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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
General Certificate of Education Advanced Level
9231/11
FURTHER MATHEMATICS
May/June 2010
Paper 1
3 hours
*1900609380*
Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF10)
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Results obtained solely from a graphic calculator, without supporting working or reasoning, will not receive
credit.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
This document consists of 5 printed pages and 3 blank pages.
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2
1
The variables x and y are such that y = −1 when x = 1 and
x 2 + y2 + Find the values of
2
dy 3
= 29.
dx
dy
d2 y
and 2 when x = 1.
dx
dx
The curve C has polar equation
[5]
r = a 1 − e−θ ,
where a is a positive constant and 0 ≤ θ < 2π .
(i) Draw a sketch of C.
[3]
(ii) Show that the area of the region bounded by C and the lines θ = ln 2 and θ = ln 4 is
1 2
2a
3
ln 2 − 13
32 .
At any point (x, y) on the curve C,
√
dx
= t (t2 + 4)
dt
and
[4]
√
dy
= −t (4 − t2 ),
dt
√
where the parameter t is such that 0 ≤ t ≤ 2. Show that the length of C is 4 2.
[3]
Given that y = 0 when t = 2, determine the area of the surface generated when C is rotated through
one complete revolution about the x-axis, leaving your answer in an exact form.
[4]
4
The sum SN is defined by SN =
N
∑ n5 . Using the identity
n=1
n + 12 − n − 12 ≡ 6n5 + 5n3 + 38 n,
6
6
find SN in terms of N . [You need not simplify your result.]
[4]
Hence find lim N −λ SN , for each of the two cases
(i) λ = 6,
N →∞
(ii) λ > 6.
5
[3]
Let
e
In = ã x(ln x)n dx,
where n ≥ 1. Show that
1
In+1 = 12 e2 − 12 (n + 1)In .
[3]
Hence prove by induction that, for all positive integers n, In is of the form An e2 + Bn , where An and
Bn are rational numbers.
[6]
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3
6
The equation
has roots α , β , γ . Use the relation x =
√
x3 + x − 1 = 0
y to show that the equation
y3 + 2y2 + y − 1 = 0
has roots α 2 , β 2 , γ 2 .
[2]
Let Sn = α n + β n + γ n .
7
(i) Write down the value of S2 and show that S4 = 2.
[3]
(ii) Find the values of S6 and S8 .
[4]
The lines l1 and l2 have vector equations
r = 4i − 2j + λ (2i + j − 4k)
and
r = 4i − 5j + 2k + µ (i − j − k)
respectively.
(i) Show that l1 and l2 intersect.
[3]
(ii) Find the perpendicular distance from the point P whose position vector is 3i − 5j + 6k to the
plane containing l1 and l2 .
[3]
(iii) Find the perpendicular distance from P to l1 .
8
The matrix A is given by
4

A=
 −4
0
Given that one eigenvector of A is
1
−1
−1
[4]
−1

4
.
5
1
−2 !, find the corresponding eigenvalue.
−1
Given also that another eigenvalue of A is 4, find a corresponding eigenvector.
[2]
[2]
1
−4 ! is an eigenvector of A, with corresponding eigenvalue 1, find matrices P
−1
and Q, together with a diagonal matrix D, such that A5 = PDQ.
[6]
Given further that
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4
9
(i) Write down the five fifth roots of unity.
(ii) Hence find all the roots of the equation
ß5 + 16 + (16
[2]
√
3)i = 0,
giving answers in the form reiqπ , where r > 0 and q is a rational number. Show these roots on an
Argand diagram.
[4]
Let w be a root of the equation in part (ii).
(iii) Show that
4
∑
k=0
√
w k 3+i 3
=
.
2
2−w
(iv) Identify the root for which | 2 − w | is least.
10
[3]
[2]
Find the set of values of a for which the system of equations
x + 4y + 12ß = 5,
2x + ay + 12ß = a − 1,
3x + 12y + 2aß = 10,
has a unique solution.
[4]
Show that the system does not have any solution in the case a = 18.
[2]
Given that a = 8, show that the number of solutions is infinite and find the solution for which
x + y + ß = 1.
[5]
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5
11
Answer only one of the following two alternatives.
EITHER
The variables ß and x are related by the differential equation
3ß2
d2 ß
dß
dß 2
+ 6ß2
+ 6ß + 5ß3 = 5x + 2.
2
dx
dx
dx
Use the substitution y = ß3 to show that y and x are related by the differential equation
dy
d2 y
+ 2 + 5y = 5x + 2.
2
d
x
dx
Given that ß = 1 and
dß
= − 23 when x = 0, find ß in terms of x.
dx
Deduce that, for large positive values of x, ß ≈ x 3 .
1
[3]
[9]
[2]
OR
The curve C has equation
y=
x(x + 1)
.
(x − 1)2
(i) Obtain the equations of the asymptotes of C .
[3]
(ii) Show that there is exactly one point of intersection of C with the asymptotes and find its
coordinates.
[2]
(iii) Find
dy
and hence
dx
(a) find the coordinates of any stationary points of C ,
(b) state the set of values of x for which the gradient of C is negative.
[6]
(iv) Draw a sketch of C.
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