1. Mathematics Intermediate Tier Paper 2 Summer 2002 (2 hours)

2. 1.(a) Calculate 26% of 34. = 26 x 34 100 =£8.84

3. (b) The total cost of 8 loaves and 12 baguettes is £15.48. One loaves cost 93p. Find the cost of one baguette. Cost of baguettes = 15.48 – 7.44 = £8.04 Cost of 8 loaves = 8 x 93 = £7.44 Cost of 1 baguette = 8.04 ÷ 12 = £0.67 = 67c

4. 2. The diagram below represents a number machine. InputAdd 7 Divide 5 Output If the input is x, write down the output in terms of x. = x + 7 5

5. 3. (a) Simplify 7a – 5 – 3a – 4. = 4a - 9 = 7a – 3a – 5 – 4

6. ( b) Find the value of 4x + 3y when x = - 6 and y = 5. = 4x - 6 + 3 x 5 = -24 + 15 = -9

7. 4. Draw, on the grid, an enlargement of the given shape, using a scale factor of 4.

8. 5. (a) A solid cuboid measures 6.4 cm by 4.8,cm by 3.5cm, as shown in the diagram. Calculate its volume, clearly stating the units of your answer. 3.5cm 4.8cm 6.4cm Volume = length x width x height = 6.4 x 4.8 x 3.5 = 107.52cm³

9. (b)Calculate the total surface area of the cuboid. 3.5cm 4.8cm 6.4cm Front and back= 2 x 6.4 x 3.5 Top and bottom = 2 x 6.4 x 4.8 Sides= 2 x 4.8 x 3.5 = 44.8cm² = 61.44cm² = 33.6cm² Total = 44.8 + 61.44 + 33.6 = 139.84cm²

10. 6. Eighty pupils were asked what they drank with their breakfast. Of these pupils, 36 drank tea, 18 drank coffee, 16 drank milk and 10 drank other drinks. (a)What is the probability that a randomly chosen pupil (i) drank coffee at breakfast? (ii) did not drink tea at breakfast? = 18 80 = 1 - 36 80 = 44 80

11. (b) Draw a pie chart to illustrate the different drink that the pupils had with their breakfast.You should show how you calculate the angles of your pie chart. Milk = 16 x 4.5 = 72° Others = 10 x 4.5 = 45° Tea = 36 x 4.5 = 162° Coffee = 18 x 4.5 = 81° 1 drink = 360 ÷ 80 = 4.5°

12. 7. Mr and Mrs Gann received their electricity bill.The details were as follows. Present meter reading 54261 Previous meter reading 52815 Charge per unit 6.52 pence per unit VAT 5% Service charge £10.56 Showing all your working, find the total cost of the electricity including VAT. Units used = 54261 - 52815 = 1446 Cost of units = 1446 x 6.52 = 9427.92c = £94.28 VAT = 5 x 104.84 100 = £5.24 Total = £104.84 + 5.24= £110.08 Cost including service charge = £94.28 + 10.56 = £104.84

13. 8. ABCD and ADEF are two parallelograms in which ADC=130 ° and DEF=60 °. Find BAF 130° A B C D E F 60° Diagram not drawn to scale. BAD = 50° DAF = 60° BAF = 50 + 60 = 110° 50° 60°

14. 9. The weight of eighty eggs were measured and the results are sumarised in the following table. Pwysau (grammau)Nifer yr wyau 50≤weight<607 60≤weight<7013 70≤weight<8029 80≤weight<9020 90≤weight<10011 (a) On the graph paper, draw a grouped frequency diagram for the data.

15. (b) Write down the modal class. 50 60 70 80 90 100 5 10 15 20 25 30 Weight (grams) Frequency 70 - 80

16. 10. Write down, in terms of n, the nth term of each of the following sequences. (a)1 x 3 2 x 4 3 x 5 4 x 6 ….. x ….. (b) 3 10 17 24 ……… n (n + 2) +7 +7 +7 7n - 4

17. 11. A gardner is making a circular lawn of radius 6m (a)Calculate the area of the lawn. (b) The gardener wishes to put an edging around the circumference of the lawn. Calculate the length of edging needed. A = πr² = 3.142 x 6 x 6 = 113.097 = 113.1 m² C = π d or C = 2 π r C = 2 x 3.142 x 6 C = 37.699 C= 37.7 m

18. 12. Beverley leaves home at 11.00 a.m. to go for a drive in her car. She travels a certain distance then stops for three quarters of an hour before starting back for home at a speed of 40 m.p.h. (a) Calculate the speed for the first part of her journey. = 68 2 = 34 m.p.h.

