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UK GCSE 2009 : Mathematics Paper II

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General Certificate of Secondary Education 2009 Paper 2 Mechanics and Statistics G0302 Additional Mathematics [G0302] FRIDAY 15 MAY, MORNING TIME 2 hours. INSTRUCTIONS TO CANDIDATES Write your Centre Number and Candidate Number on the Answer Booklet and the Supplementary Answer Booklet provided. Answer all twelve questions. At the conclusion of the examination attach the Supplementary Answer Booklet to your Answer Booklet using the treasury tag supplied. INFORMATION FOR CANDIDATES The total mark for this paper is 100. Figures in brackets printed down the right-hand side of pages indicate the approximate marks awarded to each question or part question. You may use your calculator. A copy of the formulae list is provided. Take g = 10 m/s2 when required. 4380 Answer all twelve questions 1 (Throughout this question i and j denote unit vectors parallel to a set of standard x y axes.) A body of mass 5 kg is acted on by a force F newtons. The body moves in a straight line with initial velocity (2i + 3j) m/s. After 2 seconds its velocity is (5i j) m/s. Find (i) the acceleration of the body as a vector in terms of i and j, (ii) the force F as a vector in terms of i and j, [1] (iii) the magnitude of the force F. 2 [2] [1] As part of a quality control programme, Isaac measured the mass of each apple in a consignment delivered to his store. Each reading was rounded to the nearest gram and the results are summarised in Table 1. Table 1 Mass of apple (g) Number of apples 51 100 74 101 150 104 151 180 117 181 210 96 211 260 22 Using Fig. 1 in your Supplementary Answer Booklet, draw a histogram to represent this [5] information. Label each axis clearly. 3 A car accelerates uniformly from rest to a speed V m/s in 16 seconds. It continues to move with this speed for 40 seconds. The brakes are then applied and the car retards uniformly to rest. The total time for this journey is 60 seconds. (i) On the axes drawn in Fig. 2 in your Supplementary Answer Booklet sketch the speed time graph for this journey. [2] The total distance travelled by the car in the 60 seconds is 900 metres. (ii) Calculate the value of V. 4380 [4] 2 [Turn over 4 For each of the following classes write down (a) the lower class boundary, (b) the upper class limit, (c) the class width. (i) 15 kg to 18 kg, measured correct to the nearest kg. (ii) 5.5 cm to 8.5 cm, measured correct to the nearest mm. (iii) 4 C to 2 C, measured correct to the nearest C. 5 [6] A librarian counted the number of books on each shelf in a school library. The results are summarised in Table 2. Table 2 Number of books Number of shelves 11 25 14 26 35 18 36 45 21 46 55 5 56 65 2 (i) Calculate an estimate of the mean number of books per shelf. (ii) Calculate an estimate of the median number of books per shelf. 4380 [2] [4] 3 [Turn over 6 A box of mass m kg is suspended from a fixed point O by a light inextensible string. The box is pulled aside by a force of 20.7 N which acts at 45 with the horizontal as shown in Fig. 3. O 20.7 N 45 60 Fig. 3 The box hangs in equilibrium with the string making an angle of 60 with the horizontal. Find (i) the tension T newtons in the string, (ii) the mass m kg of the box. 7 [3] [4] During the first 3 weeks in September, Dolly recorded how much money she received each day by selling snacks from the mobile refreshment service she operated. Each amount was rounded to the nearest pound and the results are summarised in Table 3. Table 3 Monday Week 1 Week 2 Week 3 Tuesday Wednesday Thursday Friday 247 250 254 244 248 252 245 249 252 249 253 257 255 258 263 These data have been plotted on the graph given in Fig. 4 in your Supplementary Answer Booklet. (i) Calculate appropriate moving averages to smooth the data. [2] (ii) Plot these averages on the graph and draw the trend line. [3] (iii) Showing clearly where any reading is taken, use the trend line to calculate an estimate of how much money she will receive on the fourth Monday in September. [4] 4380 4 [Turn over 8 A non-uniform plank AB of length 8 metres rests horizontally in equilibrium on two supports, one at A and the other at B. The reactions at A and B are 180 N and 420 N respectively. (i) Calculate the mass of the plank. [2] The centre of mass of the plank is at a point G, x metres from the end A as shown in Fig. 5. xm G B A Fig. 5 (ii) Find the value of x. [3] The two supports are removed and replaced by a single support placed at the mid-point of the plank. A child of mass m kg stands on the plank at the end A. The plank is now in equilibrium in a horizontal position as shown in Fig. 6. B A Fig. 6 (iii) Copy Fig. 6 and mark clearly on your diagram all the forces acting on the plank. (iv) Calculate the mass m kg of the child. 