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Unformatted text preview: TIME RECOGNISING ACHIEVEMENT OXFORD CAMBRIDGE AND RSA EXAMINATIONS Sixth Term Examination Papers
administered on behalf of the Cambridge Colleges MATHEMATICS III
29 JUNE 2005 9475 Wednesday Afternoon 3 hours Additional materials:
Answer paper
Graph paper
Formulae booklet Candidates may not use electronic calculators 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. WWW SP (SLM/TL) 897483
© OCR 2005 This question paper consists of 8 printed pages. Registered Charity Number: 1066969 [Turn over 2 Section A: Pure Mathematics Show that sin A :2 cos B if and only if A 2 (4n + I); :t B for some integer n. Show also that lsinrc i: cost g x/E for all values of a: and deduce that there are no solutions
to the equation sin (sin :6) Lt cos (cos Sketch, on the same axes, the graphs of y 2 sin (sinus) and y 2 cos (cos Sketch, not on the
previous axes, the graph of y 2 sin (2si111'). Find the general solution of the differential equation 3% z — 2 f 2 , where a 75 0, and show
a: m a
that it can be written in the form 112(1'2 + a2) 2 02 , where c is an arbitrary constant. Sketch
this curve.
.  d 2 2
Find an expressxon for amtr: + y ) and show that
x d2 2 2 c2 802.102 "7 :7} +1 I 2 1 “ “W77” 1“ W . dm2( J ) (a:2 + (1‘)? (3:2 + a2)3 (i) Show that, if 0 < c < a2, the points on the curve Whose distance from the origin is least
0
are (0, iw).
(1 (ii) If c > (L2, determine the points on the curve whose distance from the origin is least. Let f(a:) = 1:2 + pm + q and g(:t) = m2 + m + 3. Find an expression for f(g(:1:)) and hence
ﬁnd a necessary and sufﬁcient condition on a, b and c for it to be possible to write the quartic
expression 304 + (12:3 + ban2 + ca: + d in the form f(g(:c)), for some choiCe of values of p, q, 7' and 5. Show further that this condition holds if and only if it is possible to write the quartic expression
$4 + (L’E3 + bx? + or + d in the form (:2:2 + m: + w)2 — k, for some choice of values of v, w and k1. Find the roots of the quartic equation 1‘4 ~— 4173 + 10.232 —— 12:17 + 4 = 0. 9475 SOS RECOGNlSlNG ACHIEVEMENT ERRATUM NOTICE OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Sixth Term Examination Papers MATHEMATICS m 9475
Wednesday 29 JUNE 2005 Afternoon 3 hours Instructions to Invigilators Before the start of the examination please give each candidate a copy of this erratum
notice. Page 4 Question 6. There is an error in the sentence starting ‘Show that the equation . . .’ . This sentence
should read: ‘Show that the equation x3 — 3a2x = 2a3 cosh T is satisﬁed by 2a cosh and hence . . . b
that, if c2 2 b3 > 0 , one of the roots of the equation x3 —3bx = 2c 15 u + —, where
u u =(c+\/c2 b3)3. Page 5 Question 10(ii)
There is an error in this part question. The question should read: ‘Show that, if the discs collide at least once, their total kinetic energy just before the ﬁrst collision is gmgra #3,u). ' Any enquiry about this notice should be referred to the Information Bureau on
01223 553 998 or [email protected] JunOS/erratumOQ 3 The sequence un (n = 1, 2, . . satisﬁes the recurrence relation where la: is a constant. If m z a and Hg 2 b, where (L and b are non—zero and b # ka, prove by induction that b
“2n 2 “> “2nvl
a u2n+1 : 0713271 for n 2 1, where c is a constant to be found in terms of k, a and b. Hence express um and
112711 in terms of a, b, c and n. Find conditions on a, b and k in the three cases:
(i) the sequence on is geometric;
(ii) the sequence on has period 2; (iii) the sequence un has period 4. Let P be the point on the curve y 2 of + bx + c (where a is non—zero) at which the gradient
is m. Show that the equation of the tangent at P is (m  b)2 1—mx=c~
J 4a Show that the curves y : 0,1362 + blm + Cl and y = a2m2 + b21: + 02 (where a1 and a2 are non
zero) have a common tangent with gradient m if and only if (a,2 — al)m2 + 2(a1bQ — a2b1)m + 4ala2(62 — cl) + aw? — albg : O. Show that, in the case a1 # a2, the two curves have exactly one common tangent if and only
if they touch each other. In the case 0,} 2 a2 , ﬁnd a necessary and sufﬁcient condition for the
two curves to have exactly one common tangent. 9475 sos [Turn over 4 In this question, you may use without proof the results 4003113 y ~— 3coshy 2: cosh(3y) and arcoshy : ln + \/y2 a 1) . [ Note: arcoshy is another notation for coslf1 y] Show that the equation x3 ~ 3a2:12 : 20.3 coser is satisﬁed by 2a cosh and hence that, if . . _ i , I i
c‘2 2 b“, one of the roots of the equation 91:3 —— 3bzr 2 2c is u + —), where u z (c + \/ c2 « b3)3 .
