UCLES 2010

91**4023334091*






Sixth Term Examination Papers 9470
MATHEMATICS 2 Morning
Wednesday 23 JUNE 2010 Time: 3 hours


Additional Materials: Answer Paper
Formulae Booklet

Candidates may not use a calculator



INSTRUCTIONS TO CANDIDATES
Please read this page carefully, but do not open this question paper until you are
told that you may do so.

Write your name, centre number and candidate number in the spaces on the 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
for fragmentary answers.

You are provided with a Mathematical Formulae Booklet.

Calculators are not permitted.







Please wait to be told you may begin before turning this page.


_____________________________________________________________________________

This question paper consists of 6 printed pages and 2 blank pages.
[Turn over

g
]
g
g
g
j
Q

Section A: Pure Mathematics
1 Let P be a given point on a given curve C. The osculating circle to C at P is dened to be
the circle that satises the following two conditions at P: it touches C; and the rate of change
of its gradient is equal to the rate of change of the gradient of C.
Find the centre and radius of the osculating circle to the curve y = 1 x +tan x at the point
on the curve with x-coordinate
1
4
.
2 Prove that
cos 3x = 4 cos
3
x 3 cos x.
Find and prove a similar result for sin 3x in terms of sinx.
(i) Let
I() =

7 sin x 8 sin
3
x

dx.
Show that
I() =
8
3
c
3
+c +
5
3
,
where c = cos . Write down one value of c for which I() = 0.
(ii) Useless Eustace believes that

sin
n
xdx =
sin
n+1
x
n + 1
for n = 1, 2, 3, . . . . Show that Eustace would obtain the correct value of I() , where
cos =
1
6
.
Find all values of for which he would obtain the correct value of I().
3 The rst four terms of a sequence are given by F
0
= 0, F
1
= 1, F
2
= 1 and F
3
= 2. The
general term is given by
F
n
= a
n
+b
n
, ()
where a, b, and are independent of n, and a is positive.
(i) Show that
2
+ +
2
= 2, and nd the values of , , a and b.
(ii) Use () to evaluate F
6
.
(iii) Evaluate

n=0
F
n
2
n+1
.
2
2
9470 Jun10
4 (i) Let
I =

a
0
f(x)
f(x) + f(a x)
dx.
Use a substitution to show that
I =

a
0
f(a x)
f(x) + f(a x)
dx
and hence evaluate I in terms of a.
Use this result to evaluate the integrals

1
0
ln(x + 1)
ln(2 +x x
2
)
dx and

2
0
sin x
sin(x +

4
)
dx.
(ii) Evaluate

2
1
2
sin x
x

sin x + sin
1
x
dx.
5 The points A and B have position vectors i +j +k and 5i j k, respectively, relative to the
origin O. Find cos 2, where 2 is the angle AOB.
(i) The line L
1
has equation r = (mi +nj +pk). Given that L
1
is inclined equally to OA
and to OB, determine a relationship between m, n and p. Find also values of m, n and
p for which L
1
is the angle bisector of AOB.
(ii) The line L
2
has equation r = (ui + vj + wk). Given that L
2
is inclined at an angle
to OA, where 2 = AOB, determine a relationship between u, v and w.
Hence describe the surface with Cartesian equation x
2
+y
2
+z
2
= 2(yz +zx +xy).
6 Each edge of the tetrahedron ABCD has unit length. The face ABC is horizontal, and P is
the point in ABC that is vertically below D.
(i) Find the length of PD.
(ii) Show that the cosine of the angle between adjacent faces of the tetrahedron is 1/3.
(iii) Find the radius of the largest sphere that can t inside the tetrahedron.
3
3
9470 Jun10
[Turn over
7 (i) By considering the positions of its turning points, show that the curve with equation
y = x
3
3qx q(1 +q) ,
where q > 0 and q = 1, crosses the x-axis once only.
(ii) Given that x satises the cubic equation
x
3
3qx q(1 +q) = 0 ,
and that
x = u +q/u,
obtain a quadratic equation satised by u
3
. Hence nd the real root of the cubic equation
in the case q > 0, q = 1.
(iii) The quadratic equation
t
2
pt +q = 0
has roots and . Show that

