Sixth Term Examination Papers                    9465 MATHEMATICS 1 Morning FRIDAY 20 JUNE 2014  Time: 3 hours
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INFORMATION FOR CANDIDATES
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_____________________________________________________________________________ This question paper consists of 7 printed pages and 1 blank page.
© UCLES 2014
Section A:Pure Mathematics
1All numbers referred to in this question are non-negative integers.(i)Express each of the numbers 3,5,8,12and 16as the difference of two non-zero squares.(ii)
Prove that any odd number can be written as the difference of two squares.
(iii)Prove that all numbers of the form 4k ,where k is a non-negative integer,can be written
as the difference of two squares.(iv)Prove that no number of the form 4k +2,where k is a non-negative integer,can be
written as the difference of two squares.(v)
Prove that any number of the form pq ,where p and q are prime numbers greater than 2,can be written as the difference of two squares in exactly two distinct ways.Does this result hold if p is a prime greater than 2and q =2?
(vi)Determine the number of distinct ways in which 675can be written as the difference
of two squares.
2(i)Show that
ln(2−x )d x =−(2−x )ln(2−x )+(2−x )+c ,where x <2.
(ii)
Sketch the curve A given by y =ln |x 2−4|.
(iii)Show that the area of the finite
region enclosed by the positive x -axis,the y -axis and the curve A is 4ln(2+√3)−2√3.(iv)The curve B is given by y =
ln |x 2−4|  .Find the area between the curve B and the
x -axis with |x |<2.
[Note:you may assume that t ln t →0as t →0.]
© UCLES 2014
3The numbers a and b,where b>a 0,are such that
b
a x2d x=
b
a
x d x
2
.
(i)In the case a=0and b>0,find the value of b.
(ii)In the case a=1,show that b satisfies
3b3−b2−7b−7=0.
Show further,with the help of a sketch,that there is only one(real)value of b that
satisfies this equation and that it lies between2and3.
(iii)Show that3p2+q2=3p2q,where p=b+a and q=b−a,and express p2in terms of q.Deduce that1<b−a 43.
4An accurate clock has an hour hand of length a and a minute hand of length b(where b>a), both measured from the pivot at the centre of the clock face.Let x be the distance between the ends of the hands when the angle between the hands isθ,where0 θ<π.
Show that the rate of increase of x is greatest when x=(b2−a2)12.
In the case when b=2a and the clock starts at mid-day(with both hands pointing vertically upwards),show that this occurs for thefirst time a little less than11minutes later.
5(i)Let f(x)=(x+2a)3−27a2x,where a 0.By sketching f(x),show that f(x) 0 for x 0.
(ii)Use part(i)tofind the greatest value of xy2in the region of the x-y plane given by x 0,y 0and x+2y 3.For what values of x and y is this greatest value achieved?
(iii)Use part(i)to show that(p+q+r)3 27pqr for any non-negative numbers p,q and r.
If(p+q+r)3=27pqr,what relationship must p,q and r satisfy?
© UCLES 2014[Turn over
6(i)The sequence of numbers u0,u1,...is given by u0=u and,for n 0,
u n+1=4u n(1−u n).(∗) In the case u=sin2θfor some given angleθ,write down and simplify expressions for
u1and u2in terms ofθ.Conjecture an expression for u n and prove your conjecture.
pulleys(ii)The sequence of numbers v0,v1,...is given by v0=v and,for n 0,
v n+1=−pv2n+qv n+r,
where p,q and r are given numbers,with p=0.Show that a substitution of the form
v n=αu n+β,whereαandβare suitably chosen,results in the sequence(∗)provided
that
4pr=8+2q−q2.
Hence obtain the sequence satisfying v0=1and,for n 0,v n+1=−v2n+2v n+2.
7In the triangle OAB,the point D divides the side BO in the ratio r:1(so that BD=rDO), and the point E divides the side OA in the ratio s:1(so that OE=sEA),where r and s are both positive.
(i)The lines AD and BE intersect at G.Show that
g=
rs
1+r+rs
a+
1
1+r+rs
b,
where a,b and g are the position vectors with respect to O of A,B and G,respectively.
(ii)The line through G and O meets AB at F.Given that F divides AB in the ratio t:1,find an expression for t in terms of r and s.
8Let L a denote the line joining the points(a,0)and(0,1−a),where0<a<1.The line L b is defined similarly.
(i)Determine the point of intersection of L a and L b,where a=b.
(ii)Show that this point of intersection,in the limit as b→a,lies on the curve C given by
y=(1−√
x)2(0<x<1).
(iii)Show that every tangent to C is of the form L a for some a.© UCLES 2014
Section B:Mechanics
9A particle of mass m is projected due east at speed U from a point on horizontal ground at an angleθabove the horizontal,where0<θ<90◦.In addition to the gravitational force mg,it experiences a horizontal force of magnitude mkg,where k is a positive constant, acting due west in the plane of motio
n of the particle.Determine expressions in terms of U,θand g for the time,T H,at which the particle reaches its greatest height and the time,T L, at which it lands.
Let T=U cosθ/(kg).By considering the relative magnitudes of T H,T L and T,or otherwise, sketch the trajectory of the particle in the cases k tanθ<12,12<k tanθ<1,and k tanθ>1.
What happens when k tanθ=1?
10(i)A uniform spherical ball of mass M and radius R is released from rest with its centre
a distance H+R above horizontal ground.The coefficient of restitution between the
ball and the ground is e.Show that,after bouncing,the centre of the ball reaches a
height R+He2above the ground.
(ii)A second uniform spherical ball,of mass m and radius r,is now released from rest together with thefirst ball(whose centre is again a distance H+R above the ground
when it is released).The two balls are initially one on top of the other,with the second
ball(of mass m)above thefirst.The two balls separate slightly during their fall,with
their centres remaining in the same vertical line,so that they collide immediately after
thefirst ball has bounced on the ground.The coefficient of restitution between the
balls is also e.The centre of the second ball attains a height h above the ground.
Given that R=0.2,r=0.05,H=1.8,h=4.5and e=23,determine the value
of M/m.
© UCLES 2014[Turn over
11The diagrams below show two separate systems of particles,strings and pulleys.In both systems,the pulleys are smooth and light,the strings are light and inextensible,the particles move vertically and the pulleys labelled with P arefixed.The masses of the particles are as indicated on the diagrams.
P
M
m
P
P1
M
m1
m2 System I System II
(i)For system I show that the acceleration,a1,of the particle of mass M,measured in
the downwards direction,is given by
a1=M−m
M+m
g,
where g is the acceleration due to gravity.Give an expression for the force on the pulley due to the tension in the string.
(ii)For system II show that the acceleration,a2,of the particle of mass M,measured in the downwards direction,is given by
a2=M−4μ
M+4μ
g,
whereμ=
m1m2 m1+m2
.
In the case m=m1+m2,show that a1=a2if and only if m1=m2.© UCLES 2014

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