it Show that the other two roots of the equation 273 — 31):; :2 20 are the roots of the quadratic , b bl2
equation 162 + (u + «)3; + U2 + —~ — b = 0, and ﬁnd these roots in terms of u, b and u), where u u?
w : %(~1+i\/§). Solve completely the equation 9:3  6w 2 6. 1
u f(u) m M f I .m a V
admin) dﬁ F(1 )+ c. , where m 7E 0. Show that if/ do 2: + c , then / Find:
_ 1
(x) f x, _ 33 am; .. 1
(ll) / dIE . In this question, a and c are distinct nonzero complex numbers. The complex conjugate of
any complex number 2 is denoted by 2*. Show that
in — cl2 : aa* + cc" w ac" — ca," and hence prove that the triangle OAC in the Argand diagram, whose vertices are represented
by 0, a and c respectively, is right angled at A if and only if 2aa* 2 ac* + (11* . Points P and P’ in the Argand diagram are represented by the complex numbers ab and 1% , where b is a nonzero complex number. A Circle in the Argand diagram has centre C and
passes through the point A, and is such that 0A is tangent to the circle. Show that the point
P lies on the circle if and only if point 1” lies on the circle. . a
Conversely, show that if the pornts represented by the complex numbers ab and 5:, for some nonzero complex number b with bb" yé 1 , both lie on a circle centre 0 in the Argand diagram
which passes through A, then 0A is a tangent to the circle. 9475 $05 10 Section B: Mechanics Two particles, A and B, move without friction along a horizontal line which is perpendicular to
a vertical wall. The coefﬁcient of restitution between the two particles is e and the coefﬁcient
of restitution between particle B and the wall is also (3, where 0 < e < 1. The mass of
particle A is 4mm (with m > 0), and the mass of particle B is (1 — (3)2771. Initially, A is moving towards the wall with speed (l.  e)v (where 12 > O) and B is moving
away from the wall and towards A with speed 261). The two particles collide at a distance d from the wall. Find the speeds of A and B after the collision. When B strikes the wall, it rebounds along the same line. Show that a second collision will
take place, at a distance de from the wall. Deduce that further collisions will take place. Find the distance from the wall at which the
nth collision takes place, and show that the times between successive collisions are equal. Two thin discs, each of radius r and mass m, are held on a rough horizontal surface with their
rmg 12 ’
is wrapped once round the discs, its straight sections being parallel. The contact between the elastic band and the discs is smooth. The coefﬁcient of friction between each disc and the
horizontal surface is n, so that each disc experiences a force due to friction equal to nmg,
whether the disc is at rest or sliding. centres a distance 67‘ apart. A thin light elastic band, of natural length 27f?“ and modulus The discs are released simultaneously. If the discs collide, they rebound and a half of their
total kinetic energy is lost in the collision. (i) Show that the discs start sliding, but come to rest before colliding, if and only if §<n<1. (ii) Show that, if the discs collide at least once, their total kinetic energy just before the
ﬁrst collision is §m9(2 — 3n). (iii) Show that if 53 > n2 > 5—7 the discs come to rest exactly once after the first collision. 9475 s05 [Turn over 11 6 A horizontal spindle rotates freely in a fixed bearing, Three light rods are each attached by
one end to the spindle so that they rotate in a vertical plane. A particle of mass m is fixed
to the other end of each of the three rods. The rods have lengths a, b and c, with a > b > c
and the angle between any pair of rods is (gm The angle between the rod of length a and the
vertical axis is 6, as shown in the diagram. Find an expression for the energy of the system and Show that, if the system is in equilibrium, then (b )x/”
— c 3
tanél— ——————~«2a_bﬂc. Deduce that there are exactly two equilibrium positions and determine which of the two
equilibrium positions is stable. Show that, for the system to make complete revolutions, it must pass through its position of
stable equilibrium with an angular velocity of at least 49R
Vﬁ+w+&’ where 2R2 2 (a — b)2 + (6 ~ c)2 + (c — (1)2 . 9475 SOS 12 13 7 Section C: Probability and Statistics Five independent timers time a runner as she runs four laps of a track. Four of the timers
measure the individual lap times, the results of the measurements being the random variables
T1 to T4, each of which has variance 02 and expectation equal to the true time for the lap.
