Surname. Other Names. Centre Number. Candidate Number. Candidate Signature. General Certificate of Education Advanced Level Examination June 2014

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Surname Other Names Centre Number Candidate Number Candidate Signature Mathematics Unit Pure Core 3 MPC3 General Certificate of Education Advanced Level Examination June 2014 Leave blank Tuesday 10

1 Surname Other Names Centre Number Candidate Number Candidate Signature Mathematics Unit Pure Core 3 MPC3 General Certificate of Education Advanced Level Examination June 2014 Leave blank Tuesday 10 June am to am For this paper you must have: * the blue AQA booklet of formulae and statistical tables. You may use a graphics calculator. TIME ALLOWED * 1 hour 30 minutes At the top of the page, write your surname and other names, your centre number, your candidate number and add your signature. [Turn over] P82792/Jun14/E1

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3 INSTRUCTIONS * Use black ink or black ball-point pen. Pencil should only be used for drawing. * Answer ALL questions. * Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. * You must answer each question in the space provided for that question. If you require extra space, use an AQA supplementary answer book; do NOT use the space provided for a different question. * Show all necessary working; otherwise marks for method may be lost. * Do all rough work in this book. Cross through any work that you do not want to be marked. INFORMATION * The marks for questions are shown in brackets. * The maximum mark for this paper is 75. ADVICE * Unless stated otherwise, you may quote formulae, without proof, from the booklet. * You do not necessarily need to use all the space provided. 3 DO NOT TURN OVER UNTIL TOLD TO DO SO

4 Answer ALL questions. 4 Answer each question in the space provided for that question. 1 Use Simpson s rule, with five ordinates (four strips), to calculate an estimate for ð p 0 x 1 2 sin x dx Give your answer to four significant figures. [4 marks]

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8 2 A curve has equation y = 2 ln (2e x). (a) Find dy dx. [2 marks] (b) Find an equation of the normal to the curve y = 2 ln (2e x) at the point on the curve where x =e. [4 marks] (c) The curve y = 2 ln (2e x) intersects the line y = x at a single point, where x = a. (i) Show that a lies between 1 and 3. [2 marks] (ii) Use the recurrence relation x n þ 1 = 2 ln (2e x n ) with x 1 = 1 to find the values of x 2 and x 3, giving your answers to three decimal places. [2 marks] (iii) FIGURE 1, on page 11, shows a sketch of parts of the graphs of y = 2 ln (2e x) and y = x, and the position of x 1. On FIGURE 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of x 2 and x 3 on the x-axis. [2 marks] 8

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11 11 (c)(iii) FIGURE 1 y y = x y = 2 ln (2e x) O x 1 x [Turn over]

12 3 (a) (i) Differentiate (x ) 2 with respect to x. [2 marks] (ii) Given that y =e 2x (x ) 2, find the value of dy when x =0. dx [3 marks] 12 (b) A curve has equation y = 4x 3. Use the quotient x 2 +1 rule to find the x-coordinates of the stationary points of the curve. [5 marks]

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16 4 The sketch shows part of the curve with equation y =f(x). y 16 3 O 2 x P(4, 3) (a) On FIGURE 2 on page 17, sketch the curve with equation y = jf(x)jj. [3 marks] (b) On FIGURE 3 on page 17, sketch the curve with equation y =f(j2xj j j). j [2 marks] (c) (i) Describe a sequence of two geometrical transformations that maps the graph of y =f(x) onto the graph of y = f(2x +2). [4 marks] (ii) Find the coordinates of the image of the point P(4, 3) under the sequence of transformations given in part (c)(i). [2 marks]

17 (a) FIGURE 2 y 17 O x (b) FIGURE 3 y O x [Turn over]

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20 20 5 The functions f and g are defined with their respective domains by f(x)=x 2 6x + 5, for x 5 3 g(x)=jx j 6j, j for all real values of x (a) Find the range of f. [2 marks] (b) The inverse of f is f 1. Find f 1 (x). Give your answer in its simplest form. [4 marks] (c) (i) Find gf(x). [1 mark] (ii) Solve the equation gf(x)= 6. [4 marks]

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24 24 6 (a) By using integration by parts twice, find ð x 2 sin 2x dx [6 marks] p (b) A curve has equation y = x ffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2x, for 0 4 x 4 p 2. The region bounded by the curve and the x-axis is rotated through 2p radians about the x-axis to generate a solid. Find the exact value of the volume of the solid generated. [3 marks]

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28 7 Use the substitution u =3 x 3 to find the exact value of ð 1 0 x 5 28 dx. [6 marks] 3 x 3

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32 8 (a) Show that the expression 1 sin x cos x + cos x 1 sin x can be written as 2 sec x. [4 marks] (b) Hence solve the equation 1 sin x cos x + cos x 1 sin x = tan2 x 2 giving the values of x to the nearest degree in the interval 0 o 4 x<360 o. [6 marks] 32 (c) Hence solve the equation 1 sin(2u 30 o ) cos (2u 30 o ) + cos (2u 30 o ) 1 sin(2u 30 o ) = tan2 (2u 30 o ) 2 giving the values of u to the nearest degree in the interval 0 o 4 u o. [2 marks]

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39 39 END OF QUESTIONS

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Mathematics MPC3 (JUN14MPC301) General Certificate of Education Advanced Level Examination June Unit Pure Core TOTAL

Mathematics MPC3 (JUN14MPC301) General Certificate of Education Advanced Level Examination June Unit Pure Core TOTAL Centre Number Candidate Number For Examiner s Use Surname Other Names Candidate Signature Examiner s Initials Mathematics Unit Pure Core 3 Tuesday 10 June 2014 General Certificate of Education Advanced