2007 Paper II - IIlIIlIlIIII I 002% RECOGNISING ACHIEVEMENT...

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Unformatted text preview: IIlIIlIlIIII I 002% RECOGNISING ACHIEVEMENT Sixth Term Examination Papers administered on behalf of the Cambridge Colleges MATHEMATICS II FRIDAY 29 JUNE 2007 Additional materials: Answer paper Graph paper Formulae booklet Candidates may not use electronic calculators 9470 Morning Time: 3 hours (“geld/ans); llIIIIIIIIl T I Ti III INSTRUCTIONS TO CANDIDATES ' Write your name, Centre number and candidate number in the spaces on the answer paper/ answer booklet. 0 Begin each answer on a new page. INFORMATION FOR CANDIDATES - Each question is marked out of 20. There is no restriction of choice. 0 You will be assessed on the six questions for which you gain the highest marks. - You are advised to concentrate on no more than six questions. Little credit will be given to fragmentary answers. - You are provided with Mathematical Formulae and Tables. - Electronic calculators are not permitted. This document consists of 8 printed pages. SP (SLM) T45710/1 © OCR 2007 OCR is an exempt Charity [Turn over © OCH 2007 2 Section A: Pure Mathematics In this question, you are not required to justify the accuracy of the approximations. 1 k: 5 (i) Write down the binomial expansion of <1 —l~ in ascending powers of k, up to and including the term. (a) Use the value k: = 8 to find an apprexin'iation to five decimal places for (b) By choosing a suitable integer value of 1;}, find an approximation to five decimal places for 1 (ii) By considering the first two terms of the binomial expans1on of <1 + , show that 3029 . . « 3 ‘ is an approxnnation to 2100 A curve has equation y = 2:33 — (m2 + It has a maximum point at (p, m) and a minimum point at (q, n) where p > 0 and n > 0. Let R be the region enclosed by the curve7 the line :1? = p and the line 1/ = n. (i) Express b and c in terms of p and q. (ii) Sketch the curve. Mark on your sketch the point of inflection and shade the region R. Describe the symmetry of the curve. (iii) Show that m — n 2: (q —— 1))3. (iv) Show that the area of R is %(q — 1))4. ' 1 1 :1; By writing 17 = a, tan 6, show that, for a, # O, / 2—3 (11? = — arctai‘i — + constant . a, + :17 a, a (i) Letlz/z 0 1 + sm LI} (a) Evaluate I. 1 1 _ t2 (b) Use the substitution t = tan to show that I . ~——-—dt=l 0 1+6t2+t4 2 1 1__t2 " E‘,l t it. (11) vatuae/U 1+14t2+t4( 9470 Jun()7 3 4 Given that cos/l, cos B and [3’ are non—zero7 show that the equation a sin(A — B) + [3 cos(A + B) = ysin(A + B) reduces to the form (tanA — m)(tan B — n) : 0, where m and n are independent of A and B , if and only if 042 2 [32 + 72. Determine all values of :17, in the range () g :L‘ < 27?, for which: (i) 2sin(;I; —~ fir) + flcosfir + 211%) :- sin(:v + in); (ii) 25in(;z: ~ éw) + eos(.7; + = sin(:r: + ; (iii) 2 sin(a: + + cos(3:c) 2 sin(3;1:) . 5 In this question, f2 denotes f(f(m)), denotes , and so on. (i) The function f is defined, for :1: 7E :l:l/\/§, by (INF/5 1—\/§IL'. Find by direct calculation f2 and f3(:r), and determine f2007(w) . f(:17) 2 ii Show that f" :1; 2 tan 6 + lmr , where :1; = taut) and n is any positive integer. 3 (iii) The function is defined, for [t] g 1 by g(t) = gt + ~12—x/1 — t2. Find an expression for g“ for any positive integer n. 6 Differentiate ln + V 3 + 51:2 ) and :13\/3 + a“? and simplify your answers. Hence find I v3 + 1:2 (111‘. (ii) Find the two solutions of the differential equation that satisfy y = 0 when a: : 1. © OCR 2007 9470 Jun07 [Turn over © OCR 2007 4 A function f( is said to be concave on some interval if f” < O in that interval. Show that sinw is concave for 0 < :1: < 7r and that 1111‘ is concave for 1* > 0. Let be concave on a given interval and let 51:1, 51:2, .. inequality states that 1 ’n , 1 ” mm» «(h-Zn) ‘ 13:1 19:1 ., In lie in the interval. Jensen ’5 and that equality holds if and only if 331 = x2 = = 1:”. You may use this result without proving it. (i) Given that A, B and C are angles of a triangle, show that ‘ S sinA+sinB+sinC< ii B Choosing a suitable function f, prove that y ntlfz...f < t1+t2+“'+tn V J In \ mm TL for any positive integer 'n and for any positive numbers t1, t2, . . ., tn. Hence: (a) show that m4 + y4 + 2'4 + 16 2 8atyz7 where ac, y and z are any positive numbers; (b) find the minimum value of $5 + y5 + Z5 — 5.73342, where as, y and z are any positive numbers. The points B and C have position vectors b and c, respectively, relative to the origin A7 and A, B and C are not collinear. (i) The point X has position vector Sb + to. Describe the locus of X when 8 + t = 1. (ii) The point P has position vector [3b + 70, where B and 7 are non—zero, and ,8 + *y 75 1. The line AP cuts the line BC at D. Show that BD : D0 = *y : ,8. (iii) The line BP cuts the line CA at E, and the line CP cuts the line AB at F. Show that AFXBDXCE_1 FB DC EA_' 9470 In n07 10 © OCR 2007 9470 Juulfl Section B: Mechanics A solid right circular cone, of mass AI, has semi-vertical angle or