# MATHEMATICS Unit Pure Core 2

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1 General Certificate of Education June 2006 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Monday 22 May am to am For this paper you must have: * an 8-page answer book * the blue AQA booklet of formulae and statistical tables You may use a graphics calculator. Time allowed: 1 hour 30 minutes Instructions * Use blue or black ink or ball-point pen. Pencil should only be used for drawing. * Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MPC2. * Answer all questions. * Show all necessary working; otherwise marks for method may be lost. Information * The maximum mark for this paper is 75. * The marks for questions are shown in brackets. Advice * Unless stated otherwise, you may quote formulae, without proof, from the booklet. 6/6/6/ MPC2

2 2 Answer all questions. 1 The diagram shows a sector ofa circle ofradius 5 cm and angle y radians. 5cm 5cm y The area ofthe sector is 8.1 cm 2. (a) Show that y ˆ 0:648. (2 marks) (b) Find the perimeter ofthe sector. 2 The diagram shows a triangle ABC. B 12 cm 100 A 4.8 cm C The lengths of AC and BC are 4.8 cm and 12 cm respectively. The size ofthe angle BAC is 100. (a) Show that angle ABC ˆ 23:2, correct to the nearest 0:1. (b) Calculate the area oftriangle ABC, giving your answer in cm 2 to three significant figures.

3 3 3 The first term of an arithmetic series is 1. The common difference of the series is 6. (a) Find the tenth term ofthe series. (2 marks) (b) The sum ofthe first n terms ofthe series is (i) Show that 3n 2 2n 7400 ˆ 0. (ii) Find the value of n. (2 marks) 4 (a) The expression 1 2x 4 can be written in the form 1 px qx 2 32x 3 16x 4 By using the binomial expansion, or otherwise, find the values of the integers p and q. (b) Find the coefficient of x in the expansion of 2 x 9. (2 marks) (c) Find the coefficient of x in the expansion of 1 2x 4 2 x 9. 5 (a) Given that log a x ˆ 2 log a 6 log a 3 show that x ˆ 12. (b) Given that log a y log a 5 ˆ 7 express y in terms of a, giving your answer in a form not involving logarithms. Turn over s

4 4 6 The diagram shows a sketch ofthe curve with equation y ˆ 27 3 x. y A R O B x The curve y ˆ 27 3 x intersects the y-axis at the point A and the x-axis at the point B. (a) (i) Find the y-coordinate ofpoint A. (2 marks) (ii) Verify that the x-coordinate ofpoint B is 3. (1 mark) (b) The region, R, bounded by the curve y ˆ 27 3 x and the coordinate axes is shaded. Use the trapezium rule with four ordinates (three strips) to find an approximate value for the area of R. (4 marks) (c) (i) Use logarithms to solve the equation 3 x ˆ 13, giving your answer to four decimal places. (ii) The line y ˆ k intersects the curve y ˆ 27 3 x at the point where 3 x ˆ 13. Find the value of k. (1 mark) (d) (i) Describe the single geometrical transformation by which the curve with equation y ˆ 3 x can be obtained from the curve y ˆ 27 3 x. (2 marks) (ii) Sketch the curve y ˆ 3 x. (2 marks)

5 5 7 At the point x, y, where x>0, the gradient ofa curve is given by 1 dy dx ˆ 3x2 16 x 2 7 (a) (i) Verify that dy ˆ 0 when x ˆ 4. dx (1 mark) (ii) Write 16 x 2 in the form 16xk, where k is an integer. (1 mark) (iii) Find d2 y dx 2. (iv) Hence determine whether the point where x ˆ 4 is a maximum or a minimum, giving a reason for your answer. (2 marks) (b) The point P 1, 8 lies on the curve. (i) Show that the gradient ofthe curve at the point P is 12. (1 mark) (ii) Find an equation ofthe normal to the curve at P. (c) (i) 1 Find 3x 16 7 dx: x2 (ii) Hence find the equation of the curve which passes through the point P 1, 8. (2 8 (a) Describe the single geometrical transformation by which the curve with equation y ˆ tan 1 x can be obtained from the curve y ˆ tan x. 2 marks) (b) (c) Solve the equation tan 1 x ˆ 3 in the interval 0<x <4p, giving your answers in 2 radians to three significant figures. (4 marks) Solve the equation cos y sin y 3 cos y ˆ0 in the interval 0 < y < 2p, giving your answers in radians to three significant figures. (5 marks) END OF QUESTIONS

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