for correct use of formula A1 cao H 2 44 a b x = 1 + x(x – 3) = x – 3 + 11 x2 – 3x = x + 8 x2 – 4x – 8 = 0 if x1 = 5 x2 = 1 + = 1 + = 6.5 x3 = 1 + = 4.14286 … Show that… as workings. x = –1.46 to 2 dp M1 A1 M1 M1 A1 3 M1 for algebraic manipulation B1 for clarity of justification M1 for correct use of iteration M1 for substitutions A1 for one root found (no credit if both roots are found) Using x = 5 as the first iteration, after 19 iterations you arrive at x = –1.46 to 2 dp Likewise, if the first trial is –1, 11 iterations lead to the solution x = –1.46 to 2 dp. H 5