The ﬁfth timer measures the total time for the race, the result of the measurement being the
random variable T which has variance 02 and expectation equal to the true race time (which
is equal to the sum of the four true lap times). Find a random variable X of the form LLT + b(T1 + T2 + T3 + T1), where a and b are constants
independent of the true lap times, with the properties that: (i) whatever the true lap times, the expectation of X is equal to the true race time; (ii) the variance of X is as small as possible. Find also a random variable Y of the form cT + d(T1 + T; + T3 + T4), where c and d are
constants independent of the true lap times, with the property that, whatever the true lap
times, the expectation of Y2 is equal to 02.
In one particular race, T takes the value 220 seconds and ( ’1 + T2 + T3 + T4) takes the value
220.5 seconds. Use the random variables X and Y to estimate an interval in which true race time lies. A pack of cards consists of n + 1 cards, which are printed with the integers from 0 to n.
A game consists of drawing cards repeatedly at random from the pack until the card printed
with 0 is drawn, at which point the game ends. After each draw, the player receives £ 1 if the
card drawn shows any of the integers from 1 to 11) inclusive but receives nothing if the card drawn shows any of the integers from 'u) + l to n inclusive. (i) In one version of the game, each card drawn is replaced immediately and randomly in
the pack. Explain clearly why the probability that the player wins a total of exactly .8 3
is equal to the probability of the following event occurring: out of the ﬁrst four cards
drawn which show numbers in the range 0 to w, the numbers on the ﬁrst three are nonzero and the number on the fourth is zero. Hence show that the probability that 7113 (w + 1)4 '
Write down the probability that the player wins a total of exactly fr and hence find
the expected total win. the player wins a total of exactly .6 3 is equal to (ii) In another version of the game, each card drawn is removed from the pack. Show that
the expected total win in this version is half of the expected total win in the other version. 9475 505 [Turn OVBI‘ 14 In this question, you may use the result 00 m I .r I
/ t dt : m. (n m).
. 0 ( t+ k)n+2 (71+ 1)!kn—7n+1 ’ where m and n are positive integers with n 2 m, and where k > 0. The random variable V has density function C ka“ 1:“
2 1 !
where a is a positive integer. Show that C :— L—f—ﬁé—L .
.a. Show, by means of a suitable substitution, that U mu m “0
f0 (33 + k)2a+2 div : + M25123 (1“ and deduce that the median value of V is 1:. Find the expected value of V. The random variable V represents the speed of a randomly chosen gas molecule. The time taken for such a particle to travel a fixed distance 3 is given by the random variable T 2 Show that no (wille
I __‘ l‘ J PCT (t) d3? l and hence ﬁnd the density function of T. You may ﬁnd it helpful to make the substitution u z 3 in the integral 1' Hence ShOW that the product of the median time and the median speed is equal to the dis—
tance s, but that the product of the expected time and the expected speed is greater than 3. 9475 SOS ...
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This note was uploaded on 04/01/2012 for the course MATH 1016 taught by Professor Rotar during the Spring '12 term at Central Lancashire.
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