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UK GCSE 2007 : Mathematics Paper II

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General Certificate of Secondary Education 2007 Paper 2 Mechanics and Statistics G0302 Additional Mathematics [G0302] MONDAY 21 MAY, AFTERNOON 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. G0327 2466 Answer all twelve questions. 1 Two packages A and B, of masses 4 kg and M kg respectively, where M 4, are connected by a light inextensible string passing over a smooth fixed pulley as shown in Fig. 1. B A Fig. 1 When the packages are released from rest with the string taut the magnitude of the acceleration of each package is 2.5 m/s2. Calculate (i) the tension in the string, (ii) the value of M. 2 [2] [2] An internet information channel surveyed 180 users to determine what age-groups were accessing its site during the first week in September. Table 1 summarises the results. Table 1 Age (years) Number using site 0 9 10 14 15 19 20 24 25 34 35 59 60 8 34 29 27 54 28 0 Use the graph paper in Fig. 2 in your Supplementary Answer Booklet to draw a histogram to represent this information. [5] G0327 G0327 2466 2 [Turn over 3 (Throughout this question i and j denote unit vectors parallel to a set of standard x-y axes) Three vectors a, b and c are defined as follows a = 4i + j b = 2i 3j c = 5i 4j Calculate (i) the values of the scalar quantities p and q such that pa + qb = c, (ii) the magnitude of the single vector represented by 3a 3b + 2c giving your answer correct to 1 decimal place, [2] (iii) the angle that this vector makes with the positive x-axis. 4 [3] [1] A car which is initially at rest accelerates uniformly on a straight road for t seconds until it reaches a speed of 16 m/s. The car then travels at this speed for 40 seconds. The brakes are then applied and the car retards uniformly to rest. The magnitude of the retardation is three times the magnitude of the acceleration. (i) On the axes drawn in Fig. 3 of your Supplementary Answer Booklet sketch the speed time graph for this journey. [2] The total time taken by the car for this journey is 56 seconds. Calculate (ii) the value of t, (iii) the total distance travelled. G0327 [2] [3] 2466 3 [Turn over 5 A group of friends was debating which extra features were more popular on their mobile phones. Over a 4-week period they each recorded how many photographs they took and how often they listened to music using their mobile phones. The results are summarised in Table 2. Table 2 Name Derek Kevin Tom Lucy Rachel Rosanna Armelle Time listening to music (to nearest hour) 15 18 5 9 4 15 0 Number of photographs taken 15 8 12 23 18 3 3 (i) Work out the rank orders for the times spent listening to music and for the number of photographs taken. [2] (ii) Calculate Spearman s coefficient of rank correlation. (iii) What significance, if any, do you attach to the value you obtained in (ii)? 6 [4] [1] Tony achieved the following marks in each section of his last music exam. 24, 15, 6, 18, 12 (i) Calculate the standard deviation of these marks. [3] He needs to scale his marks so that the mean mark is 40 and the standard deviation is doubled. He does this by multiplying each score by a constant a and adding a constant b. (ii) Calculate a and b. G0327 2466 [4] 4 [Turn over 7 A uniform plank AB of length 8 metres and mass 20 kg rests horizontally in equilibrium on two supports placed at points C and D on the plank where AC = 1 m and DB = 2 m as shown in Fig. 4. C A D 1m B 2m Fig. 4 Calculate (i) the magnitude of the reaction at C, [2] (ii) the magnitude of the reaction at D. [1] A box of mass M kg is placed on the plank at A. The plank remains horizontal but is on the point of tilting about the support at C. (iii) State the reaction at D when the plank is on the point of tilting. [1] (iv) Calculate the mass M kg of the box. [2] The box is now moved from A and placed at a point on the plank x metres from B. The plank remains horizontal but is now on the point of tilting about D. (v) Find the value of x. G0327 2466 [3] 5 [Turn over 8 At a basketball training session the coach recorded the number of lay-ups scored by each member of the class. Table 3 summarises their scores. Table 3 Pupil Beth Lara Hannah Aidan Ward Larry Erin Peter Daniel Niamh Number of right-hand lay-ups scored 57 49 25 15 20 45 37 57 38 30 Number of left-hand lay-ups scored 39 38 30 21 26 31 28 33 35 25 (a) These data are plotted on the graph given in Fig. 5 in the Supplementary Answer Booklet. (i) Calculate the mean number of right-hand lay-ups scored and the mean number of left-hand lay-ups scored. [1] (ii) Draw