Cambridge A Level Mathematics - Further 9231 — 2007 May/June Paper 1 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level FURTHER MATHEMATICS 9231/01 Paper 1 May/June 2007 3 hours Additional Materials: Answer Booklet/Paper Graph Paper List of Formulae (MF10) READ THESE INSTRUCTIONS FIRST If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet. Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use a soft pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. The use of a calculator is expected, where appropriate. Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive credit. You are reminded of the need for clear presentation in your answers. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. This document consists of 5 printed pages and 3 blank pages. ©UCLES 2007 [Turn over *0222815537*

2 1 Verify that 1 n2 + 1 − 1 (n + 1)2 + 1 = 2n + 1 (n2 + 1)(n2 + 2n + 2). [1] Use the method of differences to show that, for all N ≥1, N ∑ n=1 2n + 1 (n2 + 1)(n2 + 2n + 2) < 1 2. [3] Write down the value of ∞ ∑ n=1 2n + 1 (n2 + 1)(n2 + 2n + 2). [1] 2 The curve C is defined parametrically by x = t −ln t, y = 4t 1 2, where t > 0. The arc of C joining the point where t = 1 to the point where t = 4 is rotated through one complete revolution about the x-axis. Show that the area of the surface generated is 160 3 π. [6] 3 Find the general solution of the differential equation d2y dx2 + 4dy dx + 29y = 58x + 37. [6] 4 Given that the variables x and y are related by y = x + e−xy, find the value of d2y dx2 when x = 0. [6] © UCLES 2007 9231/01/M/J/07

3 5 The diagram shows the curve C with polar equation r = eθ, where 0 ≤θ ≤1 2π. Find the maximum distance of a point of C from the line θ = 1 2π, giving the answer in exact form. [4] Find the area of the finite region bounded by C and the lines θ = 0 and θ = 1 2π, giving the answer in exact form. [3] 6 The matrix A, given by A =
7 −4 6 2 2 2 −3 4 −2  , has eigenvalues 1, 2, 4. Find a set of corresponding eigenvectors. [4] Hence find the eigenvalues of B, where B =
10 −4 6 2 5 2 −3 4 1  , and state a set of corresponding eigenvectors. [3] 7 The equation x3 + 3x −1 = 0 has roots α, β, γ . Use the substitution y = x3 to show that the equation whose roots are α3, β3, γ 3 is y3 −3y2 + 30y −1 = 0. [2] Find the value of α9 + β9 + γ 9. [5] 8 The sequence x1, x2, x3, . . . is such that x1 = 1 and xn+1 = 1 + 4xn 5 + 2xn . Prove by induction that xn > 1 2 for all n ≥1. [5] Prove also that xn > xn+1 for all n ≥1. [3] © UCLES 2007 9231/01/M/J/07 [Turn over

4 9 Let In =  1 0 1 (4 −x2)n dx, for n = 1, 2, 3, . . . . Verify that d dx x (4 −x2)n = 1 −2n (4 −x2)n + 8n (4 −x2)n+1 and hence prove that 8nIn+1 = (2n −1)In + 1 3n. [5] Find the y-coordinate of the centroid of the region bounded by the axes, the line x = 1 and the curve y = 1 4 −x2 , giving your answer correct to 3 decimal places. [5] 10 The line l1 passes through the points P and Q whose position vectors are i −j −2k and −2i + 5j + 13k respectively. The line l2 passes through the point S whose position vector is i −2j + 8k and is parallel to the vector i −j −3k. The point whose position vector is −i + 3j + 2k is on the line l3, the common perpendicular to l1 and l2. (i) Find, in the form r = a + tb, an equation for l3. [3] (ii) Find the perpendicular distance from S to l3. [4] (iii) Find the perpendicular distance from S to the plane which contains l3 and passes through P. [4] 11 (a) Use de Moivre’s theorem to show that sin 8θ can be expressed in the form sin θ cos θ(a sin6 θ + b sin4 θ + c sin2θ + d), where the value of the constant a is to be found and b, c, d are constants whose values need not be found. [6] (b) Use de Moivre’s theorem to show that N ∑ n=1 sin nθ 2n = 2N+1 sin θ + sin Nθ −2 sin(N + 1)θ 2N(5 −4 cos θ) . [7] © UCLES 2007 9231/01/M/J/07

5 12 Answer only one of the following two alternatives. EITHER The curve C has equation y = λx + x x + 2, where λ is a non-zero constant. (i) Find the asymptotes of C. [3] (ii) Show that if λ > 0 then dy dx > 0 at all points of C. [2] (iii) Show that, for λ < −1 2, C has two distinct stationary points, both to the left of the y-axis. [3] (iv) In separate diagrams draw sketches of C for each of the cases λ > 0 and λ < −1 2. [6] OR The linear transformation T : 4 →4 is represented by the matrix M = ⎛⎜⎜⎝ 1 −2 2 4 2 −4 5 9 3 −6 8 14 5 −10 12 22 ⎞⎟⎟⎠. (i) Find the rank of M. [3] (ii) Obtain a basis for the null space, K, of T. [3] (iii) Evaluate M ⎛⎜⎜⎝ −1 2 −3 4 ⎞⎟⎟⎠, and hence show that any solution of Mx = ⎛⎜⎜⎝ 5 11 17 27 ⎞⎟⎟⎠ (∗) has the form ⎛⎜⎜⎝ −1 2 −3 4 ⎞⎟⎟⎠+ λe1 + µe2, where λ and µ are constants and {e1, e2} is a basis for K. [3] (iv) Find the solution x1 of (∗) such that the first component of x1 is A, and the sum of all the components of x1 is B. [5] © UCLES 2007 9231/01/M/J/07

8 BLANK PAGE Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 9231/01/M/J/07