UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE General Certificate of Education Advanced Level FURTHER MATHEMATICS 9221/1 PAPER 1

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1 ,--, ~"\, \ "~. "',,~ \ ~ 1.f8P;)1!,'" ;i *.~ UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE General Certificate of Education Advanced Level FURTHER MATHEMATICS 9221/1 PAPER 1 MAY/JUNE SESSION hours Additional materials: Answer paper Graph paper List of Formulae TIME 3 hours INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate paper/answer booklet. There is no restriction on the number of questions which you may attempt. number in the spaces provided on the answer If a numerical answer cannot be given exactly, and the accuracy required is not specified in the question, then in the case of an angle it should be given to the nearest degree, and in other cases it should be given correct to 2 significant figures. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 120. Questions are printed in the order of their mark allocations and candidates are advised to attempt questions sequentially. The use of an electronic calculator is expected, where appropriate. You are reminded of the need for clear presentation in your answers. This question paper consists of 5 printed pages and 3 blank UCLES 2001 [Turn over

2 2 1 Solve the equation 2z3- l1z2 + 20z + 13 = 0, given that one root is 3 + 2i. [4] 2 Given that y = sinh-1x -tanh-1x ( -1 < x < 1), dy show that ~ = for only one value of x. [4] 3 Show that ~ ~+1 an[1 + (1 -a)n] -;;- -m = n(n + 1) [1] Find N ~ an[1 + (1 -a)n] ~ n(n+ 1) in terms of a and N. [2] Deduce that ~ an[1 + (1 -a)n] ~ n(n+ 1) n=l is convergent if -1 < a < 1, and state the sum to infinity. [2] 4 Given that prove that J 2 1 I n =,xnex dx, 1 1 (n + 1)ln = 2n+1e2 -e + In-1. [3] Hence find 12 in terms of e and 10. [2] 5 The sequence Ul' U2' U3'..., where U1 = 1, is such that for n ~ 1, u -n 6u +4 n+1 -U + 3. n By expressing 4- un+1 in terms of Un' prove by induction that Un < 4 for n ~ 1. [5]

3 3 6 Use de Moivre's theorem to show that cos 58 = 16coss 8-20cos cos 8. [4] By replacing 8 by!1t -8, deduce an expression for sin 58 in terms of sin 8. [1] Given now that deduce that, when 8 = ~1t, T = cosn 8 + sinn 8, n "ccc""'c-"cc,'~' 16T S -20T 3 + 5Tl =!(1 +..;3). [2] 7 The point A (2, 1) is on the curve whose equation is t",-;' 3(x + 2y)2 + 16(y -2.x) = 0. (i) Show that ~ = 1 ata. [4] (ii) Find the value of ~ ata. [4] 8 (i) Find I: sinh 2.x dx in terms of e. [3] (ii) Find I tanh!x dx. [2] (iii) Find I cosh2 x dx. [3] 9 The curve r is defined parametrically by x=t-int, y=4(..;t- 1), for t > 0. The points A and B on r correspond to t = 1 and t = 4 respectively. (i) Show that the length of the arc AB of r is n 2. [5] (ii) The arc AB is rotated through one revolution about the x-axis. Show that the area of the surface generated is ~1t(11-61n 2). [5]. 9221/1/M/J/OI [Thrn over

4 4 10 The curve C has equation y = f(x), where f, 27x+45 (x) = (x- 1)(x + 2)2' and A is the point (-1, -9) on C. (i) Express f(x) in partial fractions. [5] (ii) Hence show that, at A, ~ = O and ~ = 0. [4] (iii) It is given that when x is just less than -1, ~ is positive, and that when x is just greater than -1, ~ is negative. Draw a sketch to show the shape of C in the neighbourhood of A. [2] 11 The points A and B have position vectors 2i -9j -5k and 10i -3j -k respectively, relative to an origin 0. The line i passes through A and B. The point C on i is such that OC is perpendicular to 1. (i) Obtain a vector equation of 1. [2] (ii) Find the length of OC. [4] (iii) The point D has position vector i -2j -2k. Find the length of the perpendicular from B to the plane containing A, D and 0. [5] 12 It is given that (-1 3 a ) ( 1) L = and b = a -16 Show that the equation Lx = b (A) has a unique solution if a:#: -2. [3] Show also that (A) has an infinite number of solutions if a = -2, and give a geometrical interpretation of (A) in this case. [3] It is also given that M is a non-singular 3 x 3 matrix, and that c = Mb. Show that any solution of (A) is a solution of MLx = c, and that any solution of (B) is a solution of (A). [4] (B) Show also that the equation LMx=c does not have a unique solution if a = -2. [2]

5 5 13 The multiplicative group G has the eight elements e, a, a2, a3, b, ab, a2b, a3 b, where e is the identity element, a4 = b2 = e and ba = a3 b. (i) Prove that ba2 = a2b and that ba3 = ab. [4] (ii) Show that baba2ba3 = a2b. [2] (iii) Find the order ofa2b. [2] (iv) Find two subgroups of G of order 4. [3]!1fki (v) The group H, of order 8, has elements e4, where k = 0, 1, 2,..., 7. The group operation for H is complex multiplication. Determine whether G and H are isomorphic and justify your conclusion. [2] 14 The future performance of an investment fund is modelled by the differential equation dx -!r -+tx=e 2 dt, where x billion dollars is the magnitude of the fund at time t years. Given that x = 1 when t = 0, find x in terms of t. [5] An alternative model of future fund performance is 5dt1" d2y + 15dt dy + 10y = 3 cos 2t -sin 2t, where y = x Solve this differential equation to obtain x in terms of t, assuming that the values dx of x and dt at t = are the same for both models. [9] Describe the predicted long-term performance of the fund in the case of each model. [3]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS General Certificate of Education Advanced Level FURTHER MATHEMATICS 9231/01 Paper 1 Additional materials: Answer Booklet/Paper Graph paper List of Formulae