4380 [3] [2] 5 [Turn over 9 Each Wednesday Maeve plays a hockey match, either for her club, or for her county, but not both. If on a given Wednesday she plays a club match, the probability that she will play a county match on the following Wednesday is 0.65 If on a given Wednesday she plays a county match, the probability that she will play a club match on the following Wednesday is 0.8 (i) On Wednesday Maeve plays a match for her club. Draw a clearly labelled tree diagram to show the probabilities of playing club and county matches for the following three Wednesdays. [3] Using the tree diagram, or otherwise, calculate the probability that on these three Wednesdays she will play (ii) 3 county matches, (iii) only 1 county match, [3] (iv) at least 1 county match. 4380 [2] [3] 6 [Turn over 10 A large concrete block of mass 45 kg is lying on a rough horizontal floor. A rope is attached to the block to pull it along the floor. The rope makes an angle of 50 with the horizontal as shown in Fig. 7. T 50 Fig. 7 (i) Copy Fig. 7 and mark on it all the forces acting on the block. [2] When the tension T in the rope is 175 N, the block is on the point of moving along the floor. (ii) Calculate, correct to 2 decimal places, the normal reaction between the block and the floor. [2] (iii) Show that the coefficient of friction between the block and the floor is 0.356, correct to 3 decimal places. [3] The angle that the rope makes with the horizontal is reduced to 35 . Assume that the tension in the rope and the coefficient of friction remain unaltered. (iv) Calculate, correct to 2 decimal places, the acceleration of the block. 4380 7 [4] [Turn over 11 Two small particles A and B of masses 0.6 kg and 0.4 kg respectively are connected by a light inextensible string which passes over a smooth fixed pulley as shown in Fig. 8. A B Platform Fig. 8 The system is released from rest with the particles hanging vertically. Calculate (i) the acceleration of the system, [3] (ii) the tension in the string, [1] (iii) the force exerted by the string on the pulley when the particles are in motion. [1] When the particles have been in motion for 1.5 seconds, A strikes a fixed horizontal platform. The string becomes slack and B initially continues to rise. Assuming that B does not reach the pulley, calculate (iv) the speed of the particles at the instant A strikes the platform, [2] (v) the additional distance through which B rises after A strikes the platform, [3] (vi) the time which elapses between A striking the platform and the string becoming taut again. [2] 4380 8 [Turn over 12 Mr Caldwell asked each pupil in his Year 13 class to record the total length of time spent watching television in a week. He wanted to compare these times with the marks they achieved in a test at the end of the week. The results are shown in Table 4. Table 4 Student Times (to the nearest hour) Mark (%) Grainne Nicole Declan Marlene Charlotte Jerry Rosanna Tom Jack Luc 5 14 9 8 18 12 14 11 10 16 91 34 70 87 54 36 75 55 70 44 (i) Determine the rank orders for the times and for the marks. [2] (ii) Calculate Spearman s coefficient of rank correlation. [4] (iii) What significance, if any, do you attach to the value you obtained in (ii)? [1] The data from Table 4 are plotted on the graph given in Fig. 9 in your Supplementary Answer Booklet. (iv) Calculate the mean time and the mean mark. [1] (v) Draw your line of best fit on the graph in the Supplementary Answer Booklet. [2] (vi) Determine the equation of the line of best fit which you have drawn. [3] THIS IS THE END OF THE QUESTION PAPER 4380 9 [Turn [Turn over 4380 10 [Turn over 4380 11 [Turn over 936-011-1 12 Centre Number 71 Candidate Number General Certificate of Secondary Education 2009 Paper 2 Mechanics and Statistics [G0302] FRIDAY 15 MAY, MORNING SUPPLEMENTARY ANSWER BOOKLET 4380.02 G0302SAB Additional Mathematics 2 Using the information from Table 1, draw a histogram in Fig. 1. Label each axis clearly. Fig. 1 4380.02 2 [Turn over 3 On the axes below, sketch the speed time graph for the journey. Speed (m/s) Time (s) Fig. 2 4380.02 3 [Turn over 7 Plot the moving averages in Fig. 4 and draw the trend line. 265 260 Daily 255 amount 250 ( ) 245 240 M T W T F M T W Day T F M T W T F Fig. 4 4380.02 4 [Turn over 12 Draw the line of best fit through the data points shown in Fig. 9. 100 80 Mark (%) 60 40 20 4 6 8 10 12 14 16 18 20 Time (hours) Fig. 9 4380.02 5 [Turn over 4380.02 6 [Turn over 4380.02 7 [Turn over 936-011-2 8

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