Cambridge A Level Mathematics - Further 9231 — 2005 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 2005 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 2005 [Turn over

2 1 Use the method of differences to find SN, where SN = N2 ∑ n=N 1 n(n + 1). [3] Deduce the value of lim N→∞SN. [1] 2 Given that x = t −sin t and y = 1 −cos t, where 0 < t < 2π, show that d2y dx2 = −1 4 cosec4
1 2t. [5] 3 The points A, B, C have position vectors ai, bj, ck respectively, where a, b, c are all positive. The plane containing A, B, C is denoted by Π. (i) Find a vector perpendicular to Π. [3] (ii) Find the perpendicular distance from the origin to Π, in terms of a, b, c. [3] 4 Show that the sum of the cubes of the roots of the equation x3 + λx + 1 = 0 is −3. [3] Show also that there is no real value of λ for which the sum of the fourth powers of the roots is negative. [3] 5 A curve is defined parametrically by x = t −8t 1 2 and y = 16 3 t 3 4. The arc of this curve joining the point where t = 1 to the point where t = 4 is denoted by C. (i) Show that the length of C is 11. [5] (ii) Find, correct to 3 significant figures, the area of the surface generated when C is rotated through one complete revolution about the x-axis. [3] © UCLES 2005 9231/01/M/J/05

3 6 The curve C has polar equation r = π −θ θ , where 1 2π ≤θ ≤π. (i) Draw a sketch of C. [3] (ii) Show that the area of the region bounded by the line θ = 1 2π and C is π
3 4 −ln 2. [5] 7 The planes Π1 and Π2 have equations x + 2y −3 + 4 = 0 and 2x + y −4 −3 = 0 respectively. Show that, for all values of λ, every point which is in both Π1 and Π2 is also in the plane x + 2y −3 + 4 + λ(2x + y −4 −3) = 0. [2] The planes Π1 and Π2 meet in the line l. (i) Find the equation of the plane Π3 which passes through l and the point whose position vector is ak. [3] (ii) Find the value of a if Π2 is perpendicular to Π3. [3] 8 The integral In, where n is a non-negative integer, is defined by In = 1 0 e−x(1 −x)n dx. (i) Show that In+1 = 1 −(n + 1)In. [3] (ii) Use induction to show that In is of the form An + Bne−1, where An and Bn are integers. [4] (iii) Express Bn in terms of n. [2] © UCLES 2005 9231/01/M/J/05 [Turn over

4 9 Find the eigenvalues and a corresponding set of eigenvectors of the matrix M given by M =  a 2 1 0 b −1 0 0 c  , where a, b, c are all different. [6] Find a matrix P and a diagonal matrix D such that (M −kI)n = PDP−1, where I is the identity matrix, k is a constant scalar and n is a positive integer. [4] [You are not required to evaluate P−1.] 10 (i) Write down, in any form, all the complex roots of the equation w12 = 1. [2] (ii) Explain why the equation ( + 2)12 = 12 (*) has exactly 10 non-real roots and show that they may be expressed in the form −1 −i cot
1 12kπ, where k = ±1, ±2, ±3, ±4, ±5. [6] (iii) Show that −1 −i cot
1 12kπ−1 + i cot
1 12kπ = cosec2
1 12kπ. [1] (iv) Given that the product of the roots of (*) is −512 3 , find the value of sin2
1 12π sin2
2 12π sin2
3 12π sin2
4 12π sin2
5 12π. [2] © UCLES 2005 9231/01/M/J/05

5 11 The matrix A is defined by A =  1 3 2 1 −1 −1 2 2 θ  . Find the rank of A, distinguishing between the cases θ ≠1 and θ = 1. [4] Consider the system S of equations: x + 3y + 2 = 1, x −y − = 0, 2x + 2y + θ = 3θ + φ −2. (i) Show that if θ ≠1 then S has a unique solution. Find this solution in the case φ = 0. [3] (ii) Show that if θ = 1 and φ = 0 then S has an infinite number of solutions. [3] (iii) Show that if θ = 1 and φ ≠0 then S has no solution. [2] 12 Answer only one of the following two alternatives. EITHER The curve Γ, which has equation y = ax2 + bx + c x2 + px + q , has asymptotes x = 1, x = 4 and y = 2. Find the values of a, p and q. [4] It is given that Γ has a stationary point at x = 2. (i) Find the value of c. [3] (ii) Show that if b ≠−10 then Γ has exactly 2 stationary points. [2] (iii) Draw a sketch of Γ for the case where b = −6. [4] OR It is given that d2y dx2 + (2a −1)dy dx + a(a −1)y = 2a −1 + a(a −1)x, where a is a constant. Find y in terms of a and x, given that y and dy dx are both zero when x = 0. [8] Hence show that if a > 1 then y ≈x as x →∞. [2] It is given that d2 dx2 + (2a −1)d dx + a(a −1) = ex, where the constant a is positive. Find lim x→∞e−x . [3] © UCLES 2005 9231/01/M/J/05

8 BLANK PAGE Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they 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/M/J/05