2005 Paper I

# 2005 Paper I - OXFORD CAMBRIDGE AND RSA EXAMINATIONS Sixth...

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Unformatted text preview: OXFORD CAMBRIDGE AND RSA EXAMINATIONS Sixth Term Examination Papers administered on behalf of the Cambridge Colleges MATHEMATICS I Wednesday 29 JUNE 2005 Afternoon Additional materials: Answer paper Graph paper Formulae booklet Candidates may not use electronic calculators TIME 3 hours INSTRUCTIONS TO CANDIDATES - Write your name, Centre number and candidate number in the spaces on the answer paper/ answer booklet. - Begin each answer on a new page. INFORMATION FOR CANDIDATES - Each question is marked out of 20. There is no restriction of choice. - 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. RECOGNISING ACHIEVEMENT 9465 3 hours SP (SLM) 896535/1 © com 2005 This question paper consists of 8 printed pages. Registered Charity Number: 1066969 [Turn over Section A: Pure Mathematics 47231 is a five-digit number whose digits sum to 4 + 7 + 2 + 3 + 1 2 17. (i) Show that there are 15 ﬁve—digit numbers whose digits sum to 43. You should explain your reasoning clearly. (ii) How many ﬁve-digit numbers are there whose digits sum to 39‘? The point P has coordinates (pg, 21)) and the point Q has COOI‘dlIlELteS ((12,211), where p and q are non-zero and p 7S q. The curve C is given by y2 2 4m. The point B is the intersection of the tangent to C at P and the tangent to C at Q. Show that R has coordinates (pq, p + q), The point S is the intersection of the normal to C at P and the normal to at Q. If p and q are such that (1,0) lies on the line PQ, show that S has coordinates (p2 + (12 + 1, p + q), and that the quadrilateral PSQR is a rectangle. In this question a and b are distinct, non~zero real numbers, and c is a real number. (i) Show that, if a and b are either both positive or both negative, then the equation __M + z 1 r — a x m b has two distinct real solutions. (ii) Show that the equation :c m ~ + = 1 + c :1: — a :1: — b . . 2 40b . . . has exactly one real solution if c = ~m§ . Show that this condition can be a __ a—l—b a—b 2 written 02 = 1 — < > and deduce that it can only hold if 0 < 02 g 1 . 9465 805 5 1. 3 37r , 24 Given that c056 : ~ and that —— g 9 g 27r , show that 311126 :- ——~— and evaluate 2 2 ’ cos30. 5 3 tan 6 — tan3 9 Pro rtlr 'de t't‘ t 393 ve rel 1113’ 3“ 1~3tan29 7r 1 Hence evaluate tan 9, given that tan 30 z 3— and that a- < 0 g g . Evaluate the integral 1 f (x +1),H do: 0 in the cases 1:: 0 and k 2 0, 2’“ —‘ 1 Deduce that ~75“ z 1112 when k- z 0. Evaluate the integral 1 / + 1)?” dm 0 in the different cases that arise according to the value of m. The point A has coordinates (5,16) and the point B has coordinates (—4,4). The variable point P has coordinates (33,31) and moves on a path such that AP = QBP. Show that the Cartesian equation of the path of P is (\$+7)2+y2=100. The point C has coordinates (a ,0) and the point D has coordinates (b , 0). The variable point Q moves on a path such that QC = k x QD , where k > 1 . Given that the path of Q is the same as the path of P, show that a+7 a2+51 b+7" b2«+51' Show further that (a + 7)(b + 7) = 100, in the case a % b. 9465 s05 [Tu rn OVEI‘ 4 n 7 The notation f(7’) denotes the product Ni) x f(2) >< f(3) x >< T21 Simplify the following products as far as possible: (i) ; Tl 2 . 2 2' ~1 . (111) H (cos ~71 + Sln ~75 cot , where n 18 even. 71 n n l d 8 Show that, if y2 : :ckf(:1:), then 2rcyagj: : 163/2 + as“ 19E dx. (i) By setting k = 1 in this result, ﬁnd the solution of the differential equation 1 Zmyfﬂ = yz +272 ——1 dcc for which y = 2 when x = 1. Describe geometrically this solution. (ii) Find the solution of the differential equation 2:0 yew = 2111(27) w mg? for which y = 1 when ac = 1 . 