and smooth surfaces. It stands with its base on a smooth horizontal table. A particle of mass m is projected so that it strikes the curved surface of the cone at speed it. The coefficient of restitution between the particle and the cone is e. The impact has no rotational effect on the cone and the cone has no vertical velocity after the impact. (i) The particle strikes the cone in the direction of the normal at the point of impact Explain why the trajectory of the particle immediately after the impact is parallel to the normal to the surface of the cone. Find an expression, in terms of N], m, a, e and u, for the speed at which the cone slides along the table immediately after impact. (ii) If instead the particle falls vertically onto the cone, show that the speed to at which the cone slides along the table immediately after impact is given by mu(1 + e) sin (1 cos a w = ———-~—~—2 A! + m cos a Show also that the value of (r for which to is greatest is given by (0 / 1W : so = —~—— . 21% + m A solid figure is composed of a uniform solid cylinder of density p and a uniform solid hemi— sphere of density 3p. The cylinder has circular cross-section, with radius 7“, and height 37“, and the hemisphere has radius 7‘. The flat face of the hemisphere is joined to one end of the cylinder, so that their centres coincide. The figure is held in equilibrium by a force P so that one point of its flat base is in contact with a rough horizontal plane and its base is inclined at an angle or to the horizontal. The force P is horizontal and acts through the highest point of the base. The coeflicient of friction between the solid and the plane is ,u. Show that M} lg—écotal. [Turn over 6 11 In this question take the acceleration due to gravity to be 10 in s‘2 and neglect air resistance. The point 0 lies in a horizontal field. The point B lies 50 In east of O. A particle is projected from B at speed 25 m s"1 at an angle arctané above the horizontal and in a direction that makes an angle 600 with OB; it passes to the north of O. (i) Taking unit vectors i, j and k in the directions east, north and vertically upwards, respectively, find the position vector of the particle relative to O at time t seconds after the particle was projected, and show that its distance from O is 5(t2 — “515+ 10)1n. When this distance is shortest, the particle is at point P. Find the position vector of P and its horizontal hearing from 0. (ii) Show that the particle reaches its maximum height at P. (iii) When the particle is at P, a marksman fires a bullet from 0 directly at P. The initial speed of the bullet is 350 m s"1. Ignoring the effect of gravity on the bullet show that, when it passes through P, the distance between P and the particle is approximately 3 In. © OCR 2007 9470 Jun()7 12 13 14 7 Section C: Probability and Statistics I have two identical dice. When I thron either one of them, the probability of it showing a 6 is p and the probability of it not showing a 6 is (1, where p + q = 1. As an experiment to determine )3, I throw the dice simultaneously until at least one die shows a 6. If both dice show a six on this throw, I stop. If just one die shows a six, I throw the other die until it shows a 6 and then stop. (i) Show that the probability that I stop after 7‘ throws is pq'r‘lt2 —- q"~1 —— q’"), and find an expression for the expected number of throws. 00 [Note: You may use the result 2 7"er = {5(1 — 104.] 1‘20 (ii) In a large number of such experiments, the mean number of throws was m. Find an estimate for p in terms of m. "HT _T/n me Given that 0 < 7' < n and r is much smaller than n, show that 71 There are k guests at a party. Assuming that there are exactly 365 days in the year, and that the birthday of any guest is equally likely to fall on any of these days, show that the probability that there are at least two guests with the same birthday is approximately 1 — e”k(k_1)/730. Using the approximation % ln 2, find the smallest value of [<3 such that the probability that at least two guests share the same birthday is at least How many guests must there be at the party for the probability that at least one guest has the same birthday as the host to be at least %? The random variable X has a continuous probability density function given by 0 forwgl lnrc forlgirék = ink for k. £53 < 2 a—-b'r, for2k<m<4k where k, a and b are constants. (i) Sketch the graph of y = (ii) Determine a and b in terms of k: and find the numerical values of k, (I, and 15. (iii) Find the median value of X. © OCR 2007 9470 J\1n()7 Wm“— 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 (OCR) to trace copyright holders, but if any items requiring clearance have unwittingiy been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR 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. © OCH 2007 9470 Jun07 i a i r i l ...
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2007 Paper II - IIlIIlIlIIII I 002% RECOGNISING ACHIEVEMENT...

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