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UK GCSE 2010 : Mathematics Paper I

20 pages, 36 questions, 3 questions with responses, 3 total responses,

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General Certificate of Secondary Education 2010 Paper 1 Pure Mathematics G0301 Additional Mathematics [G0301] MONDAY 17 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. 5467 Answer all eleven questions 1 (i) Using the axes and scales in Fig. 1 in your Supplementary Answer Booklet, sketch the graph of y = 2 sin x for 360 < x < 360 . [2] (ii) Using the axes and scales in Fig. 2 in your Supplementary Answer Booklet, sketch the 1 graph of y = sin (2 x) for 360 < x < 360 . 2 [2] (i) Solve the equation tan x = 3 for 180 , x < 180 . [2] (ii) Hence solve the equation x tan 10 = 3 2 _ + for 360 , x < 360 . 3 [3] 4 3 (i) Find A 1 where A = 6 5 [2] (ii) Hence, using a matrix method, solve the following simultaneous equations for x and y. 4x 3y = 18 6x + 5y = 29 5467 2 [4] [Turn over 4 2 dy (a) Find when y = 2x5 2 dx 5x (b) Find 5 [2] [4] . Fig. 3 shows a sketch of the graph of y = 2x3 + 4x 7 y x Fig. 3 P is the point on this curve whose x-coordinate is 1 (i) Find the equation of the tangent to this curve at P. [4] (ii) Find the coordinates of the other point on this curve at which the gradient is the same as the gradient at P. [3] 5467 3 [Turn over 6 (i) Show that x+2 x 1 2 x + 3 3x 1 can be written as x 2 + 4x + 1 6x 2 + 7x 3 [4] (ii) Hence, or otherwise, solve the equation [4] x + 2 1 x 1 = 2 x + 3 2 3x 1 7 (a) Solve the equation 9 2 x 3 5 [4] =7 (b) If logy 5 = 0.5 what is the value of y? [1] (c) If log3 5 = a and log3 8 = b express the following in terms of a and b. (i) log3 1.6 (ii) log3 120 5467 [1] [2] 4 [Turn over 8 Julie s house J is 3.00 km from a helipad H. Ken s house K is 1.00 km from Julie s house. H, J and K all lie in a straight line on horizontal ground. A helicopter is descending towards the helipad along a flight path FH which is at a constant angle to the horizontal. This flight path passes directly over Ken s house and also over Julie s house. When the helicopter is at a position X, Julie measures the angle of elevation of the helicopter as 21.20 from her house J and Ken measures it as 32.90 from his house K, as shown in Fig. 4. X 21.20 H 3.00 km J 1.00 km F 32.90 K Fig. 4 (i) Write down the sizes of the angles JKX and JXK. (ii) Calculate the distance JX. [3] (iii) Write down the size of the angle HJX. [1] (iv) Calculate the distance HX. [3] (v) Calculate the size of the angle XHJ. [2] (vi) Calculate the height of the helicopter as it passes over Julie s house. 5467 [1] [2] 5 [Turn over 9 Lesley recorded the price P, in pounds, and the number of days D for five different holidays at a given hotel. The details are given in Table 1. Table 1 Price P ( ) Days D 79.50 2 150.98 5 262.19 11 355.60 17 398.44 20 She believes that a relationship of the form P = aDb exists between P and D, where a and b 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 3 decimal places. Label the axes clearly. [6] (ii) Hence, or otherwise, obtain values for a and b. Give your answers correct to 1 decimal place. [4] (iii) Use the formula P = aDb with the values you obtained for a and b to calculate the price of a 4 week holiday in this hotel. Give your answer correct to the nearest penny. State any assumption which you make. [2] (iv) Monica has saved 330 and wants to book a holiday at this hotel. Use the formula P = aDb to calculate the maximum number of complete days she can stay. 5467 6 [2] [Turn over 10 Alison had twelve 2 coins, eight 1 coins and twenty 20p coins. The total mass of the coins was 320 g. Let x, y and z represent the masses, in grams, of a 2 coin, a 1 coin and a 20p coin respectively. (i) Show that x, y and z satisfy the equation 3x + 2y + 5z = 80 [1] Brian had twenty-five 2 coins, thirty 1 coins and fifteen 20p coins. The total mass of these coins was 660 g. (ii) Show that x, y and z also satisfy the equation 5x + 6y + 3z = 132 [1] Christine had eighteen 50p coins, twelve 1 coins and twenty-seven 20p coins. The total mass of these coins was 393 g. 2 The mass of a 50p coin is 3 that of a 2 coin. (iii) Show that x, y and z also satisfy the equation 4x + 4y + 9z = 131 [2] (iv) Solve these equations to find the masses of all four coins, i.e. a 2 coin, a 1 coin, a 50p coin and a 20p coin. Show clearly each stage of your solution. [8] David had twenty 2 coins, some of which were counterfeit. Each counterfeit coin has a mass of 10 g. The total mass of David s coins was 228 g. (v) Calculate how many counterfeit coins David had. 5467 7 [2] [Turn over 11 A curve is defined by the equation y = 14x + 3x2 2x3 (i) Find the coordinates of the points where this curve crosses the x-axis. [2] (ii) Find the coordinates of the turning points, giving your answers correct to 2 decimal places. [6] (iii) Identify each turning point as either a maximum or a minimum point. You must show working to justify your answer. [2] (iv) Using your answers from parts (i) to (iii), sketch this curve using Fig. 6 in your Supplementary Answer Booklet. [3] (v) Find the area enclosed between this curve and the negative x-axis. [3] THIS IS THE END OF THE QUESTION PAPER 5467 8 [Turn over 1847-036-1 [Turn over Centre number 71 Candidate number General Certificate of secondary education Paper 1 Pure Mathematics [G0301] monDay 17 may, aFternoon supplementary answer booklet 5467.02 G0301 additional mathematics *G0301* 2010 1 (i) sketch the graph of y = 2 sin x on the axes in Fig. 1 below. y 2 1 360 270 180 0 90 x 90 180 270 360 1 2 Fig. 1 1 (ii) sketch the graph of y = sin ( x) on the axes in Fig. 2 below. 2 y 2 1 360 270 180 0 90 90 180 270 360 x x 1 2 Fig. 2 5467.02 2 [turn over 9 Draw a suitable straight line graph using the axes and scales in Fig. 5 below. label the axes. 2.6 2.4 2.2 2.0 1.8 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Fig. 5 5467.02 3 [turn over 11 sketch the graph of y = 14x + 3x2 2x3 in Fig. 6. y 30 20 10 4 x 0 2 2 4 10 20 Fig. 6 x 5467.02 4 [turn over 1847-036-2

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