Edexcel GCE Further Pure Mathematics FP3 Advanced/Advanced Subsidiary

Centre No. Candidate No. Surname Signature Paper Reference(s) 6669/01R Edexcel GCE Further Pure Mathematics FP3 Advanced/Advanced Subsidiary Monday 24 June 2013 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink) Paper Reference 6669 01 R Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. Initial(s) Examiner s use only Team Leader s use only Question Number Blank 1 2 3 4 5 6 7 8 Instructions to Candidates In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper. Answer ALL the questions. You must write your answer to 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 paper is 75. There are 32 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 should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy. 2013 Pearson Education Ltd. Printer s Log. No. P42956A W850/R6669/57570 5/5/5/5/ *P42956A0132* Total Turn over

1. The hyperbola H has foci at (5, 0) and ( 5, 0) and directrices with equations x 9 9 = and x =. 5 5 Find a cartesian equation for H. (7) 2 *P42956A0232*

Question 1 continued Q1 (Total 7 marks) *P42956A0332* 3 Turn over

2. Two skew lines l 1 and l 2 have equations l 1 : r = (i j + k) + (4i + 3j + 2k) l 2 : r = (3i + 7j + 2k) + ( 4i + 6j + k) respectively, where and are real parameters. (a) Find a vector in the direction of the common perpendicular to l 1 and l 2 (2) (b) Find the shortest distance between these two lines. (5) 4 *P42956A0432*

Question 2 continued *P42956A0532* 5 Turn over

Question 2 continued 6 *P42956A0632*

Question 2 continued Q2 (Total 7 marks) *P42956A0732* 7 Turn over

3. The point P lies on the ellipse E with equation 2 2 x y + = 1 36 9 N is the foot of the perpendicular from point P to the line x = 8 M is the midpoint of PN. (a) Sketch the graph of the ellipse E, showing also the line x = 8 and a possible position for the line PN. (1) (b) Find an equation of the locus of M as P moves around the ellipse. (4) (c) Show that this locus is a circle and state its centre and radius. (3) 8 *P42956A0832*

Question 3 continued *P42956A0932* 9 Turn over

Question 3 continued 10 *P42956A01032*

Question 3 continued Q3 (Total 8 marks) *P42956A01132* 11 Turn over

4. 1 has vector equation r = 1 1 + 2 1 1 s 1 + t 2, 0 2 where s and t are real parameters. 1 2 by the transformation represented by the matrix T, where 2 0 3 T = 0 2 1 0 1 2 2 in the form r.n = p (9) 12 *P42956A01232*

Question 4 continued *P42956A01332* 13 Turn over

Question 4 continued 14 *P42956A01432*

Question 4 continued Q4 (Total 9 marks) *P42956A01532* 15 Turn over

5. 5 n In = x x 1 ( 2 1) 2 dx, n 0 1 (a) Prove that, for n 1, n ( 2n + 1) I = ni + 3 5 1 n n 1 (5) (b) Using the reduction formula given in part (a), find the exact value of I 2 (5) 16 *P42956A01632*

Question 5 continued *P42956A01732* 17 Turn over

Question 5 continued 18 *P42956A01832*

Question 5 continued Q5 (Total 10 marks) *P42956A01932* 19 Turn over

6. It is given that 1 2 0 is an eigenvector of the matrix A, where 4 2 3 A = 2 b 0 a 1 8 and a and b are constants. (a) Find the eigenvalue of A corresponding to the eigenvector (b) Find the values of a and b. 1 2. 0 (3) (3) (c) Find the other eigenvalues of A. (5) 20 *P42956A02032*

Question 6 continued *P42956A02132* 21 Turn over

Question 6 continued 22 *P42956A02232*

Question 6 continued Q6 (Total 11 marks) *P42956A02332* 23 Turn over

7. y O x Figure 1 The curves shown in Figure 1 have equations y = 6 cosh x and y = 9 2 sinh x (a) Using the definitions of sinh x and cosh x in terms of e x, find exact values for the x-coordinates of the two points where the curves intersect. (6) The finite region between the two curves is shown shaded in Figure 1. (b) Using calculus, find the area of the shaded region, giving your answer in the form a ln b + c, where a, b and c are integers. (6) 24 *P42956A02432*

Question 7 continued *P42956A02532* 25 Turn over

Question 7 continued 26 *P42956A02632*

Question 7 continued Q7 (Total 12 marks) *P42956A02732* 27 Turn over

8. y C O 1 8 Figure 2 x The curve C, shown in Figure 2, has equation y = 2x 1 2, 1 x 8 (a) Show that the length s of curve C is given by the equation s = 1 8 1 1 + dx x (2) (b) Using the substitution x = sinh 2 u, or otherwise, find an exact value for s. Give your answer in the form a 2 + ln(b + c 2) where a, b and c are integers. (9) 28 *P42956A02832*

Question 8 continued *P42956A02932* 29 Turn over

Question 8 continued 30 *P42956A03032*

Question 8 continued *P42956A03132* 31 Turn over

Question 8 continued Q8 (Total 11 marks) TOTAL FOR PAPER: 75 MARKS END 32 *P42956A03232*