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Unformatted text preview: OCRﬁ RECOGNISING ACHIEVEMENT Sixth Term Examination Papers administered on behalf of the Cambridge Colleges MATHEMATICS I WEDNESDAY 27 JUNE 2007 Afternoon Time: 3 hours lg Additional materials: Answer paper OE Graph paper C Formulae booklet c Candidates may not use electronic calculators ;; W  w _  _ _ , __ , _
w . _ m 7 , ___m , W _ , ﬂ INSTRUCTIONS TO CANDIDATES 3——  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) T43390 © OCR 2007 OCR is an exempt Charity [Turn over Section A: 2 Pure Mathematics 1 A positive integer with 271 digits the first of which must not be 0) is called a balanced number
if the sum of the ﬁrst n digits equals the sum of the last n digits. For example, 1634 is a
4—digit balanced number, but 123401 is not a balanced number. (i) (ii) (ii) Show that seventy 4—digit balanced numbers can be made using the digits 0, 11 2, 3
and 4. Show that %k + 1) (41;: + 5) 4—digit balanced numbers can be made using the digits
0 t0 kt. 77/
You may use the identity 2 r2 E én (n + 1,) (271 + 1.) . r=0 Given that A : arctané and that B : arctané considering tan (A + B), that A + B :: (where A and B are acute) show, by The non—zero integers p and q satisfy 1 7r
arctan — + arctan —— = — .
p q 4 Show that (p — 1) (q — 1) = 2 and hence determine 1) and q. Let r, s and t be positive integers such that the highest common factor of s and t is 1. Show that, if
S arctan 4 + arctan 7r
1" 5+1‘ 4’ then there are only two possible values for t, and give 7' in terms of s in each case. 3 Prove the identities c0846 — sin40 E cos 26 and cos“? + sin46 E 1 —— $511926. Hence or
otherwise evaluate 3 7r 2 5 . 4
/ cos 6 (16 and 5111 6 (:16.
0 0 Evaluate also © OCR 2007 / 2 cos6 6 (:10 and / 2 sin6 0 (:19.
0 0 9465 JunO7 3 4 Show that 3:3 — 3;:7bc: + b3 + (:3 can be written in the form + b + c) :16), where is a
quadratic expression. Show that 2Q( can be written as the sum of three expressions7 each
of which is a perfoet square. . . . " 9
It is given that the equations (13/3 + by + (r 2 0 and by“ + (:y + a = 0 have a conn‘non root i . Q \
The coeffiments a, b and c are real, (1 and b are both non—zero, and (LC 75 b“. Show that 9. ‘2
(an — b”) h : be  a‘
. . ‘ . . . O
and determine 21 Similar expressmn invoiV1ng k“. Hence ShOW‘ that
0 2
(ac M b2) (ab — C“) 2 (he — a2) and that (13 — Babe + b3 + (:3 = 0. Deduce that either kt = 1 or the two equations are identical. 5 Note: a regular octahedron is a polyhedron with eight faces each of which is an equilateral
triangle. (i) Show that the angle between any two faces of a regular octahedron is arecos (ii) Find the ratio of the volume of a regular octahedron to the volume of the cube whose
vertices are the centres of the faces of the octahedron. 6 Given that 1:2 — 3/2 = (:1; — y);3 and that m — y = d (Where (1 ¢ 0), express each of a: and
y in tennis of (1. Hence ﬁnd a pair of integers m and n satisﬁring m — n :2 (Mm  Where m > n > 100. (ii) Given that 1173—y3 = (37 — 3/)4 and that 30—31 = cl (Where d 75 O), Show that 3333/ = d3 —d'3. Hence show that
2:11: didﬁ and determine a pair of distinct posmve integers m and n such that m3 —n3 z (m — n)/ . oocn 2007 9465111m37 [Turn over © OCR 2007 1 2
(i) The line L1 has vector equation F 2 (l + /\ 2
2 —3 4 1 The line L2 has vector equation 1‘ z —2 + H 2 9 *2 Show that the distance D between a point on L1 and a point on L2 can he expressed in
the form 1)? : (3y,4)\_5)2+ (A»1)Q+ 36. Hence determine the minimum distance between these two lines and ﬁnd the coordinates
of the points on the two lines that are the minimum distance apart. .2 .0,
(ii) The line L3 has vector equation I‘ = 3 + (r 1
5 U
3 4A:
The line L41 has vector equation r = 3 + ,8 1  k:
—2 —3k Determine the minimum distance between these two lines7 explaining geometrically the
two different cases that arise according to the value of A7. A curve is given by the equation
3/ 2 (Mfg —~ Gating + (12a + 12) :10 ~— (8u, + 16) , where a is a real number. Show that this curve touches the curve with equation 21 = 3:3 W)
at (2 , 8). Determine the coordinates of any other point of intersection of the two curves.
(i) Sketch on the same axes the curves and when a = 2.
(ii Sketch on the same axes the curves and when a = 1.