19. (b) On the graph paper, draw lines to represent (ii) Her ¾ hour stop and (ii) her return journey home. Distance (miles) from home 70 20 30 40 50 10 60 11:00 12:00 13:00 14:00 15:00 Time

20. 13. (a) The population of a country increased from 56 000 000 to 59 500 000. What percentage increase is this? = 59 500 000 – 56 000 000 = 3 500 000 Increase Percentage increase = 3 500 000 x 100 56 000 000 = 6.25%

21. (b) What will be the amount if £5000 is invested for 3 years at the rate of 4% compound interest per annum? Interest 1st year = 4 x 5000 100 = £200 Amount in the bank = 5000 + 200 = £5200 Interest 2nd year = 4 x 5200 100 = £208 Amount in the bank = 5000 + 208 = £5408 Interest 3rd year = 4 x 5408 100 = £216.32 Amount in the bank = 5408 + 216.32 = £5624.32

22. 14. Solve the following equation. 15-4x = 3 7 15 – 4x = 3 x 7 15 – 4x = 21 15 – 21 = 4x -6 = 4x - 6 = x 4 - 3 = x 2 -1 ½ = x x = -1 ½

23. Use the method of trial and improvement to find this solution correct to one decimal place. 15. A solution to the equation x³ + 5x – 30 = 0 lies between 2 and 3. Try x = 2.5 2.5³ + 5 x 2.5 – 30 = -1.875 Too small Try x = 2.6 2.6³ + 5 x 2.6 – 30= 0.576 Too big Try x = 2.55 2.55³ + 5 x 2.55 – 30 = -0.6686 Too small Felly x = 2.6 (1 ll.d.)

24. 16. (a) The batting scores of 100 cricketers were recorded and the results are summarised in the following table. Batting scoreFrequency 0 – 1920 20 – 3945 40 – 5924 60 – 799 80 - 992 On the graph paper, draw a frequency polygon for the data. Midpoint x 9.5 29.5 49.5 69.5 89.5

25. Frequency Batting Score 0 20 40 60 80100 10 20 30 40 50 (b) Find an estimate for the mean of the batting scores. = Σfx Σf = 20x9.5+45x29.5+24x49.5+9x69.5+2x89.5 20+45+24+9+2 = 3510 100 =35

26. 17. The diameter of a circle, AB is of length 8.7cm, BC has length 5.4cm and ACB = 90 °. Calculate the length of AC. A B C 5.4cm 8.7cm Diagram not drawn to scale. AB² = AC² + CB² 8.7² = AC² + 5.4² 8.7² - 5.4² = AC² 46.53 = AC² AC = √ 46.53 AC = 6.82cm Neu / or AC² = AB² - BC² AC² = 8.7² - 5.4²

27. 18. ABCD is a rectangle. (c) Draw the locus of all the points inside the rectangle whose distance from A is the same as the length of AB. A B C D (a)Draw the locus of all the points inside the rectangle whose distance from AB is the same as their distance from AD. (b) Draw the locus of all the points inside the rectangle which are 6cm from DC.

28. 19. Find, in standard form, the value of (a) (7.4 x 10 -5 ) x (3.9 x 10 -4 ) (b) 59639 0.087 = 2.889 x 10 -8 = 6.86 x 10 5 (3 sig.fig)

29. 20 (a) Simplify the expression (4x 3 y 2 ) x (2x 4 y 5 ) (b) Expand and simplify (x + 5) (x – 6) (c) Make d the subject of the following formula: h = √ t - d = 8x 7 y 7 = x² -6x +5x - 30 = x² -x - 30 First Outside Inside Last h² = t - d d = t - h²

30. 21. Solve the following equation: 4x – 8 – x = 2 3 6 6 (4x – 8) – 6 x = 6 x 2multiply every term by 6 3 6 2(4x – 8) – x = 12expand brackets 8x – 16 – x = 12 8x - x = 12 + 16collect terms 7x = 28 x = 28 7 x = 4

31. 22. A vertical flagpole, BDC, stands on horizontal ground ABE. It is supported by two ropes AC and DE. The length of AC is 13.5m, and the distance CD is 4.7m. The rope AC makes an angle of 62° with the ground and the rope DE is fixed to the ground at E such that BE is 8.4m. Calculate the size of BDE. 4.7m 8.4m 13.5m 62° A B E D C Diagram not drawn to scale. Sin 62° = BC 13.5 13.5 Sin 62° = BC BC = 11.9m Tan BDE = 8.4 7.2 Tan BDE = 1.16667 BDE = Tan -1 1.16667 BDE = 49.4° BD = 11.9 – 4.7 = 7.2m

32. 23. Two bags contain some coloured balls, which are identical except for their colour. One ball is taken at random from each bag and their colours noted. The probability of the selected ball from the first bag being red is ¼. The probability of the selected ball from the second bag NOT being red is ⅔. Red Not red First bag Second bag Red Not red Red Not red ⅔ (a) Complete the following tree diagram. (b) Calculate the probability that both balls are red. (c) Calculate the probability that only one ball is red. ¼ ¾ ⅓ ⅔ ⅓ = 1 x 1 4 3 = 1 12 = 1 x 2 + 3 x 1 4 3 4 3 = 2 + 3 12 12 = 5 12 P(red, red) = P (red, not red) + P(not red, red)