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UK GCSE 2006 : Mathematics Paper I

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General Certificate of Secondary Education 2006 Paper 1 Pure Mathematics G0301 Additional Mathematics [G0301] TUESDAY 16 MAY, AFTERNOON TIME 2 hours. INSTRUCTIONS TO CANDIDATES 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS 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 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. G0316 837 Answer all eleven questions. 1 (i) Using the axes and scales in Fig. 1 in your Supplementary Answer Booklet, sketch the [2] graph of y = sin x for 0 x 360 (ii) Hence, using the axes and scales in Fig. 2 in your Supplementary Answer Booklet, sketch the graph of y = sin x 1 for 0 x 360 2 [2] Solve the equation sin (1 x + 80 ) = 0.6 2 for 180 x 180 Give your answers correct to 2 decimal places. 3 [5] 3 8 (i) Find A 1 where A = 2 6 [2] (ii) Hence, using a matrix method, solve the following simultaneous equations for x and y. [4] (a) Find dy 5 when y = 2 x 7 4 7 dx x [3] (b) Find 4 [3] 2 5 x 4 dx . x7 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS 3x + 8y = 13 2x + 6y = 9 G0316 837 2 [Turn over 5 (i) Show that 3x 1 2 x 1 + x+2 x+4 can be written as 5 x 2 + 14 x 6 x2 + 6x + 8 [4] (ii) Hence, or otherwise, solve the equation 3x 1 2 x 1 + =3 x+2 x+4 6 [4] (a) If log4 64 = a what is the value of a? [1] (b) If log3 2 = b express log3 18 in terms of b. [3] (c) Solve the equation 1 5(1 + 2 x) = 21 [4] 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS giving your answer correct to 3 decimal places. G0316 837 3 [Turn over 7 Fig. 3 shows a sketch of the graph of the function y = x2 4x + 7 y A(1, 4) x Fig. 3 [4] (ii) Find the equation of the tangent to the curve which has a gradient of 6 [3] (iii) Verify that both tangents meet at the point (3, 0). [1] 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS (i) Find the equation of the tangent at the point A(1, 4). G0316 837 4 [Turn over 8 The points A, O and B lie in a straight line on horizontal ground, with AO = 1200 m and OB = 500 m, as shown in Fig. 4. X 65.0 15.0 A O 1200m 500m B Fig. 4 A hot-air balloon takes off from A and rises at a constant angle of 15.0 to the horizontal in a plane vertically above the line AOB. When it is at the point X its angle of elevation to an observer at O is 65.0 [4] (i) Calculate the distance OX. The balloon then descends along the path XB, reaching the ground at B, as shown in Fig. 4. [4] (iii) Calculate the angle to the horizontal at which the balloon descends. [2] 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS (ii) Calculate the distance BX. G0316 837 5 [Turn over 9 With her digital camera Ann can take high, medium and low resolution photos. One day she took 10 high, 15 medium and 20 low resolution photos and she used 45 Mb (megabytes) of storage on her memory stick. Let x, y and z represent the amounts of storage, in megabytes, used by a high, a medium and a low resolution photograph respectively. (i) Show that x, y and z satisfy the equation 2x + 3y + 4z = 9 [1] On the next day Ann took 14 high, 12 medium and 20 low resolution photographs which used 50 Mb of storage. (ii) Show that x, y and z also satisfy the equation 7x + 6y + 10z = 25 [1] On the third day Ann took 15 high, 12 medium and 36 low resolution photographs which used 60 Mb of storage. (iii) Show that x, y and z also satisfy the equation 5x + 4y + 12z = 20 [1] (iv) Solve these equations, showing clearly each stage of your solution. [9] The total amount of storage on the memory stick is 256 Mb. (v) Show that, in addition to all the photographs taken so far, Ann could get just over 5 minutes of video before the memory stick would be full. [2] 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS The camera has a video mode. Six seconds of video recording uses the same amount of storage on the memory stick as a high resolution photograph. G0316 837 6 [Turn over 10 At an Arctic weather station the wind chill index W was recorded for various wind speeds V and the results are given in Table 1. Table 1 Wind speed V (km/h) Wind chill index W 5 10 25 50 80 25.5 28.5 33.0 36.9 39.8 It is believed that a relationship of the form W = kV n exists between W and V, where k and n are constants. (i) Using Fig. 5 in your Supplementary Answer Booklet verify this relationship by drawing a suitable straight line graph, using values correct to three decimal places. Label the [6] axes clearly. (ii) Hence, or otherwise, obtain values for k and n. [4] (iii) Use the formula W = kV n with the values you obtained for k and n to calculate the wind chill index when the wind speed is 100 km/h. State any assumption which you make. [3] (iv) Use the formula W = kV n to determine the wind speed which would give a wind chill index of 35 [2] 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS When the wind chill index reaches 35 there is a high risk of frostbite. G0316 837 7 [Turn over 11 A curve is defined by the equation y = x(x + 1)(x 2) (i) Write down the coordinates of the points where the curve crosses the x-axis. [1] (ii) Find, correct to 2 decimal places, the coordinates of the turning points and identify each as either a maximum or a minimum point. [8] [3] (iv) Find the area above the x-axis which is enclosed by the curve and the x-axis. [4] 1 27.4.05BP 2 4.7.05EA 3 2.11.05ES 4 4.2.06RTS (iii) Sketch the curve using Fig. 6 in your Supplementary Answer Booklet. S 04/05 7600 302507(10) [Turn over

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