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UK GCSE 2008 : Mathematics Paper I

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General Certificate of Secondary Education 2008 Paper 1 Pure Mathematics G0301 Additional Mathematics [G0301] TUESDAY 13 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. 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. G031S8 3505 BLANK PAGE G031S8 3505 2 [Turn over Answer all eleven questions 1 (i) Using the axes and scales in Fig. 1 in your Supplementary Answer Booklet, sketch the graph of y = sin x for 180 x 180 . [2] (ii) Using the axes and scales in Fig. 2 in your Supplementary Answer Booklet, sketch the graph of y = sin 2x for 180 x 180 . [2] 2 (i) Solve the equation tan = 2 for 0 < 360 . Give your answers correct to 2 decimal places. [2] (ii) Hence solve the equation tan (4x + 5 ) = 2 for 0 3 x < 90 . Give your answers correct to 2 decimal places. 4 3 (i) Find A 1 where A = 7 6 [3] [2] (ii) Hence, using a matrix method, solve the following simultaneous equations for x and y. 4x 3y = 5 7x + 6y = 8 G031S8 3505 3 [4] [Turn over 5 (a) Find dy 4 when y = 3 x 4 + 5 1 dx x (b) Find 4 3x 4 + [2] 4 1 dx 5 x [4] Fig. 3 shows a sketch of the graph of y = x 3 + 6x 2 5x y 15 10 5 A(1, 0) 1 2 3 4 5 x Fig. 3 (i) Find the equation of the tangent to the curve at the point A(1,0). [4] The tangent at another point B on the curve is parallel to the tangent at A. (ii) Find the coordinates of B. G031S8 3505 [4] 4 [Turn over 6 (i) Show that 2x 1 2x + 1 3x + 4 4 x can be written as 2(4 x 2 + x + 4) 3 x 2 8 x 16 [4] (ii) Hence, or otherwise, solve the equation 2x 1 2x + 1 =2 3x + 4 4 x [4] (a) If loga 32 = 5 what is the value of a? [1] (b) If log2 5 = b express log2 50 in terms of b. Give your answer in its simplest form. 7 [3] (c) Solve the equation x (7 2 ) 3 =8 giving your answer correct to 2 decimal places. G031S8 3505 5 [4] [Turn over 8 At a given instant one boat is at A and another is at B. The boat at A is heading at a constant speed in a straight line towards a harbour at Y. The boat at B is heading at the same speed in a straight line towards another harbour at X. The distances AY and BY are 22.00 km and 18.50 km respectively. The boat at B is due north of Y and the angle AYB is 56.45 , as shown in Fig 4. B A O 56.45 X Y Fig. 4 (i) Show that the distance AB is 19.40 km. [3] (ii) Calculate the size of the angle BAY. [3] (iii) Calculate the size of the angle ABY. [1] The harbour at X is 6.11 km due west of the harbour at Y. (iv) Calculate the size of the angle XBY. [2] (v) Show that the angles BAY and ABX are approximately equal. [1] The lines AY and BX intersect at the point O. (vi) Explain why the boats will meet each other at O. G031S8 3505 6 [1] [Turn over 9 Cepheid variables are stars whose brightness fluctuates regularly with time. An astronomer measured the period P and the average luminosity (intrinisic brightness relative to the sun) L of several stars and the results are given in Table 1. Table 1 Period P (days) Average luminosity L (suns) 5.32 11.61 25.18 38.49 51.42 2017 4947 12 050 19 630 27 390 It is believed that a relationship of the form L = kPn exists between L and P, 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 two decimal places. Label the axes clearly. [6] (ii) Hence, or otherwise, obtain values for k and n. [4] (iii) Use the formula L = kPn with the values you obtained for k and n to calculate the average luminosity of the star X-Cygni, which has a period of 16.39 days. [1] (iv) Use the formula L = kPn to calculate the period of the star Y-Cygni, whose average luminosity is 1394 suns. State any assumption about the formula which you make. [3] G031S8 3505 7 [Turn over 10 Becky, Catriona and Grace went shopping in a continental market. They each bought identical sunglasses, bag and watch. By bargaining with the stall holders Becky bought her sunglasses for 60% of the marked price, her bag for 40% of the marked price and her watch for 40% of the marked price. She spent a total of 40 euro. Let x, y and z represent the marked prices, in euro, of the sunglasses, the bag and the watch respectively. (i) Show that x, y and z satisfy the equation 3x + 2y + 2z = 200 [2] Catriona bought her sunglasses, bag and watch for 40%, 30% and 50% of the marked prices respectively. She spent a total of 35 euro. (ii) Show that x, y and z also satisfy the equation 4x + 3y + 5z = 350 [1] Grace bought her sunglasses, bag and watch for 50%, 50% and 30% of the marked prices respectively. She spent a total of 39 euro. (iii) Show that x, y and z also satisfy the equation 5x + 5y + 3z = 390 (iv) Solve these equations, showing clearly each stage of your solution. [1] [8] Another friend, Laura, had asked Grace to buy her a pair of sunglasses and a watch. Grace bought these at the same prices as she herself had paid for them. (v) Would Laura have spent less money if she had asked one of the other girls to buy the sunglasses and watch at the same prices as they had paid for them? [2] G031S8 3505 8 [Turn over 11 A curve is defined by the equation y = x3 8x2 + 16x (i) Find the coordinates of the two points where the curve meets the x-axis. [2] (ii) Find the coordinates of the turning points and identify each as either a maximum or a minimum point. [8] (iii) Sketch the curve using Fig. 6 in your Supplementary Answer Booklet. [3] (iv) Find the area enclosed by the curve and the x-axis. [3] THIS IS THE END OF THE QUESTION PAPER G031S8 3505 9 [Turn over S 2/07 528-012-1 [Turn over Centre Number 71 Candidate Number General Certificate of Secondary Education 2008 Paper 1 Pure Mathematics [G0301] TUESDAY 13 MAY, AFTERNOON SUPPLEMENTARY ANSWER BOOKLET G031S8 3505.02 G0301SAB Additional Mathematics 1 (i) Sketch the graph of y = sin x on the axes in Fig. 1 below. y 2 1 0 180 90 180 x 90 90 180 x 1 2 Fig. 1 (ii) Sketch the graph of y = sin 2x on the axes in Fig. 2 below. y 2 1 0 180 90 1 2 Fig. 2 G031S8 3505.02 2 [Turn over 9 Draw a suitable straight line graph using the axes and scales in Fig. 5 below. Label the axes. 4.8 4.6 4.4 4.2 4.0 3.8 3.6 3.4 3.2 0.5 1.0 1.5 2.0 Fig. 5 G031S8 3505.02 3 [Turn over 11 Sketch the graph of y = x3 8x2 + 16x in Fig. 6 below. y 10 5 0 1 2 3 Fig. 6 S 2/07 528-012-2 4 5 x

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