your line of best fit on the graph in the Supplementary Answer Booklet. [2] (iii) Determine the equation of the line of best fit which you have drawn. [4] (iv) Estimate the number of left-hand lay-ups likely to be scored by a player who scored 50 right-hand lay-ups during the session. [1] (b) What is the median number of left-hand lay-ups ? G0327 2466 6 [1] [Turn over 9 The practice manager at a local surgery recorded the number of patients treated in the surgery by the doctors over the past 3 years. Table 4 summarises the results. Table 4 Jan Mar Apr Jun Jul Sep Oct Dec 2004 1003 841 728 900 2005 1055 905 784 968 2006 1111 965 836 1036 These data have been plotted on the graph given in Fig. 6 in your Supplementary Answer Booklet. (i) Calculate appropriate moving averages to smooth the data. [3] (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 the number of patients likely to be treated in the period January to March 2007. [4] 10 A boy is standing at the edge of the flat roof of a building. He is 10 metres above the pavement. He throws a ball vertically upwards with a speed of 8 m/s. (i) Find the maximum height above the pavement reached by the ball. [3] The boy fails to catch the ball as it falls back down again towards the pavement. Calculate (ii) the speed of the ball, in m/s, at the instant when it strikes the pavement, giving your answer correct to 1 decimal place. [2] (iii) the time, in seconds, which elapses from the instant the ball leaves the boy s hand until it strikes the pavement, giving your answer correct to 1 decimal place. [4] The ball rebounds with half the speed with which it strikes the pavement. (iv) Calculate the height to which the ball rebounds after the first bounce, giving your answer correct to 1 decimal place. [2] G0327 2466 7 [Turn over 11 A child throws a red die and a blue die. The red die is numbered 1 to 6 and the blue die is numbered 1, 2, 2, 3, 3, 3. Using a probability space diagram or otherwise calculate the probability that (i) both dice show prime numbers, [2] (ii) the sum of the scores on both dice is prime, [2] (iii) both dice show the number 1 given that the sum of the scores is prime, [3] (iv) the score on the blue die is greater than the score on the red die. [2] A double occurs when both dice show the same score. Both dice are thrown 6 times. (v) Calculate the probability of getting no doubles. (vi) Hence calculate the probability of scoring at least one double. G0327 [2] [1] 2466 8 [Turn over 12 A crate of mass 20 kg slides down a rough chute in a warehouse. The chute is inclined at 25 to the horizontal as shown in Fig. 7. A B 25 Fig. 7 The frictional resistance to the motion is constant throughout. (i) Copy Fig. 7 and mark on it the forces acting on the crate. [2] The crate passes two markers A and B, 4 metres apart on the chute, with speeds of 2 m/s and 4 m/s respectively. Calculate (ii) the acceleration of the crate, [2] (iii) the magnitude of the frictional resistance, giving your answer correct to 1 decimal place, [4] (iv) the coefficient of friction between the crate and the surface of the chute, giving your answer correct to 1 decimal place. [3] The crate reaches the bottom of the chute 2 seconds after passing A. Calculate (v) the speed of the crate when it reaches the bottom of the chute, [1] (vi) the distance of the bottom of the chute from A. [1] THIS IS THE END OF THE QUESTION PAPER G0327 2466 9 [Turn over G0327 2466 10 [Turn [Turn over G0327 2466 10 [Turn [Turn over G0327 2466 11 [Turn [Turn over S 02/06 0000 7-028-1 [Turn over Centre Number 71 Candidate Number General Certificate of Secondary Education 2007 Paper 2 Mechanics and Statistics [G0302] MONDAY 21 MAY, AFTERNOON SUPPLEMENTARY ANSWER BOOKLET G0327 2466SAB G0302SAB Additional Mathematics 2 Using the information from Table 1 draw a histogram in Fig. 2. Label each axis clearly. Fig. 2 G0327 2466SAB 2 [Turn over 4 On the axes below sketch the speed time graph for the journey. Speed (m/s) Time (s) Fig. 3 G0327 2466SAB 3 [Turn over 8 Draw your line of best fit through the data points shown in Fig. 5. 50 40 Number of left-hand lay-ups scored 30 20 10 10 20 30 40 50 60 Number of right-hand lay-ups scored Fig. 5 G0327 2466SAB 4 [Turn over 9 Plot the moving averages in Fig. 6 and draw the trend line. 1200 1100 Number of 1000 patients 900 2004 2005 Dec Oct ep Jul S Jun Apr Mar Jan Dec Oct ep Jul S Jun Apr Mar Jan Dec Oct ep Jul S Jun Apr Jan Mar 800 2006 Fig. 6 G0327 2466SAB 5 [Turn over G0327 G0327 2466SAB 6 [Turn over G0327 G0327 2466SAB 7 [Turn over S 2/06 0000 7-029-1 [Turn over

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