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UK GCSE 2007 : Mathematics Paper I

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General Certificate of Secondary Education 2007 Paper 1 Pure Mathematics G0301 Additional Mathematics [G0301] TUESDAY 15 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 eleven questions. At the conclusion of this examination attach the Supplementary Answer Booklet to your Answer Booklet using the treasury tag supplied. 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA INFORMATION FOR CANDIDATES The total mark for this paper is 100 Figures in brackets printed down the right-hand side of pages indicate the marks awarded to each question or part question. You may use a calculator. A copy of the formulae list is provided. G031S7 2348 Answer all eleven questions 1 Using the axes and scales in Fig. 1 in your Supplementary Answer Booklet, sketch the graph of y = 2 cos x for 0 x 360 . [4] 2 (i) Solve the equation cos = 0.1 for 180 < 180 . Give your answers correct to 2 decimal places. [2] (ii) Hence solve the equation cos (2x 10 ) = 0.1 for 90 < x 3 90 . Give your answers correct to 2 decimal places. (i) Find A 1 where A = [] 3 5 2 1 [2] [2] (ii) Hence, using a matrix method, solve the following simultaneous equations for x and y. 3x + 2y = 16 5x + y = 5 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA 4 dy 2 (a) Find when y = 3x2 + dx x (b) Find G031S7 2348 (x5 [4] [2] 1 + 2 ) dx . x2 [4] 2 [Turn over 5 8 Fig. 2 shows a sketch of the graph of y = 4 for x > 0 x y x P Fig. 2 (i) Find the coordinates of the point P where the curve cuts the x-axis. [1] (ii) Find the equation of the tangent to the curve at P. [2] (iii) Find the coordinates of the point Q on the above curve where the gradient of the tangent is 1. [2] 2 (iv) Verify that the tangent to the curve at Q passes through the origin. 6 [2] (i) Show that 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA 1 2x x+1 x 5 2x + 5 can be written as 5x2 4x + 10 2x2 5x 25 [4] (ii) Hence, or otherwise, solve the equation 1 2x x+1 = 2 x 5 2x + 5 G031S7 2348 3 [4] [Turn over 7 (a) Solve the equation 4(2x+3) = 3 giving your answer correct to 3 decimal places. [4] [1] (c) If log 6 = p and log 4 = q express log 36 in terms of p and hence log 9 in terms of p and q. [3] 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA (b) If log2 a = 3 what is the value of a? G031S7 2348 4 [Turn over 8 Mary entered a forest, which was on level horizontal ground, at a point O. She intended to walk along a straight path to a camp site at X. The distance OX was 6.25 km and the path OX was in a direction of 47.50 east of north, as shown in Fig. 3. N X m 5k Z 6.2 P 47.50 3.50 km O Fig. 3 As the path OX was closed near O she had to take an alternative route. She travelled along another straight path from O to P, where the distance OP was 3.50 km and the direction of OP was 75.00 east of north. From P she walked along a straight path which went due north and joined the path OX at Z. [1] (ii) Calculate the distance PX. [3] (iii) Calculate the size of the angle OX P. [3] (iv) Show that the angle XZ P is 132.50 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA (i) Calculate the size of the angle XO P. [1] (v) Find the size of the angle XP Z. [1] (vi) Calculate the distance ZX that Mary still had to walk after she reached the path OX. [3] G031S7 2348 5 [Turn over 9 In an area prone to hurricanes the wind pressure P (kg/m2) was measured for various hurricane wind speeds V (km/h) and the results are given in Table 1. Table 1 Wind speed V (km/h) Wind pressure P (kg/m2) 119.1 154.5 178.6 210.8 249.5 106.9 180.2 240.7 334.9 469.2 It is believed that a relationship of the form P = kVn exists between P and V, where k and n are constants. (i) Using Fig. 4 in your Supplementary Answer Booklet verify this relationship by drawing a suitable straight line graph, using values correct to three decimal places. Label the axes clearly. [6] (ii) Hence, or otherwise, obtain values for k and n. [4] A builder states that his buildings can withstand a wind pressure of 400 kg/m2 without structural damage. (iii) Use the formula P = kVn with the values you obtained for k and n to calculate the wind speed that his buildings can withstand. [2] A tropical storm has a wind speed of 62.8 km/h. [2] 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA (iv) Calculate the wind pressure resulting from this tropical storm. State any assumption which you make. G031S7 2348 6 [Turn over 10 Jack and Jill were on different mobile phone tariffs. One day Jack sent 20 texts and 8 pictures and made 12 minutes of calls for a total cost of 8.60 Let x and y represent the costs, in pence, of sending a text and a picture respectively, and let z represent the cost, in pence per minute, of making a call. (i) Show that x, y and z satisfy the equation 5x + 2y + 3z = 215 [1] The next day Jack sent 9 texts and 9 pictures and made 6 minutes of calls for a total cost of 5.70. (ii) Show that x, y and z also satisfy the equation 3x + 3y + 2z = 190 [1] On Jill s tariff texts were 20% dearer, pictures cost 6p more but calls were 5p per minute cheaper. She sent 10 texts and 10 pictures and made 10 minutes of calls for a total cost of 7.80. (iii) Show that x, y and z also satisfy the equation 6x + 5y + 5z = 385 [3] [8] (v) How much would Jack have spent over the two days if he had been on Jill s tariff? [2] 1 2.2.06AC 2 22.3.06EA 3 3.6.06EA (iv) Solve these equations, showing clearly each stage of your solution. G031S7 2348 7 [Turn over 11 A curve is defined by the equation y = x (3x2 + 2x 5) The curve crosses the x-axis at the origin and at two other points. (i) Find the coordinates of these two points. [2] (ii) Find the coordinates of the turning points and identify each as either a maximum or a minimum point. [8] (iii) Using your results from (i) and (ii) sketch the curve using Fig. 5 in your Supplementary Answer Booklet. [3] (iv) Find the area enclosed by the curve and the positive x-axis. S 2/06 0000 7-026-1 [3] [Turn over

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