Paper Reference. Pure Mathematics P2 Advanced/Advanced Subsidiary. Monday 20 June 2005 Morning Time: 1 hour 30 minutes. Mathematical Formulae (Lilac)

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1 Centre No. Candidate No. Paper Reference(s) 6672/01 Edexcel GCE Pure Mathematics P2 Advanced/Advanced Subsidiary Monday 20 June 2005 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Lilac) Paper Reference Surname Signature Items included with question papers Nil Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP 48G. Initial(s) Examiner s use only Team Leader s use only Question Number Blank Instructions to Candidates In the boxes above, write your centre number, candidate number, your surname, initial(s) and signature. Check that you have the correct question paper. You must write your answer for each question in the space following the question. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 8 questions in this question paper. The total mark for this question paper is 75. There are 24 pages in this question paper. Any pages are indicated. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit. This publication may be reproduced only in accordance with Edexcel Limited copyright policy Edexcel Limited. Printer s Log. No. N21140A W850/R6672/ /7/3/3/3/3/47,300 *N21140A0124* Total Turn over

2 1. Solve (a) 5 x = 8, giving your answer to 3 significant figures, (b) log 2 (x + 1) log 2 x = log 2 7. (3) (3) 2 *N21140A0224*

3 Question 1 continued Q1 (Total 6 marks) *N21140A0324* 3 Turn over

4 2. (a) Write down the first three terms, in ascending powers of x, of the binomial expansion of (1 + px) 12, where p is a non-zero constant. (2) Given that, in the expansion of (1 + px) 12, the coefficient of x is ( q) and the coefficient of x 2 is 11q, (b) find the value of p and the value of q. (4) 4 *N21140A0424*

5 Question 2 continued Q2 (Total 6 marks) *N21140A0524* 5 Turn over

6 3. A river, running between parallel banks, is 20 m wide. The depth, y metres, of the river measured at a point x metres from one bank, is given by the formula 1 y = x (20 x), 0 x (a) Complete the table below, giving values of y to 3 decimal places. x y (b) Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. (4) Given that the cross-sectional area is constant and that the river is flowing uniformly at 2 m s 1, (c) estimate, in m 3, the volume of water flowing per minute, giving your answer to 3 significant figures. (2) (2) 6 *N21140A0624*

7 Question 3 continued Q3 (Total 8 marks) *N21140A0724* 7 Turn over

8 4. The function f is defined by (a) Show that 2 f( x) =, x> 1. x 1 5x f: x, x> x x x (4) (b) Find f 1 (x). (3) The function g is defined by g:x x 2 + 5, x R. 1 (c) Solve fg(x) =. 4 (3) 8 *N21140A0824*

9 Question 4 continued Q4 (Total 10 marks) *N21140A0924* 9 Turn over

10 5. Figure 1 y C (1, 2) y = x + 1 x O 1 3 x x + 1 Figure 1 shows part of the curve C with equation y =, x> 0. x The finite region enclosed by C, the lines x = 1, x = 3, and the x-axis is rotated through 360 about the x-axis to generate a solid S. (a) Using integration, find the exact volume of S. (7) Figure 2 y T C (1, 2) R y = x + 1 x O 1 3 x The tangent T to C at the point (1, 2) meets the x-axis at the point (3, 0). The shaded region R is bounded by C, the line x = 3 and T, as shown in Figure 2. (b) Using your answer to part (a), find the exact volume generated by R when it is rotated through 360 about the x-axis. (3) 10 *N21140A01024*

11 Question 5 continued *N21140A01124* 11 Turn over

12 Question 5 continued 12 *N21140A01224*

13 Question 5 continued Q5 (Total 10 marks) *N21140A01324* 13 Turn over

14 6. f(x) = 3e x 1 ln x 2, x > 0. 2 (a) Differentiate to find f (x). (3) The curve with equation y = f(x) has a turning point P. The x-coordinate of P is α. 1 (b) Show that α = e α. 6 (2) The iterative formula xn is used to find an approximate value for α. = 1 e x n, 1, x = 0 (c) Calculate the values of x 1, x 2, x 3 and x 4, giving your answers to 4 decimal places. (2) (d) By considering the change of sign of f (x) in a suitable interval, prove that α = correct to 4 decimal places. (2) 14 *N21140A01424*

15 Question 6 continued Q6 (Total 9 marks) *N21140A01524* 15 Turn over

16 7. Figure 3 y 1 O 3 b (1, a) x Figure 3 shows part of the graph of y = f(x), x R. The graph consists of two line segments that meet at the point (1, a), a < 0. One line meets the x-axis at (3, 0). The other line meets the x-axis at ( 1, 0) and the y-axis at (0, b), b < 0. In separate diagrams, sketch the graph with equation (a) y = f(x + 1), (b) y = f( x ). (2) (3) Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that f(x) = x 1 2, find (c) the value of a and the value of b, (d) the value of x for which f(x) = 5x. (2) (4) 16 *N21140A01624*

17 Question 7 continued *N21140A01724* 17 Turn over

18 Question 7 continued 18 *N21140A01824*

19 Question 7 continued Q7 (Total 11 marks) *N21140A01924* 19 Turn over

20 8. (a) Given that 2 sin(θ + 30) = cos(θ + 60), find the exact value of tan θ. (5) (b) (i) Using the identity cos(a + B) cosa cosb sin A sinb, prove that cos 2A 1 2 sin 2 A. (2) (ii) Hence solve, for 0 x < 2π, cos 2x = sin x, giving your answers in terms of π. 1 (iii) Show that sin 2y tan y + cos 2y 1, for 0 y < π. 2 (5) (3) 20 *N21140A02024*

21 Question 8 continued *N21140A02124* 21 Turn over

22 Question 8 continued 22 *N21140A02224*

23 Question 8 continued Q8 (Total 15 marks) TOTAL FOR PAPER: 75 MARKS END *N21140A02324* 23

24 BLANK PAGE 24 *N21140A02424*