(iii) Sketch on the same axes the curves and when a = ~2. 9465 1111107 Section B: Mechanics 9 A particle of weight it" is placed on a rough plane inclined at an angle of 9 to the horizontal.
The coefﬁcient of friction between the, particle and the plane is a. A horizontal. force X acting
on the particle is just suflicient to prevent the particle from sliding down the plane; when a
horizontal force kX acts on the particle, the particle is about to slide up the plane. Both
horizontal forces act in the vertical plane containing the line of greatest slope. Prove that
(k: — i) (1 + if) sinﬁcosd : a (k +1) (1 + W and hence that k? 2 2 .
(1 — M) 10 The Norman army is advancing with constant speed it towards the Saxon army, which is at
rest. When the armies are d apart, a Saxon horseman rides from the Saxon army directly
towards the Norman army at constant speed Sinniltaneously a Norman horseman rides
from the Norman army directly towards the Saxon army at constant speed y, where 3/ > u.
The horsemen ride their horses so that y — 21L“ < u < 2y — :1;. When each horseman reaches the opposing army, he immediately rides straight latch to his
own army without changing his speed. Represent this information on a displacen‘ient—time
graph, and show that the two horsemen pass each other at distanms wd :1:d(2y — — u)
and ——
17 + y (u + 3;)(1: + y) from the Saxon army. Explain briefly what will happen in the cases u > 23/ —— :17 and (ii) a < y — 2.1:. 11 A smooth, straight, narrow tube of length L is ﬁxed at an angle of 300 to the horizontal.
A particle is fired up the tube, from the lower end, with initial velocity a. When the particle
reaches the upper end of the tube, it continues its motion until it returns to the same level
as the lower end of the tube, having travelled a horizontal distance D after leaving the tube.
Show that D satisﬁes the equation 49132 — m (U2 — Lg) D — 3L (1,? — 9L) 2 0 and hence that (if) 2\/§gD — 3(u2 — 29L) dL 89D — 2{\/§ (u? — 9L) ' The ﬁnal horizontal displacement of the particle from the lower end of the tube is R. Show
(11% that —— :2 0 when 2D : Lx/g, and determine, in terms of a and g, the correspomling value dL
of R . @009 2007 9465Junﬁ7 {Turn over Section C: 12 (i) 6 Probability and Statistics A bag contains N sweets (where N 2 2), of which a are red. Two sweets are drawn
from the bag without replacement. Show that the probability that the ﬁrst sweet is red
is equal to the probability that the second sweet is red. There are two bags, each containing N sweets (where N 2 2). The ﬁrst bag contains
(L red sweets, and the second bag cm'itains b red sweets. There is also a biased coin,
showing Heads with probability p and Tails With probability (1, Where p + q x l. The coin is tossed. If it shows Heads then a sweet is chosen from the first bag and
transferred to the second bag; if it shows Tails then sweet is chosen from the second
bag and transferred to the ﬁrst bag. The coin is then tossed a second time: if it shows
Heads then a sweet is chosen from the ﬁrst bag, and if it shows Tails then a sweet is
chosen from the second bag. Show that the probability that the ﬁrst sweet is red is equal to the probability that the
second sweet is red. 13 A bag contains eleven small discs, which are identical except that six of the discs are blank
and ﬁve of the discs are numbered, using the numbers 1, 2, 3, 4 and 5. The bag is shaken,
and four discs are taken one at a time without replacement. Calculate the probability that: © OCR 2007 all four discs taken are numbered; all four discs taken are numbered, given that the disc numbered “3” is taken ﬁrst;
exactly two numbered discs are taken, given that the disc numbered “3” is taken ﬁrst;
exactly two numbered discs are taken, given that the disc numbered “"3” is taken;
exactly two numbered discs are taken, given that a numbered disc is taken ﬁrst; exactly two numbered discs are taken, given that a numbered disc is taken. 9465 J unU7 14 ( © OCR 2007 p... ) H. 1—. 7 The discrete random variable X has a Poisson (:listribution with mean A. Sketch the graph y z + 1) e71", stating the coordinates of the turning point and the
points of intersection with the axes. It is known that P(X 2 2) = 1  p, where p is a. given number in the range 0 < p < 1.
Show that this information determines a unique value (which you should not attempt to ﬁnd) of /\. It known instead) that P (X = 1) q, where q is a given number in the range
0 < q < 1. Show that this information (:leteri’nines unique value of /\ (which you should ﬁnd) for exactly one Value of (1 (which you should also ﬁnd).
It is known (instead) that P (X z 1  X g 2) z 7*, where r is a given number in the range 0 < ‘r < 1. Show that this information determines a unique value of /\ (which you should
ﬁnd) for exactly one value of 1' (which you should also ﬁnd). 9465 JunO7 Permission to reproduce items where thirdnparty 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 unwittingly 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 itsell a department of the University of Cambridge. © OCH 2007 9465 .lunl)7 ...
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