Edexcel GCE Further Pure Mathematics FP3 Advanced/Advanced Subsidiary

Transcription

1 Centre No. Candidate No. Surname Signature Paper Reference(s) 6669/01 Edexcel GCE Further Pure Mathematics FP3 Advanced/Advanced Subsidiary Friday 24 June 2011 Afternoon Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink) Paper Reference 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 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 28 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 Edexcel Limited copyright policy Edexcel Limited. Printer s Log. No. P35414A W850/R6669/ /5/3/4 *P35414A0128* Total Turn over

2 3 1. The curve C has equation y = 2 x, 0 x 2. The curve C is rotated through 2 radians about the x-axis. Using calculus, find the area of the surface generated, giving your answer to 3 significant figures. (5) 2 *P35414A0228*

3 Question 1 continued Q1 (Total 5 marks) *P35414A0328* 3 Turn over

4 2. (a) Given that y = xarcsin x, 0 x 1, find (i) an expression for d y dx, (ii) the exact value of d y dx when x = 1. 2 (3) (b) Given that 2x y = arctan(3e ), show that dy 3 = dx 5cosh 2x+ 4sinh 2x (5) 4 *P35414A0428*

5 Question 2 continued Q2 (Total 8 marks) *P35414A0528* 5 Turn over

6 3. Show that 8 1 (a) dx= kπ, giving the value of the fraction k, 2 5 x 10x+ 34 (5) (b) ( x 10x+ 34) dx= ln( A+ n), giving the values of the integers A and n. (4) 6 *P35414A0628*

7 Question 3 continued Q3 (Total 9 marks) *P35414A0728* 7 Turn over

8 4. (a) Prove that, for n 1, n In = e 2 x (ln x) d x, n 0 1 I n n = I n 3 3 e 3 1 (4) (b) Find the exact value of I 3. (4) 8 *P35414A0828*

9 Question 4 continued *P35414A0928* 9 Turn over

10 Question 4 continued 10 *P35414A01028*

11 Question 4 continued Q4 (Total 8 marks) *P35414A01128* 11 Turn over

12 2x 5. The curve C 1 has equation y = 3sinh2 x, and the curve C 2 has equation y = 13 3e. (a) Sketch the graph of the curves C 1 and C 2 on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. (4) (b) Solve the equation k is an integer. 2x 3sinh e, x = giving your answer in the form 1 2 ln k, where (5) 12 *P35414A01228*

13 Question 5 continued *P35414A01328* 13 Turn over

14 Question 5 continued 14 *P35414A01428*

15 Question 5 continued Q5 (Total 9 marks) *P35414A01528* 15 Turn over

16 6. The plane P has equation r = 1 + λ 2 + μ (a) Find a vector perpendicular to the plane P. (2) The line l passes through the point A (1, 3, 3) and meets P at (3, 1, 2). The acute angle between the plane P and the line l is. (b) Find to the nearest degree. (4) (c) Find the perpendicular distance from A to the plane P. (4) 16 *P35414A01628*

17 Question 6 continued *P35414A01728* 17 Turn over

18 Question 6 continued 18 *P35414A01828*

19 Question 6 continued Q6 (Total 10 marks) *P35414A01928* 19 Turn over

20 7. The matrix M is given by k 1 1 M = 1 0 1, k 1 (a) Show that det M = 2 2k. (b) Find M 1, in terms of k. (2) (5) The straight line l 1 is mapped onto the straight line l 2 by the transformation represented by the matrix The equation of l 2 is ( r a) b = 0, where a= 4i+ j+ 7k and b= 4i+ j+ 3 k. (c) Find a vector equation for the line l 1. (5) 20 *P35414A02028*

21 Question 7 continued *P35414A02128* 21 Turn over

22 Question 7 continued 22 *P35414A02228*

23 Question 7 continued Q7 (Total 12 marks) *P35414A02328* 23 Turn over

24 8. The hyperbola H has equation x a y b 2 2 = (a) Use calculus to show that the equation of the tangent to H at the point ( acosh θ, bsinh θ ) may be written in the form xb coshθ ya sinhθ = ab (4) The line l 1 is the tangent to H at the point ( acosh θ, bsinh θ ), θ. Given that l 1 meets the x-axis at the point P, (b) find, in terms of a and, the coordinates of P. (2) The line l 2 is the tangent to H at the point (a, 0). Given that l 1 and l 2 meet at the point Q, (c) find, in terms of a, b and, the coordinates of Q. (2) (d) Show that, as varies, the locus of the mid-point of PQ has equation x(4 y + b ) = ab (6) 24 *P35414A02428*

25 Question 8 continued *P35414A02528* 25 Turn over

26 Question 8 continued 26 *P35414A02628*

27 Question 8 continued *P35414A02728* 27 Turn over

28 Question 8 continued Q8 END (Total 14 marks) TOTAL FOR PAPER: 75 MARKS 28 *P35414A02828*