9465 SOS 10 Section B: Mechanics A non—uniform rod AB has weight W and length 3l. When the rod is suspended horizontally in equilibrium by vertical strings attached to the ends A and B, the tension in the string attached to A is T. When instead the rod is held in equilibrium in a horizontal position by means of a smooth pivot at a distance I from A and a vertical string attached to B, the tension in the string is ’I‘. Show that 5T :2 2W. When instead the end B of the rod rests on rough horizontal ground and the rod is held in equilibrium at an angle 6 to the horizontal by means of a string that is perpendicular to the rod and attached to A, the tension in the string is éT. Calculate 9 and ﬁnd the smallest value of coefficient of friction between the rod and the ground that will prevent slipping. Three collinear, non-touching particles A, B and C have masses o, b and c, respectively, and are at rest on a smooth horizontal surface. The particle A is given an initial velocity it towards B . These particles collide, giving B a velocity 12 towards C. These two particles then collide, giving C a velocity w. The coefﬁcient of restitution is e in both collisions. Determine an expression for v, and show that 2 abu (1 + e) “’2 (a,+b)(b+c)' Determine the ﬁnal velocities of each of the three particles in the cases: . a_b__ (I) E“C”ei (ii) 22-226. 9465 s05 [Turn over & ; 2 i l l 2‘ i i l 11 A particle moves so that 1‘, its displacement from a ﬁxed origin at time t, is given by r = (sin2t) i + (2 cos t)j, where 0 g t < 27r. (i) Show that the particle passes through the origin exactly twice. (ii) Determine the times when the velocity of the particle is perpendicular to its displace» ment. (iii) Show that, when the particle is not at the origin, its velocity is never parallel to its displacement. (iv) Determine the maximum distance of the particle from the origin, and sketch the path of the particle. 0405 505 7 Section C: Probability and Statistics 12 The probability that a hobbit smokes a pipe is 0.7 and the probability that a hobbit wears a hat is 0.4. The probability that a hobbit smokes a pipe but does not wear a hat is p. Determine the range of values of p consistent with this information. (b) The probability that a wizard wears a hat is 0.7 ; the probability that a wizard wears a cloak is 0.8 ; and the probability that a wizard wears a ring is 0.4 . The probability that a wizard does not wear a hat, does not wear a cloak and does not wear a ring is 0.05. The probability that a wizard wears a hat, a cloak and also a ring is 0.1 . Determine the probability that a wizard wears exactly two of a hat, a cloak, and a ring. The probability that a wizard wears a hat but not a ring, given that he wears a cloak, is q. Determine the range of values of q consistent with this information. 13 The random variable X has mean a and standard deviation 0'. The distribution of X is symmetrical about a and satisﬁes: P(X<n+0)==a and P(X<u+%a):b, where a and b are ﬁxed numbers. Do not assume that X is Normally distributed. (a) Determine expressions (in terms of a and b) for P(lrw%a<X<n+a) and P(X<n+%a|X,>,n~—%0). (b) My local supermarket sells cartons of skimmed milk and cartons of full~fat milk: 60% of the cartons it sells contain skimmed milk, and the rest contain full-fat milk. The volume of skimmed milk in a carton is modelled by X ml, with p = 500 and a = 10. The volume of full fat milk in a carton is modelled by X ml, with n = 495 and a = 10. (i) Today, I bought one carton of milk, chosen at random, from this supermarket. When I get home, I ﬁnd that it contains less than 505 ml. Determine an expression (in terms of a and b) for the probability that this carton of milk contains more than 500 ml. (ii) Over the years, I have bought a very large number of cartons of milk, all chosen at random, from this supermarket. 70% of the cartons I have bought have contained at most 505 ml of milk. Of all the cartons that have contained at least 495 ml of milk, one third of them have contained full-fat milk. Use this information to estimate the values of a and b. 9465 s05 [Turn over .. , M A. .. ,ww, 8 14 The random variable X can take the value X = — l, and also any value in the range 0 g X < 00. The distribution of X is given by P(X:—1):m, P(O<X<cv):k(1——e”m), for any non-negative number 1n, where k: and m are constants, and m < é— . (i) Find It in terms of m. (ii) Show that E(X) = 1 —~ 2m. (iii) Find, in terms of m, Var and the median value of X. (iv) Given that 00 a 2 /0y26 Mimi-WT, ﬁnd in terms of m. 9465 SOS ...
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