Cambridge A Level Mathematics - Further 9231 — 2004 Oct/Nov Paper 1 · Variant 1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level FURTHER MATHEMATICS 9231/01 Paper 1 October/November 2004 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 on both sides of the paper. 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. 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. 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. This document consists of 5 printed pages and 3 blank pages. © UCLES 2004 [Turn over

2 1 The linear transformation T :
4 →
4 is represented by the matrix  1 5 2 6 2 0 −1 7 3 −1 −2 10 4 10 13 29 . Find the dimension of the null space of T. [4] 2 The curve C is defined parametrically by x = a cos3 t, y = a sin3 t, 0 ≤t ≤1 2π, where a is a positive constant. Find the area of the surface generated when C is rotated through one complete revolution about the x-axis. [5] 3 Given that α + β + γ = 0, α2 + β2 + γ 2 = 14, α3 + β3 + γ 3 = −18, find a cubic equation whose roots are α, β, γ. [4] Hence find possible values for α, β, γ . [2] 4 The curve C has polar equation r = e 1 5θ, 0 ≤θ ≤3 2π. (i) Draw a sketch of C. [2] (ii) Find the length of C, correct to 3 significant figures. [4] 5 Let SN = N ∑ n=1 (−1)n−1n3. Find S2N in terms of N, simplifying your answer as far as possible. [4] Hence write down an expression for S2N+1 and find the limit, as N →∞, of S2N+1 N3 . [3] 6 Write down all the 8th roots of unity. [2] Verify that  −eiθ −e−iθ ≡ 2 −(2 cos θ) + 1. [1] Hence express 8 −1 as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form. [5] © UCLES 2004 9231/01/O/N/04

3 7 The curve C has equation xy + (x + y)5 = 1. (i) Show that dy dx = −5 6 at the point A (1, 0) on C. [3] (ii) Find the value of d2y dx2 at A. [5] 8 The sequence of real numbers a1, a2, a3, . . . is such that a1 = 1 and an+1 = an + 1 an λ , where λ is a constant greater than 1. Prove by mathematical induction that, for n ≥2, an ≥2g(n), where g(n) = λ n−1. [6] Prove also that, for n ≥2, an+1 an > 2(λ−1)g(n). [3] 9 It is given that In =  1 0 (1 + x3)−n dx, where n > 0. (i) Show that d dx x(1 + x3)−n = −(3n −1)(1 + x3)−n + 3n(1 + x3)−n−1, and hence, or otherwise, show that In+1 = 2−n 3n + 1 −1 3n In. [5] (ii) By considering the graph of y = 1 1 + x3 , show that I1 < 1. [2] (iii) Deduce that I3 < 53 72. [3] © UCLES 2004 9231/01/O/N/04 [Turn over

4 10 The curve C has equation y = x2 + 2x −3 (λx + 1)(x + 4), where λ is a constant. (i) Find the equations of the asymptotes of C for the case where λ = 0. [4] (ii) Find the equations of the asymptotes of C for the case where λ is not equal to any of −1, 0, 1 4, 1 3. [3] (iii) Sketch C for the case where λ = −1. Show, on your diagram, the equations of the asymptotes and the coordinates of the points of intersection of C with the coordinate axes. [4] 11 The line l1 passes through the point A, whose position vector is 3i −5j −4k, and is parallel to the vector 3i + 4j + 2k. The line l2 passes through the point B, whose position vector is 2i + 3j + 5k, and is parallel to the vector i −j −4k. The point P on l1 and the point Q on l2 are such that PQ is perpendicular to both l1 and l2. The plane Π1 contains PQ and l1, and the plane Π2 contains PQ and l2. (i) Find the length of PQ. [4] (ii) Find a vector perpendicular to Π1. [2] (iii) Find the perpendicular distance from B to Π1. [3] (iv) Find the angle between Π1 and Π2. [3] © UCLES 2004 9231/01/O/N/04

5 12 Answer only one of the following two alternatives. EITHER The variable y depends on x, and the variables x and t are related by x = et. Show that xdy dx = dy dt and x2d2y dx2 = d2y dt2 −dy dt . [5] (i) Given that y satisfies the differential equation 4x2 d2y dx2 + 16xdy dx + 25y = 50(ln x) −1, find a differential equation involving only t and y. [2] (ii) Show that the complementary function of the differential equation in t and y may be written in the form Re −3 2t sin(2t + φ), where R and φ are arbitrary constants. [3] (iii) Find a particular integral of the differential equation in t and y. [3] (iv) Hence find the general solution of the differential equation in x and y. [1] OR The matrix A has λ as an eigenvalue with e as a corresponding eigenvector. Show that if A is non-singular then (i) λ ≠0, [2] (ii) the matrix A−1 has λ −1 as an eigenvalue with e as a corresponding eigenvector. [2] The matrices A and B are given by A = 1 −1 2 0 −2 4 0 0 −3  and B = (A + 4I)−1. Find a non-singular matrix P, and a diagonal matrix D, such that B = PDP−1. [10] © UCLES 2004 9231/01/O/N/04

8 BLANK PAGE Every reasonable effort has been made to trace all copyright holders. The publishers would be pleased to hear from anyone whose rights we have unwittingly infringed. University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 9231/01/O/N/04