3
+
3
= p
3
3qp .
It is given that one of these roots is the square of the other. By considering the expression
(
2
)(
2
), nd a relationship between p and q. Given further that q > 0, q = 1
and p is real, determine the value of p in terms of q.
8 The curves C
1
and C
2
are dened by
y = e
x
(x > 0) and y = e
x
sinx (x > 0),
respectively. Sketch roughly C
1
and C
2
on the same diagram.
Let x
n
denote the x-coordinate of the nth point of contact between the two curves, where
0 < x
1
< x
2
< , and let A
n
denote the area of the region enclosed by the two curves
between x
n
and x
n+1
. Show that
A
n
=
1
2
(e
2
1)e
(4n+1)/2
and hence nd

n=1
A
n
.
4
4
9470 Jun10
Section B: Mechanics
9 Two points A and B lie on horizontal ground. A particle P
1
is projected from A towards B
at an acute angle of elevation and simultaneously a particle P
2
is projected from B towards
A at an acute angle of elevation . Given that the two particles collide in the air a horizontal
distance b from B, and that the collision occurs after P
1
has attained its maximum height h,
show that
2hcot < b < 4hcot
and
2hcot < a < 4hcot ,
where a is the horizontal distance from A to the point of collision.
10 (i) In an experiment, a particle A of mass m is at rest on a smooth horizontal table.
A particle B of mass bm, where b > 1, is projected along the table directly towards A
with speed u. The collision is perfectly elastic.
Find an expression for the speed of A after the collision in terms of b and u, and show
that, irrespective of the relative masses of the particles, A cannot be made to move at
twice the initial speed of B.
(ii) In a second experiment, a particle B
1
is projected along the table directly towards A
with speed u. This time, particles B
2
, B
3
, . . . , B
n
are at rest in order on the line
between B
1
and A. The mass of B
i
(i = 1, 2, . . . , n) is
n+1i
m, where > 1. All
collisions are perfectly elastic. Show that, by choosing n suciently large, there is no
upper limit on the speed at which A can be made to move.
In the case = 4, determine the least value of n for which A moves at more than 20u.
You may use the approximation log
10
2 0.30103.
11 A uniform rod AB of length 4L and weight W is inclined at an angle to the horizontal. Its
lower end A rests on a xed support and the rod is held in equilibrium by a string attached
to the rod at a point C which is 3L from A. The reaction of the support on the rod acts in a
direction to AC and the string is inclined at an angle to CA. Show that
cot = 3 tan + 2 cot .
Given that = 30

and = 45

, show that = 15

.
5
5
9470 Jun10
[Turn over
Section C: Probability and Statistics
12 The continuous random variable X has probability density function f(x), where
f(x) =

a for 0 x < k
b for k x 1
0 otherwise,
where a > b > 0 and 0 < k < 1. Show that a > 1 and b < 1.
(i) Show that
E(X) =
1 2b +ab
2(a b)
.
(ii) Show that the median, M, of X is given by M =
1
2a
if a + b 2ab and obtain an
expression for the median if a +b 2ab.
(iii) Show that M < E(X) .
13 Rosalind wants to join the Stepney Chess Club. In order to be accepted, she must play a
challenge match consisting of several games against Pardeep (the Club champion) and Quentin
(the Club secretary), in which she must win at least one game against each of Pardeep and
Quentin. From past experience, she knows that the probability of her winning a single game
against Pardeep is p and the probability of her winning a single game against Quentin is q,
where 0 < p < q < 1.
(i) The challenge match consists of three games. Before the match begins, Rosalind must
choose either to play Pardeep twice and Quentin once or to play Quentin twice and
Pardeep once. Show that she should choose to play Pardeep twice.
(ii) In order to ease the entry requirements, it is decided instead that the challenge match
will consist of four games. Now, before the match begins, Rosalind must choose whether
to play Pardeep three times and Quentin once (strategy 1), or to play Pardeep twice and
Quentin twice (strategy 2) or to play Pardeep once and Quentin three times (strategy 3).
Show that, if q p >
1
2
, Rosalind should choose strategy 1.
If q p <
1
2
give examples of values of p and q to show that strategy 2 can be better or
worse than strategy 1.
6
6
9470 Jun10



BLANK PAGE
7
9470 Jun10
[Turn over
BLANK PAGE
















































Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate, a department of the University of Cambridge.
Cambridge Assessment is a not-for-profit organisation.

8
9470 Jun10